Merge pull request 'release: v0.10.0 — reviewer-ready' (#20) from release/v0.10.0 into main

Post-merge consistency commit:

  * CITATION.cff version 0.9.0 → 0.10.0, date 2026-05-26
  * CHANGELOG.md — new "[0.10.0]" section listing all 13 commits
    that landed across PRs #17 / #18 / #19
  * Post-merge gate fixes:
      - doxygen_groups.h + doxygen_namespaces.h gain \\file briefs
      - .codespellrc extended (honour, thead, optimiser)
      - compile-time.md: unbalanced backtick on line 102 fixed
        (was confusing Doxygen's verbatim-block detector)

Final gate state on the merged main:
  ✅ 259/259 tests pass
  ✅ test-count consistency
  ✅ markdown links 166/166 resolve (43 .md files)
  ✅ CGAL conventions 0/8 violations
  ✅ license-headers 68/68 carry MIT SPDX
  ✅ codespell 0 typos
  ✅ shellcheck 0 findings (18 scripts)
  ✅ Doxygen 396/396 symbols, 0 warnings

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>

.clang-format

@@ -0,0 +1,76 @@
+# conformallab++ formatting policy
+#
+# Captures the style already present in code/include/.  Documented here
+# so clang-format can enforce it locally (scripts/quality/clang-format.sh)
+# and so new contributors get the same output their editor would on save.
+#
+# This is NOT the upstream CGAL clang-format (there isn't one published);
+# it's the style our tree already uses, mechanically extracted.
+
+BasedOnStyle: LLVM
+Language: Cpp
+Standard: c++17
+
+IndentWidth: 4
+TabWidth: 4
+UseTab: Never
+ColumnLimit: 100                  # loose; readability over hard wrap
+
+# Brace placement — matches the project tree:
+#   functions / methods → opening brace on a new line  (CGAL convention)
+#   structs / classes   → opening brace on a new line
+#   else / catch        → on the same line as the closing brace of the preceding block
+BreakBeforeBraces: Custom
+BraceWrapping:
+  AfterClass:            true
+  AfterStruct:           true
+  AfterEnum:             true
+  AfterFunction:         true
+  AfterNamespace:        false
+  AfterUnion:            true
+  AfterControlStatement: false
+  BeforeElse:            false
+  BeforeCatch:           false
+  IndentBraces:          false
+  SplitEmptyFunction:    false
+  SplitEmptyRecord:      false
+  SplitEmptyNamespace:   true
+
+# Reference & pointer modifiers attach to the type (`int& x`, not `int &x`).
+PointerAlignment: Left
+ReferenceAlignment: Left
+
+# Aligned `using = ...` blocks are intentional in the trait classes.
+AlignConsecutiveDeclarations: AcrossEmptyLines
+AlignConsecutiveAssignments: AcrossEmptyLines
+AlignTrailingComments: true
+AlignAfterOpenBracket: Align
+
+AllowShortFunctionsOnASingleLine: Inline
+AllowShortIfStatementsOnASingleLine: Never
+AllowShortLoopsOnASingleLine: false
+AllowShortBlocksOnASingleLine: Never
+AllowShortLambdasOnASingleLine: Inline
+
+# Template-related: break before each parameter when the template line
+# would otherwise exceed ColumnLimit (matches existing
+# `template <typename TriangleMesh, typename ...>` patterns).
+BreakBeforeBinaryOperators: NonAssignment
+BinPackParameters: false
+BinPackArguments: false
+AlwaysBreakTemplateDeclarations: Yes
+SpaceAfterTemplateKeyword: true
+
+NamespaceIndentation: None
+AccessModifierOffset: -4
+IndentCaseLabels: false
+
+# Includes: keep manual ordering — re-ordering can break Eigen / CGAL
+# transitive-include assumptions in subtle ways.  We just enforce no
+# accidental duplicate blank lines.
+SortIncludes: Never
+MaxEmptyLinesToKeep: 1
+KeepEmptyLinesAtTheStartOfBlocks: false
+
+# Comments: don't touch.
+ReflowComments: false

.clang-tidy

@@ -0,0 +1,45 @@
+# conformallab++ clang-tidy policy
+#
+# Curated, deliberately small.  The CGAL header tree triggers tens of
+# thousands of warnings under default settings (CGAL's chosen style is
+# pre-C++17 in many places).  Restricting to checks that fire on OUR
+# code, not on transitive CGAL/Eigen/Boost headers, keeps the signal
+# meaningful.
+#
+# Promotion gate: a check moves into this list only when (a) it fires
+# on code we authored AND (b) the fix is mechanical (no algorithmic
+# rewrite required).  Anything algorithmic belongs in a code review,
+# not in a static analyser.
+
+Checks: >
+  -*,
+  bugprone-too-small-loop-variable,
+  bugprone-use-after-move,
+  bugprone-undefined-memory-manipulation,
+  bugprone-integer-division,
+  bugprone-suspicious-string-compare,
+  bugprone-misplaced-widening-cast,
+  bugprone-sizeof-expression,
+  cppcoreguidelines-init-variables,
+  cppcoreguidelines-pro-type-member-init,
+  performance-for-range-copy,
+  performance-implicit-conversion-in-loop,
+  performance-unnecessary-copy-initialization,
+  performance-unnecessary-value-param,
+  readability-misleading-indentation,
+  readability-redundant-smartptr-get,
+  modernize-use-nullptr,
+  modernize-use-override,
+  modernize-deprecated-headers
+
+# Only emit warnings on our own headers.  CGAL/Eigen/etc. live under
+# `code/deps/` (vendored) or are installed system-wide; we never want
+# clang-tidy fixes for them.
+HeaderFilterRegex: '^.*/code/include/(?!deps/).*$'
+
+WarningsAsErrors: ''
+
+CheckOptions:
+  - { key: cppcoreguidelines-init-variables.IgnoreArrays, value: true }
+  - { key: performance-for-range-copy.WarnOnAllAutoCopies, value: true }
+  - { key: performance-unnecessary-value-param.AllowedTypes, value: 'Eigen::Vector.*;Eigen::Matrix.*' }

.cmake-format.yaml

@@ -0,0 +1,28 @@
+# conformallab++ cmake-format policy
+#
+# Drives the cmake-format / cmake-lint tools used by
+# scripts/quality/cmake-format.sh.  The defaults are deliberately
+# permissive — the goal is to catch the obvious style drift (mixed
+# 2-vs-4-space indent, inconsistent argument wrapping, undocumented
+# options) without forcing a rewrite of every CMakeLists.txt we have.
+
+format:
+  line_width: 100               # match .clang-format
+  tab_size: 4
+  use_tabchars: false
+  separate_ctrl_name_with_space: false
+  separate_fn_name_with_space: false
+  dangle_parens: false
+  command_case: lower           # lowercase commands (cgal/upstream convention)
+  keyword_case: upper           # KEYWORDS like PUBLIC/PRIVATE/INTERFACE in caps
+
+lint:
+  # Whitelist the disables we explicitly accept.
+  disabled_codes:
+    - C0103   # invalid variable name — we use CGAL_/conformallab_ prefixes
+    - C0301   # line too long — handled by line_width, not as a lint error
+    - C0111   # missing docstring on a function — most of ours are obvious
+
+  # Maximum allowed nesting of conditional blocks.  3 is conservative;
+  # raise if we ever genuinely need deeper.
+  max_conditionals_custom_parser: 3

.codespellrc

@@ -0,0 +1,87 @@
+# conformallab++ codespell policy
+#
+# Driven by scripts/quality/codespell.sh.  We scan code comments + docs
+# for common typos; vendored dependencies + the build tree are excluded.
+#
+# False positives go into ignore-words-list (lowercase, comma-separated).
+# Math-heavy projects accumulate them quickly — names of mathematicians,
+# differential operators, etc.
+
+[codespell]
+skip = code/deps,build,build-*,build_T_*,test-reports,doc/doxygen,.git,*.svg,*.lock,*.pdf,*.png,*.jpg,Doxyfile,*.bib
+
+# Words codespell considers misspellings but we intentionally keep:
+#   bessel    — Bessel functions (math)
+#   ist       — German for "is", appears in German doc paragraphs
+#   sinces    — appears in "sinces 1858" style historical refs (false positive)
+#   nd        — short-form ordinal, e.g. "2nd"
+#   te        — appears in greek transliteration "θ → te"
+#   inout     — common parameter direction word
+#   nin       — math symbol ∉ accidental match
+#   numer     — "numerical/numerator" abbreviation in headers
+#   neet      — German "neet" / accidental matches
+#   anc       — appears in "anc(ient)" math literature refs
+#   sinks     — "sinks" can hit Sinkhorn
+ignore-words-list = bessel,ist,sinces,nd,te,inout,nin,numer,neet,anc,sinks,doubleClick,
+    centre,centres,centered,centering,centring,
+    behaviour,behaviours,behavioural,
+    analogue,analogues,
+    initialise,initialised,initialises,initialising,initialisation,
+    normalise,normalised,normalises,normalising,normalisation,
+    centralise,centralised,centralises,centralising,
+    serialise,serialised,serialises,serialising,serialisation,
+    parameterise,parameterised,parameterises,parameterising,
+    parametrise,parametrised,parametrises,parametrising,
+    realise,realised,realises,realising,realisation,
+    optimise,optimised,optimises,optimising,optimisation,
+    sanitise,sanitised,sanitises,sanitising,
+    generalise,generalised,generalises,generalising,
+    amortise,amortised,amortises,amortising,
+    factorise,factorised,factorises,factorising,
+    discretise,discretised,discretises,discretising,
+    summarise,summarised,summarises,summarising,
+    colour,colours,coloured,colouring,
+    artefact,artefacts,
+    iff,
+    dof,dofs,
+    browseable,
+    re-use,re-uses,re-used,re-using,
+    specialise,specialised,specialises,specialising,specialisation,specialisations,
+    visualise,visualised,visualises,visualising,visualisation,visualisations,
+    model,modeled,modelled,modelling,
+    minimise,minimised,minimises,minimising,minimisation,
+    maximise,maximised,maximises,maximising,maximisation,
+    organise,organised,organises,organising,organisation,
+    characterise,characterised,characterises,characterising,
+    emphasise,emphasised,emphasises,emphasising,
+    analyse,analysed,analyses,analysing,analyser,analysers,
+    organise,organisation,organisational,
+    parameterise,parameterisation,
+    centre,centred,centres,
+    catalogue,catalogues,
+    maths,
+    generalisation,generalisations,
+    realisation,realisations,
+    specialisation,specialisations,
+    visualisation,visualisations,
+    minimisation,maximisation,characterisation,
+    groupes,fuchsiens,théorie,théorème,
+    iff,
+    honour,honoured,honours,honouring,thead,optimiser,optimisers,
+    categorise,categorised,categorises,categorising,
+    optimisation,optimisations,
+    acknowledgement,acknowledgements,acknowledging,
+    neighbour,neighbours,neighbouring,neighboured,
+    labelled,labelling,labels,labelled,
+    fulfil,fulfils,fulfilled,fulfilling,
+    endcode,
+    deklaration,deklarationen,
+    recognise,recognised,recognises,recognising,recognisation,
+    signalled,signalling,
+    travelled,travelling,
+    cancelled,cancelling,
+    modelled,modelling
+
+# Words we explicitly DO want flagged (override the default skip list).
+# Keep empty for now; add as we hit real-but-not-flagged typos.
+builtin = clear,rare,informal,usage,code,en-GB_to_en-US,names

.editorconfig

@@ -0,0 +1,56 @@
+# conformallab++ EditorConfig
+#
+# Honoured natively by VSCode (with the EditorConfig extension), CLion,
+# Vim, Emacs, Sublime, …  Covers the basics that .clang-format /
+# .cmake-format don't catch (Markdown, Python, YAML, shell, JSON, …)
+# and acts as a cross-IDE fallback when clang-format isn't installed.
+#
+# Authoritative formatting for C++ source still comes from .clang-format;
+# this file just keeps the editor's defaults from fighting it.
+
+root = true
+
+[*]
+charset = utf-8
+end_of_line = lf
+insert_final_newline = true
+trim_trailing_whitespace = true
+indent_style = space
+indent_size = 4
+
+# C++ — match .clang-format
+[*.{h,hpp,cpp,c,cc}]
+indent_size = 4
+max_line_length = 100
+
+# CMake — match .cmake-format.yaml
+[{CMakeLists.txt,*.cmake}]
+indent_size = 4
+max_line_length = 100
+
+# Python — PEP-8 default
+[*.py]
+indent_size = 4
+max_line_length = 100
+
+# Shell — Google shell style
+[*.sh]
+indent_size = 4
+max_line_length = 100
+
+# YAML — community convention
+[*.{yml,yaml}]
+indent_size = 2
+
+# JSON
+[*.json]
+indent_size = 2
+
+# Markdown — preserve trailing spaces (used for line breaks); don't strip
+[*.md]
+trim_trailing_whitespace = false
+max_line_length = off
+
+# Makefiles must use tabs
+[Makefile]
+indent_style = tab

.gitea/docker/Dockerfile.ci-cpp

@@ -2,15 +2,19 @@ FROM --platform=linux/arm64 ubuntu:22.04
 
 # Node.js 20 from NodeSource (Ubuntu Jammy ships v12 which is too old
 # for actions/checkout@v4 — static class blocks require Node.js >= 16).
+#
+# libboost-dev  — header-only Boost required by CGAL 6.x (-DWITH_CGAL=ON)
 RUN apt-get update -qq && \
     apt-get install -y --no-install-recommends \
       curl ca-certificates && \
     curl -fsSL https://deb.nodesource.com/setup_20.x | bash - && \
     apt-get install -y --no-install-recommends \
       nodejs \
+      doxygen \
       cmake \
       build-essential \
       git \
+      libboost-dev \
     && rm -rf /var/lib/apt/lists/*
 
 WORKDIR /workspace

.gitea/workflows/cpp-tests.yml

@@ -6,10 +6,16 @@ on:
       - main
       - dev
       - "claude/**"
+      - "feature/**"
   pull_request:
 
+# ─────────────────────────────────────────────────────────────────────────────
+# Job 1 — test-fast
+#   Pure-math tests (Clausen, ImLi₂, Hyper-ideal geometry).
+#   No CGAL, no Boost. Eigen + GTest only. Runs on ALL branches.
+# ─────────────────────────────────────────────────────────────────────────────
 jobs:
-  test:
+  test-fast:
     runs-on: eulernest
     container:
       image: git.eulernest.eu/conformallab/ci-cpp:latest
@@ -17,11 +23,11 @@ jobs:
     steps:
       - uses: actions/checkout@v4
 
-      - name: Configure (tests-only mode)
+      - name: Configure (tests-only)
         run: cmake -S code -B build -DCMAKE_BUILD_TYPE=Release
 
-      - name: Build test binary
-        run: cmake --build build --target conformallab_tests -j$(nproc)
+      - name: Build
+        run: nice -n 19 cmake --build build --target conformallab_tests -j$(nproc)
 
       - name: Run tests
         run: >
@@ -29,7 +35,7 @@ jobs:
           --output-on-failure
           --output-junit test-results.xml
 
-      - name: Show test summary
+      - name: Summary
         if: always()
         run: |
           if [ -f test-results.xml ]; then
@@ -37,5 +43,120 @@ jobs:
             failed=$(grep -o 'failures="[0-9]*"' test-results.xml | grep -o '[0-9]*' | head -1)
             skipped=$(grep -o 'skipped="[0-9]*"' test-results.xml | grep -o '[0-9]*' | head -1)
             passed=$(( ${total:-0} - ${failed:-0} - ${skipped:-0} ))
-            echo "TOTAL: ${total:-0} | PASSED: $passed | FAILED: ${failed:-0} | SKIPPED: ${skipped:-0}"
+            echo "FAST  ▸ TOTAL ${total:-0} | PASSED $passed | FAILED ${failed:-0} | SKIPPED ${skipped:-0}"
           fi
+
+# ─────────────────────────────────────────────────────────────────────────────
+# Job 2 — test-cgal
+#   Full CGAL test suite (Phase 3–7, 158 tests).
+#   Runs ONLY on pull requests (not on direct pushes to dev/main).
+#   Starts only after test-fast succeeds.
+#
+#   Uses -DWITH_CGAL_TESTS=ON (not -DWITH_CGAL=ON) to avoid building
+#   Viewer/GLFW — the CI container has no wayland-scanner.
+#
+#   Boost (libboost-dev) is already present in the container since the image rebuild.
+# ─────────────────────────────────────────────────────────────────────────────
+  test-cgal:
+    needs: test-fast
+    if: github.event_name == 'pull_request'
+    runs-on: eulernest
+    container:
+      image: git.eulernest.eu/conformallab/ci-cpp:latest
+      # Memory bumped from 1400m → 1600m to avoid OOM during CGAL header
+      # compilation on ARM64 (CGAL + Eigen templates allocate ~700 MB per
+      # cc1plus instance; -j1 leaves a small margin).
+      # memory-swap == memory disables swap entirely so OOM fails fast
+      # rather than thrashing on the SD card.
+      options: "--memory=1600m --memory-swap=1600m"
+
+    steps:
+      - uses: actions/checkout@v4
+
+      - name: Configure (WITH_CGAL_TESTS — no viewer, no wayland-scanner)
+        run: cmake -S code -B build -DWITH_CGAL_TESTS=ON -DCMAKE_BUILD_TYPE=Release
+
+      - name: Build CGAL-Tests
+        run: nice -n 19 cmake --build build --target conformallab_cgal_tests -j1
+
+      - name: Run CGAL-Tests
+        run: >
+          ctest --test-dir build
+          -R "^cgal\."
+          --output-on-failure
+          --output-junit cgal-results.xml
+
+      - name: Summary
+        if: always()
+        run: |
+          if [ -f cgal-results.xml ]; then
+            total=$(grep -o 'tests="[0-9]*"'     cgal-results.xml | grep -o '[0-9]*' | head -1)
+            failed=$(grep -o 'failures="[0-9]*"' cgal-results.xml | grep -o '[0-9]*' | head -1)
+            skipped=$(grep -o 'skipped="[0-9]*"' cgal-results.xml | grep -o '[0-9]*' | head -1)
+            passed=$(( ${total:-0} - ${failed:-0} - ${skipped:-0} ))
+            echo "CGAL  ▸ TOTAL ${total:-0} | PASSED $passed | FAILED ${failed:-0} | SKIPPED ${skipped:-0}"
+          fi
+
+      # ── Structural gate: doc/api/tests.md totals match ctest reality ───
+      # Single source of truth for test counts (see doc/release-policy.md).
+      # Reuses the already-built ./build dir via BUILD_DIR env var, so this
+      # adds ~5 s on top of the existing CGAL job.
+      - name: Verify test-count consistency (doc/api/tests.md)
+        run: BUILD_DIR=build bash scripts/check-test-counts.sh
+
+      # ── Structural gate: end-to-end smoke (try_it.sh) ──────────────────
+      # The user-facing quick-start script: configure + build + run the
+      # full ctest + run the Euclidean example on a bundled mesh.  If
+      # this regresses, README quick-start instructions are broken.
+      # try_it.sh creates its own build-try/ — accept the ~3 min cost as
+      # the price of guaranteeing the documented workflow stays working.
+      - name: End-to-end smoke test (scripts/try_it.sh)
+        run: bash scripts/try_it.sh
+
+# ─────────────────────────────────────────────────────────────────────────────
+# Job 3 — quality-gates  (style + convention block)
+#
+# Cheap, deterministic checks that should never break unless a contributor
+# introduces a regression.  Each gate is a script under scripts/quality/
+# and exits 0 only when its tree is clean.  These ran for weeks locally
+# at zero findings before being promoted here.
+#
+# Tools installed at job-start (the ci-cpp image already has python3 +
+# bash; we add codespell + shellcheck on top).  Total wall-time: ~30 s
+# on the eulernest runner.
+#
+# Strictly required for merges into main/dev — a regression fails the PR.
+# ─────────────────────────────────────────────────────────────────────────────
+  quality-gates:
+    needs: test-fast
+    runs-on: eulernest
+    container:
+      image: git.eulernest.eu/conformallab/ci-cpp:latest
+
+    steps:
+      - uses: actions/checkout@v4
+
+      - name: Install codespell + shellcheck (job-local)
+        run: |
+          apt-get update -qq
+          apt-get install -y --no-install-recommends \
+            codespell shellcheck
+
+      - name: License headers (every C++ source carries MIT SPDX)
+        run: bash scripts/quality/license-headers.sh
+
+      - name: CGAL conventions (6 rules over CGAL public headers)
+        run: python3 scripts/quality/cgal-conventions.py
+
+      - name: codespell (docs + source comments + script messages)
+        run: bash scripts/quality/codespell.sh
+
+      - name: shellcheck (scripts/**/*.sh, severity=warning, strict)
+        run: bash scripts/quality/shellcheck.sh --strict
+
+      - name: Summary
+        if: always()
+        run: |
+          echo "QUALITY ▸ all four gates passed."
+          echo "  see scripts/quality/README.md for the full catalogue"
+          echo "  (sanitizers, clang-tidy, coverage, etc. are local-only)"

.gitea/workflows/doc-build.yaml

@@ -0,0 +1,56 @@
+name: API Docs
+
+on:
+  push:
+    branches:
+      - main
+  pull_request:
+
+# ─────────────────────────────────────────────────────────────────────────────
+# Doc-build — informational only
+#
+# Generates Doxygen HTML from the public headers and reports warning
+# statistics.  Does NOT block merges: `continue-on-error: true` ensures
+# warnings or extraction issues never fail the CI gate.  When Doxygen
+# coverage is denser (Phase 8c), this job can be promoted to a hard
+# requirement and the HTML deployed to Pages.
+#
+# Note: Gitea Actions on GHES does not support `actions/upload-artifact@v4`,
+# so HTML artifact upload is intentionally omitted.  The warning summary
+# in the job log is the primary reviewer signal; reviewers who want the
+# HTML can rebuild it locally with `cmake --build build --target doc`.
+# ─────────────────────────────────────────────────────────────────────────────
+jobs:
+  doc-build:
+    if: github.event_name == 'pull_request'
+    runs-on: eulernest
+    container:
+      image: git.eulernest.eu/conformallab/ci-cpp:latest
+    continue-on-error: true   # never block the merge
+    steps:
+      - uses: actions/checkout@v4
+
+      - name: Generate API documentation
+        run: doxygen Doxyfile 2>&1 | tee doxygen.log
+
+      - name: Summarise warnings
+        if: always()
+        run: |
+          if [ -f doc/doxygen/doxygen-warnings.log ]; then
+            warn=$(wc -l < doc/doxygen/doxygen-warnings.log)
+            echo "DOC ▸ Doxygen warnings: $warn"
+            echo ""
+            echo "First 20 warnings:"
+            head -20 doc/doxygen/doxygen-warnings.log
+          else
+            echo "DOC ▸ No warning log produced — check that Doxyfile WARN_LOGFILE points to doc/doxygen/doxygen-warnings.log"
+          fi
+
+      - name: Report HTML output
+        if: always()
+        run: |
+          if [ -d doc/doxygen/html ]; then
+            files=$(find doc/doxygen/html -type f | wc -l)
+            size=$(du -sh doc/doxygen/html | cut -f1)
+            echo "DOC ▸ HTML output: $files files, $size total"
+          fi

.gitea/workflows/doxygen-pages.yml

@@ -0,0 +1,115 @@
+name: Doxygen → Codeberg Pages
+
+# Auto-publish Doxygen HTML to https://tmoussa.codeberg.page/ConformalLabpp/
+# every time the public API or docs source changes on main.
+#
+# Pattern: mirrors mirror-to-codeberg.yml — reuses the existing
+# CODEBERG_TOKEN secret + HTTPS push.  No new secret setup required.
+#
+# Trigger: push to main that touches code/include/**, Doxyfile, the
+# filter script, doc/**/*.md, README.md, or this workflow file.  Also
+# manually triggerable via workflow_dispatch.
+
+on:
+  push:
+    branches:
+      - main
+    paths:
+      - "code/include/**"
+      - "Doxyfile"
+      - "scripts/doxygen-md-filter.sh"
+      - "doc/**/*.md"
+      - "README.md"
+      - "CLAUDE.md"
+      - ".gitea/workflows/doxygen-pages.yml"
+  workflow_dispatch: {}
+
+jobs:
+  publish:
+    runs-on: eulernest
+    container:
+      image: git.eulernest.eu/conformallab/ci-cpp:latest
+
+    steps:
+      - uses: actions/checkout@v4
+
+      - name: Configure CMake (Doxygen target only — no compiler needed)
+        run: cmake -S code -B build
+
+      - name: Build Doxygen HTML
+        run: |
+          cmake --build build --target doc
+          test -f doc/doxygen/html/index.html
+          warnings=$(wc -l < doc/doxygen/doxygen-warnings.log)
+          echo "DOC ▸ Doxygen warnings: $warnings"
+          if [ "$warnings" -gt 0 ]; then
+            echo "::warning::Doxygen produced $warnings warning(s) — review doc/doxygen/doxygen-warnings.log"
+            head -30 doc/doxygen/doxygen-warnings.log
+          fi
+
+      - name: Enforce Doxygen coverage 100%
+        # Coverage is measured against every public symbol under
+        # code/include/ (the `detail::` namespaces are excluded).  As of
+        # the `docs/doxygen-coverage-100` PR the baseline is 100 %, so
+        # the gate fires only on regressions.
+        run: bash scripts/doxygen-coverage.sh --threshold 100
+
+      - name: Regenerate doc/api/headers.md from XML
+        run: |
+          python3 scripts/gen-headers-md.py
+          # If the auto-generated headers.md drifted from main, note it.
+          # This job runs on every main push so a drift only persists
+          # for the duration of one push — the next push that lands
+          # will fold the new headers.md back into main (via the
+          # codeberg pages branch).  For deterministic regeneration
+          # within main itself, run `bash scripts/regen-docs.sh`
+          # locally before pushing.
+          if ! git diff --quiet -- doc/api/headers.md; then
+            echo "::warning::doc/api/headers.md drifted — run scripts/regen-docs.sh locally and commit before next push"
+            git --no-pager diff -- doc/api/headers.md | head -30
+          fi
+
+      - name: Publish HTML to codeberg pages branch
+        env:
+          CODEBERG_TOKEN: ${{ secrets.CODEBERG_TOKEN }}
+        run: |
+          set -eu
+          # Build the publish payload in a clean scratch dir so the
+          # orphan branch contains only the reviewer hub + Doxygen
+          # output (and a marker README), never any build/source
+          # artefacts.
+          publish_dir=$(mktemp -d)
+          cp -r doc/doxygen/html/. "$publish_dir/"
+
+          # ── Reviewer hub override ─────────────────────────────────
+          # If doc/reviewer/hub.html is present, install it as the
+          # publish landing page and demote the auto-generated Doxygen
+          # index to /doxygen.html.  The hub is hand-curated and lives
+          # under source control; this step keeps it visible after
+          # every push to main, surviving the auto-publish cycle.
+          if [ -f doc/reviewer/hub.html ]; then
+              mv "$publish_dir/index.html" "$publish_dir/doxygen.html"
+              cp doc/reviewer/hub.html "$publish_dir/index.html"
+              echo "DOC ▸ reviewer hub installed; Doxygen index now at /doxygen.html"
+          fi
+
+          cat > "$publish_dir/README.txt" <<EOF
+          conformallab++ — Doxygen HTML API documentation + reviewer hub.
+          Auto-generated by .gitea/workflows/doxygen-pages.yml from
+          commit ${GITHUB_SHA:-$(git rev-parse HEAD)} on $(date -Iseconds).
+          Source: https://codeberg.org/TMoussa/ConformalLabpp
+          Reviewer hub: doc/reviewer/hub.html (in-repo)
+          Doxygen index: /doxygen.html
+          EOF
+
+          cd "$publish_dir"
+          git init -q -b pages
+          git config user.email "ci@eulernest"
+          git config user.name  "conformallab CI"
+          git add -A
+          git commit -q -m "Auto-publish: Doxygen HTML for ${GITHUB_SHA:-HEAD}"
+          # Force-push: the pages branch is a publish target, history
+          # is not interesting (we only ever serve the latest snapshot).
+          git push -f \
+            "https://TMoussa:${CODEBERG_TOKEN}@codeberg.org/TMoussa/ConformalLabpp.git" \
+            pages:pages

.gitea/workflows/markdown-links.yml

@@ -0,0 +1,40 @@
+name: Markdown link check
+
+# Verify every internal markdown link in the repo resolves to an existing
+# file (or anchor).  External http(s) links are also probed but with a
+# loose timeout — flaky third-party hosts must not break our CI.
+#
+# Trigger: PRs that touch any *.md file, plus a weekly cron so external
+# link rot is caught even when nobody is editing docs.
+
+on:
+  pull_request:
+    paths:
+      - "**/*.md"
+      - ".gitea/workflows/markdown-links.yml"
+  push:
+    branches:
+      - main
+    paths:
+      - "**/*.md"
+      - ".gitea/workflows/markdown-links.yml"
+  schedule:
+    - cron: "0 5 * * 1"   # Monday 05:00 UTC weekly link-rot check
+  workflow_dispatch: {}
+
+jobs:
+  check:
+    runs-on: eulernest
+    container:
+      image: git.eulernest.eu/conformallab/ci-cpp:latest
+
+    steps:
+      - uses: actions/checkout@v4
+
+      # ── Pure-python internal link check (no external network needed) ────
+      # We use the same logic that found the 2 broken links before the
+      # reviewer meeting: parse every [text](path) link, check that the
+      # target file exists relative to the source file's directory.  Skips
+      # http(s)://, mailto:, and pure-anchor (#fragment) links.
+      - name: Internal link check (all *.md files)
+        run: python3 scripts/check-markdown-links.py

.gitea/workflows/mirror-to-codeberg.yml

@@ -4,7 +4,6 @@ on:
   push:
     branches:
       - main
-      - dev
 
 jobs:
   mirror:
@@ -14,10 +13,11 @@ jobs:
       - name: Mirror all branches to Codeberg
         env:
           CODEBERG_TOKEN: ${{ secrets.CODEBERG_TOKEN }}
+          GITEA_MIRROR_TOKEN:  ${{ secrets.MIRROR_TOKEN }}
         run: |
           git clone --bare \
-            https://oauth2:${GITHUB_TOKEN}@git.eulernest.eu/conformallab/ConformalLabpp.git \
+            https://oauth2:${GITEA_MIRROR_TOKEN}@git.eulernest.eu/conformallab/ConformalLabpp.git \
             repo.git
           cd repo.git
           git push --mirror \
-            https://TMoussa:${CODEBERG_TOKEN}@codeberg.org/TMoussa/ConformalLabpp.git
+            https://TMoussa:${CODEBERG_TOKEN}@codeberg.org/TMoussa/ConformalLabpp.git

.gitea/workflows/perf-compile-time.yml

@@ -0,0 +1,175 @@
+name: Compile-time perf bench
+
+# Cross-platform compile-time benchmark.  Validates the predictions
+# made in doc/architecture/compile-time.md against the eulernest CI
+# runner (Linux + g++ on ARM64).
+#
+# Specifically tests whether:
+#   1. FAST_TEST_BUILD=ON delivers the ~40 % wall-time reduction on
+#      Linux + g++ that was predicted from `-ftime-trace` profiling
+#      (recall: on Apple clang + Apple M1 it was net-neutral).
+#   2. ccache hit rate is in the predicted 80%+ range on a warm
+#      rerun (where the macOS-local hit rate was 0 % due to
+#      Apple-clang + PCH friction).
+#
+# When run:
+#   * push to main (after PR #19 lands)
+#   * workflow_dispatch (manual trigger for ad-hoc verification)
+#
+# NOT run on every PR — this is a perf data-collection job, not a
+# correctness gate.  Pollutes the summary with timings but does not
+# block merges.
+
+on:
+  push:
+    branches:
+      - main
+    paths:
+      - "code/CMakeLists.txt"
+      - "code/tests/**/CMakeLists.txt"
+      - "code/include/**"
+      - ".gitea/workflows/perf-compile-time.yml"
+  workflow_dispatch: {}
+
+jobs:
+  compile-time-matrix:
+    runs-on: eulernest
+    container:
+      image: git.eulernest.eu/conformallab/ci-cpp:latest
+      options: "--memory=2400m --memory-swap=2400m"
+
+    steps:
+      - uses: actions/checkout@v4
+
+      - name: Install ccache (idempotent)
+        run: |
+          which ccache >/dev/null 2>&1 || apt-get install -y --no-install-recommends ccache
+
+      # ─── Run 1: baseline (PCH OFF, Unity OFF, ccache cleared) ──────
+      - name: "Run 1: cold baseline (no PCH, no Unity, no ccache)"
+        run: |
+          ccache -C >/dev/null 2>&1 || true
+          rm -rf build-baseline
+          cmake -S code -B build-baseline -G Ninja \
+              -DWITH_CGAL_TESTS=ON \
+              -DCONFORMALLAB_USE_PCH=OFF \
+              -DCMAKE_UNITY_BUILD=OFF \
+              -DCONFORMALLAB_USE_CCACHE=OFF
+          start=$(date +%s)
+          nice -n 19 cmake --build build-baseline --target conformallab_cgal_tests -j1
+          end=$(date +%s)
+          echo "PERF baseline_wall=$((end - start)) s"
+          echo "PERF_BASELINE_WALL=$((end - start))" >> $GITHUB_ENV
+
+      # ─── Run 2: PCH only ──────────────────────────────────────────
+      - name: "Run 2: PCH only (Unity off, ccache off)"
+        run: |
+          ccache -C >/dev/null 2>&1 || true
+          rm -rf build-pch
+          cmake -S code -B build-pch -G Ninja \
+              -DWITH_CGAL_TESTS=ON \
+              -DCONFORMALLAB_USE_PCH=ON \
+              -DCMAKE_UNITY_BUILD=OFF \
+              -DCONFORMALLAB_USE_CCACHE=OFF
+          start=$(date +%s)
+          nice -n 19 cmake --build build-pch --target conformallab_cgal_tests -j1
+          end=$(date +%s)
+          echo "PERF pch_only_wall=$((end - start)) s"
+          echo "PERF_PCH_WALL=$((end - start))" >> $GITHUB_ENV
+
+      # ─── Run 3: default (PCH + Unity Build + #6 Dense→Core) ───────
+      - name: "Run 3: default config (PCH + Unity + Dense→Core)"
+        run: |
+          ccache -C >/dev/null 2>&1 || true
+          rm -rf build-default
+          cmake -S code -B build-default -G Ninja \
+              -DWITH_CGAL_TESTS=ON \
+              -DCONFORMALLAB_USE_CCACHE=OFF
+          start=$(date +%s)
+          nice -n 19 cmake --build build-default --target conformallab_cgal_tests -j1
+          end=$(date +%s)
+          echo "PERF default_wall=$((end - start)) s"
+          echo "PERF_DEFAULT_WALL=$((end - start))" >> $GITHUB_ENV
+
+      # ─── Run 4: + FAST_TEST_BUILD (-O0 -g) ────────────────────────
+      - name: "Run 4: default + FAST_TEST_BUILD=ON (-O0 -g for tests)"
+        run: |
+          ccache -C >/dev/null 2>&1 || true
+          rm -rf build-fast
+          cmake -S code -B build-fast -G Ninja \
+              -DWITH_CGAL_TESTS=ON \
+              -DCONFORMALLAB_FAST_TEST_BUILD=ON \
+              -DCONFORMALLAB_USE_CCACHE=OFF
+          start=$(date +%s)
+          nice -n 19 cmake --build build-fast --target conformallab_cgal_tests -j1
+          end=$(date +%s)
+          echo "PERF fast_test_wall=$((end - start)) s"
+          echo "PERF_FAST_WALL=$((end - start))" >> $GITHUB_ENV
+
+      # ─── Run 5: ccache hit-rate validation ─────────────────────────
+      - name: "Run 5: ccache hit-rate (rebuild build-default)"
+        run: |
+          ccache -C >/dev/null 2>&1 || true
+          ccache --zero-stats >/dev/null
+          # First rebuild: populate ccache.
+          rm -rf build-cc
+          cmake -S code -B build-cc -G Ninja \
+              -DWITH_CGAL_TESTS=ON \
+              -DCONFORMALLAB_USE_CCACHE=ON
+          nice -n 19 cmake --build build-cc --target conformallab_cgal_tests -j1 >/dev/null
+          ccache_first=$(ccache -s 2>&1 | grep -E "^\s*Hits" | head -1 | awk '{print $2}')
+          # Second rebuild: expect cache hits.
+          rm -rf build-cc-warm
+          cmake -S code -B build-cc-warm -G Ninja \
+              -DWITH_CGAL_TESTS=ON \
+              -DCONFORMALLAB_USE_CCACHE=ON
+          start=$(date +%s)
+          nice -n 19 cmake --build build-cc-warm --target conformallab_cgal_tests -j1
+          end=$(date +%s)
+          warm_wall=$((end - start))
+          ccache_stats=$(ccache -s 2>&1 | grep -E "Hits|Misses" | head -4)
+          echo "── ccache stats after warm rebuild ──"
+          echo "$ccache_stats"
+          echo "PERF ccache_warm_wall=${warm_wall} s"
+          echo "PERF_CCACHE_WARM_WALL=$warm_wall" >> $GITHUB_ENV
+
+      # ─── Test correctness (last gate; perf data already collected) ─
+      - name: Verify all configs produced working binaries
+        if: always()
+        run: |
+          for build in build-baseline build-pch build-default build-fast; do
+              if [ -d "$build" ]; then
+                  ctest --test-dir "$build" -R "^cgal\." --output-on-failure --timeout 120 \
+                      | tail -3
+              fi
+          done
+
+      # ─── Final summary ─────────────────────────────────────────────
+      - name: Compile-time perf summary
+        if: always()
+        run: |
+          echo "══════════════════════════════════════════════════════"
+          echo "  COMPILE-TIME PERF BENCH — Linux ARM64 / g++ / -j1"
+          echo "══════════════════════════════════════════════════════"
+          printf "  %-30s  %4s s\n" "Run 1: cold baseline"          "${PERF_BASELINE_WALL:-?}"
+          printf "  %-30s  %4s s\n" "Run 2: + PCH"                  "${PERF_PCH_WALL:-?}"
+          printf "  %-30s  %4s s\n" "Run 3: + PCH + Unity (default)" "${PERF_DEFAULT_WALL:-?}"
+          printf "  %-30s  %4s s\n" "Run 4: + FAST_TEST_BUILD"      "${PERF_FAST_WALL:-?}"
+          printf "  %-30s  %4s s\n" "Run 5: + ccache warm rerun"    "${PERF_CCACHE_WARM_WALL:-?}"
+          echo "──────────────────────────────────────────────────────"
+          echo "Predictions to validate vs Apple-M1 baseline:"
+          echo "  ┃ FAST_TEST_BUILD: expected ~40 % faster than default"
+          echo "  ┃ ccache warm:     expected ≤ 10 s (vs Apple's 55 s)"
+          echo "──────────────────────────────────────────────────────"
+          # Compute relative deltas
+          if [ -n "${PERF_DEFAULT_WALL:-}" ] && [ -n "${PERF_FAST_WALL:-}" ]; then
+              pct=$(awk -v d="${PERF_DEFAULT_WALL}" -v f="${PERF_FAST_WALL}" \
+                  'BEGIN { printf "%.0f", 100.0 * (d - f) / d }')
+              echo "  Δ FAST_TEST_BUILD vs default: ${pct} % wall reduction"
+          fi
+          if [ -n "${PERF_DEFAULT_WALL:-}" ] && [ -n "${PERF_CCACHE_WARM_WALL:-}" ]; then
+              pct=$(awk -v d="${PERF_DEFAULT_WALL}" -v c="${PERF_CCACHE_WARM_WALL}" \
+                  'BEGIN { printf "%.0f", 100.0 * (d - c) / d }')
+              echo "  Δ ccache warm vs default:     ${pct} % wall reduction"
+          fi
+          echo "══════════════════════════════════════════════════════"

.gitignore

@@ -0,0 +1,33 @@
+# macOS
+.DS_Store
+.AppleDouble
+.LSOverride
+
+# Build directories
+build/
+build-*/
+build_*/
+code/build*/
+
+# CMake
+CMakeCache.txt
+CMakeFiles/
+cmake_install.cmake
+CTestTestfile.cmake
+
+# Test output
+*.xml
+Testing/
+
+# IDE
+.idea/
+.vscode/
+*.user
+*.suo
+
+# Claude Code worktrees
+.claude/
+
+# Doxygen output
+doc/doxygen/
+*.dox.tmp

CHANGELOG.md

@@ -0,0 +1,241 @@
+# Changelog
+
+All notable changes to **conformallab++** are recorded here.  Format
+follows [Keep a Changelog](https://keepachangelog.com/en/1.1.0/); the
+project uses [Semantic Versioning](https://semver.org).
+
+---
+
+## [0.10.0] — 2026-05-26
+
+The **"reviewer-ready"** release.  Three PRs (#17 + #18 + #19, 13
+thematic commits total) landed: a 100 %-Doxygen-covered public API,
+a 14-gate structural quality suite (4 of them required CI), a
+researcher-targeted reviewer materials package, the `output_uv_map`
+named parameter extended to four of the five DCE solvers, six new
+roadmap phases from a full Java-library scan, thirteen new Tier-1/2
+literature citations, three RESEARCH-only phases with acceptance
+criteria, and a six-mode compile-time workflow matrix.
+
+### Added — reviewer materials (PR #19)
+
+* `doc/reviewer/{briefing,questions,agenda,README}.md` — one-page
+  reviewer briefing + seven scoped questions (Q1–Q2 research-track
+  alignment; Q3–Q4 porting decisions; Q5–Q6 process; Q7 the "no"
+  question) + internal meeting agenda + landing index.
+* `doc/reviewer/hub.html` — hand-curated reviewer landing page,
+  in-repo so the publish URL survives every merge.
+* `code/deps/THIRD-PARTY-LICENSES.md` — per-vendored-dep SPDX with
+  MIT-compatibility analysis (CGAL LGPL §3 vs §4 distinction).
+* `doc/architecture/dependencies.md` — required vs optional deps;
+  standalone-verification recipe.
+
+### Added — new roadmap content (PR #19)
+
+* Six new phases from full Java-library scan: Phase 9d (cones),
+  9d.4 (variational Möbius centring), 9e (circle-pattern layout),
+  10d (Koebe circle-domain), 10e (quasi-isothermic, ~800 lines, 6
+  classes), 10f (Koebe polyhedra), 10g (cyclic-symmetry quotients).
+* Three RESEARCH phases with acceptance criteria: 9d.2 (non-Euclidean
+  cone extensions), 9f (polygon Laplacian on non-triangular meshes,
+  no Java parent), 10c′ (Koebe polyhedron rigidity).
+* 13 new Tier-1/Tier-2 citations in `doc/math/references.md`.
+
+### Added — compile-time workflow matrix (PR #19)
+
+* PCH + Unity Build defaults: CGAL test wall-time **78 s → 55 s**
+  (−30 %), CPU time 676 s → 167 s (−75 %).
+* Five new opt-in workflow modes (`BUILD_TESTING=OFF`,
+  `CONFORMALLAB_HEADERS_CHECK`, `CONFORMALLAB_DEV_BUILD`,
+  `CONFORMALLAB_FAST_TEST_BUILD`, `CONFORMALLAB_USE_CCACHE`).
+* `doc/architecture/compile-time.md` — full measurement + workflow
+  matrix + macOS-vs-Linux honesty notes.
+* `.gitea/workflows/perf-compile-time.yml` — Linux CI bench.
+
+### Added — output_uv_map covers 4 of 5 DCE entries (PR #19)
+
+* `CGAL::discrete_inversive_distance_map` honours `output_uv_map`
+  via Bowers-Stephenson edge-length reconstruction.
+* `CGAL::discrete_circle_packing_euclidean` rejects `output_uv_map`
+  with a clear `std::runtime_error` (face-based DOFs, Phase 9c).
+
+### Added — structural quality gates (PR #18)
+
+Four scripts promoted to required CI: `license-headers.sh`,
+`cgal-conventions.py`, `codespell.sh`, `shellcheck.sh --strict`.
+Seven additional local-only gates: clang-format, cmake-format,
+cppcheck, sanitizers (ASan + UBSan), clang-tidy, multi-compiler,
+reproducible-build, CGAL-version-matrix.
+
+### Added — Doxygen 100 % public-API coverage (PR #17)
+
+* Doxygen coverage: **24 % → 100 %** (396/396 public symbols).
+* Fixed `EXCLUDE_PATTERNS` bug that previously silently excluded
+  every `.hpp`/`.h` — pre-fix HTML had ~0 % API surface.
+* MathJax + CGAL `\cgalParam*` aliases.
+* New scripts: `scripts/doxygen-coverage.sh` (CI-gateable),
+  `scripts/gen-headers-md.py` (auto-regenerates `doc/api/headers.md`).
+
+### Changed
+
+* Three headers `<Eigen/Dense>` → `<Eigen/Core>` (none use Eigen
+  decompositions): `projective_math.hpp`,
+  `hyper_ideal_visualization_utility.hpp`, `mesh_utils.hpp`.
+* `doc/api/tests.md` — CGAL suite 234 → 236 tests.
+* `code/.gitignore` — un-ignore `code/deps/THIRD-PARTY-LICENSES.md`.
+
+### Numbers at release
+
+* 259 / 259 tests pass, 0 skipped.
+* 100 % Doxygen coverage on public API, 0 warnings.
+* 14 / 15 quality gates green, 1 SKIP (no CGAL tarballs locally).
+* CI build wall: ~55 s on Apple M1 (−30 % vs v0.9.0).
+* 13 Tier-1 / Tier-2 literature citations integrated.
+
+---
+
+## [0.9.0] — 2026-05-22
+
+The “DCE-complete + CGAL-surface-complete” release.  Two new discrete-
+conformal models, the analytic-Hessian optimisation for HyperIdeal, the
+CGAL public API surface for all five models, and a full documentation
+audit that corrects four pre-existing port-vs-research mis-labels.
+
+### Added — new functionals (Phase 9a)
+
+* `code/include/cp_euclidean_functional.hpp` —
+  **CP-Euclidean** functional (face-based circle packing),
+  Bobenko-Pinkall-Springborn 2010.  Direct port of
+  `CPEuclideanFunctional.java` (260 Java lines + 88-line test).
+  Analytic 2×2-per-edge Hessian `h_jk = sin θ / (cosh Δρ − cos θ)`.
+* `code/include/inversive_distance_functional.hpp` —
+  **Inversive-Distance** functional (vertex-based, Luo 2004 + Glickenstein
+  2011).  No Java original — implemented from the literature with
+  Bowers-Stephenson 2004 initialisation.  Cross-validated against the
+  Euclidean functional at the natural initial geometry (Glickenstein §5).
+
+### Added — Newton solvers (Phase 9a-Newton)
+
+* `newton_cp_euclidean()` — uses the analytic Hessian.
+* `newton_inversive_distance()` — uses FD Hessian; analytic via
+  Glickenstein 2011 eq. (4.6) tracked in `research-track.md` as
+  Phase 9a.2-analytic.
+
+### Added — Hessian optimisation (Phase 9b)
+
+* `hyper_ideal_hessian_block_fd()` — per-face 6×6 block-local Hessian
+  for HyperIdeal.  **96.5× speed-up measured on a 200-face mesh
+  (V=202, 603 DOFs)**, full-FD 226 ms → block-FD 2.3 ms.
+* Java parity note: `HyperIdealFunctional.java:295-298` declares
+  `hasHessian() == false`; both FD variants are conformallab++
+  research extensions beyond the Java port.
+
+### Added — CGAL public API surface (Phase 8b-Lite)
+
+* `<CGAL/Discrete_conformal_map.h>` extended with
+  `discrete_conformal_map_spherical()` and
+  `discrete_conformal_map_hyper_ideal()`.
+* `<CGAL/Discrete_circle_packing.h>` — `Default_cp_euclidean_traits` +
+  `discrete_circle_packing_euclidean()`.
+* `<CGAL/Discrete_inversive_distance.h>` — `Default_inversive_distance_traits`
+  + `discrete_inversive_distance_map()`.
+* `<CGAL/Conformal_layout.h>` — thin CGAL-namespace re-exports of
+  `euclidean_layout`, `spherical_layout`, `hyper_ideal_layout`.
+
+All five DCE models are now reachable from a single
+`#include <CGAL/Discrete_*.h>`.
+
+### Added — documentation
+
+* `doc/roadmap/research-track.md` — new consolidated catalogue of
+  every conformallab++ item that goes beyond the Java port, with full
+  literature citations and acceptance criteria.  Includes the
+  Phase 9b-analytic plan (Schläfli 1858 + Springborn 2020 §4 +
+  Cho-Kim 1999 + Glickenstein 2011 §4).
+* `doc/architecture/phase-9a-validation.md` — line-by-line mapping
+  CPEuclideanFunctional.java ↔ C++ port, plus three special-case
+  verifications of Luo’s edge-length formula.
+* `doc/roadmap/phases.md` — Phase 9 split into 9a.1 (Java port) /
+  9a.2 (research) / 9b (research); new Phase 11+ section with
+  optional Schottky uniformisation and Riemann-map sub-packages.
+* `doc/math/references.md` — five new primary literature entries
+  (Bowers-Stephenson 2004, Glickenstein 2011, BPS 2010,
+  Schläfli 1858/60, plus a reframed Luo 2004 entry).
+
+### Changed
+
+* **Four port-vs-research mis-labels** corrected (full audit
+  documented in `research-track.md`):
+  - `InversiveDistanceFunctional.java` does not exist in the Java
+    repo; the C++ implementation is research, not a port.
+  - HyperIdeal Hessian: Java has `hasHessian()==false`; C++ Hessians
+    are research, not ports.
+  - `add-inversive-distance.md` tutorial rewritten end-to-end.
+  - `references.md` and `java-parity.md` reframed.
+* `Discrete_conformal_map.h` (Phase 8a MVP wrapper) now deduces the
+  kernel from `TriangleMesh::Point` via `CGAL::Kernel_traits` rather
+  than hard-coding `Simple_cartesian<double>`.  Regression-guarded by
+  `KernelIsDeducedFromMeshPointType` test.
+
+### Removed
+
+* Three stale stub test files in `code/tests/` (15 GTEST_SKIPs total):
+  - `test_spherical_functional.cpp`
+  - `test_hyper_ideal_functional.cpp`
+  - `test_hyper_ideal_hyperelliptic_utility.cpp`
+  They referenced a "HDS port (Phase 4)" that never happened —
+  CoHDS was intentionally replaced by `CGAL::Surface_mesh`, and the
+  functional tests live in `code/tests/cgal/test_*_functional.cpp`.
+
+### CI / Infrastructure
+
+* `.gitea/workflows/cpp-tests.yml` — test-cgal memory fixed
+  (1400→1600 MB, `-j2 → -j1`).  Addresses OOM on ARM64 runner.
+* `.gitea/workflows/doc-build.yaml` — soft-fail Doxygen job
+  (no merge-blocking).
+* `Doxyfile` + CMake `doc` target — `cmake --build build --target doc`.
+* 12 macOS Finder-duplicate files removed from `code/include/`.
+
+### Test counts
+
+```
+v0.7.0:  176 CGAL + 36 non-CGAL = 212 total,  13 skipped (HDS stubs)
+v0.9.0:  227 CGAL + 23 non-CGAL = 250 total,   0 skipped   (+38 net, +51 CGAL)
+```
+
+Non-CGAL count dropped from 36 → 23 because three stale HDS-port stubs
+were removed (see "Removed" above) — the functionality is fully covered
+in the CGAL test suite where it actually lives.
+
+Five test suites added: `CGALConformalTraits`, `CGALDiscreteConformalMap`,
+`CPEuclideanFunctional`, `InversiveDistanceFunctional`, `HyperIdealHessian`,
+`NewtonPhase9a`, `CGALPhase8bLite`.
+
+---
+
+## [0.7.0] — 2026-05-18
+
+The “mathematician-ready” release.  See the v0.7.0 announcement in
+README.md (legacy) or `CITATION.cff` for the corresponding citation
+entry.  Phases 1–7 complete: three DCE geometry modes (Euclidean /
+Spherical / HyperIdeal), Newton solver, BFS-trilateration layout,
+Gauss-Bonnet, tree-cotree cut graph, Möbius holonomy, period matrix
+for genus 1, fundamental domain (genus 1), texture atlas.
+
+---
+
+## How to update this file
+
+Every new release adds a new top-level section above the previous one.
+For non-trivial PRs that don't trigger a release, add an entry under
+an `[Unreleased]` section at the top; promote it to the next release
+header at tag time.
+
+Categories (Keep-A-Changelog convention):
+
+* **Added** — new features / files / public APIs.
+* **Changed** — behaviour-altering changes to existing features.
+* **Deprecated** — features still present but slated for removal.
+* **Removed** — deleted features / files.
+* **Fixed** — bug fixes.
+* **Security** — security-relevant fixes.

CITATION.cff

@@ -0,0 +1,68 @@
+cff-version: 1.2.0
+message: "If you use this software in your research, please cite it as below."
+
+authors:
+  - family-names: Moussa
+    given-names: Tarik
+    email: Tarik.moussa95@gmail.com
+
+title: "conformallab++"
+version: 0.10.0
+date-released: 2026-05-26
+url: "https://codeberg.org/TMoussa/ConformalLabpp"
+repository-code: "https://codeberg.org/TMoussa/ConformalLabpp"
+license: MIT
+
+abstract: >
+  conformallab++ is a C++17 implementation of discrete conformal maps on
+  triangulated surfaces, covering Euclidean, Spherical, and Hyper-ideal geometry
+  modes. It provides a Newton solver for discrete conformal equivalence (DCE),
+  tree-cotree cut graphs, Möbius holonomy, period matrix computation with
+  SL(2,ℤ) reduction, and fundamental domain construction. The long-term goal is
+  a CGAL package for discrete conformal geometry.
+
+keywords:
+  - discrete conformal geometry
+  - conformal maps
+  - surface parameterization
+  - period matrix
+  - Teichmüller theory
+  - CGAL
+  - C++
+
+references:
+  - type: thesis
+    authors:
+      - family-names: Sechelmann
+        given-names: Stefan
+    title: >
+      Variational Methods for Discrete Surface Parameterization:
+      Applications and Implementation
+    institution:
+      name: Technische Universität Berlin
+    year: 2016
+    doi: 10.14279/depositonce-5415
+    notes: "Primary algorithmic source for this implementation"
+
+  - type: article
+    authors:
+      - family-names: Springborn
+        given-names: Boris
+    title: "Ideal Hyperbolic Polyhedra and Discrete Uniformization"
+    journal: "Discrete & Computational Geometry"
+    year: 2020
+    doi: 10.1007/s00454-019-00132-8
+    notes: "Mathematical basis for the HyperIdeal geometry mode"
+
+  - type: article
+    authors:
+      - family-names: Bobenko
+        given-names: Alexander I.
+      - family-names: Springborn
+        given-names: Boris A.
+    title: >
+      Variational Principles for Circle Patterns and Koebe's Theorem
+    journal: "Transactions of the American Mathematical Society"
+    year: 2004
+    doi: 10.1090/S0002-9947-03-03239-2
+    notes: "Variational framework underlying all three geometry modes"

CLAUDE.md

@@ -0,0 +1,401 @@
+# CLAUDE.md
+
+This file provides guidance to Claude Code (claude.ai/code) when working with code in this repository.
+
+## Project purpose and long-term goal
+
+conformallab++ is a C++17 reimplementation of [ConformalLab](https://github.com/varylab/conformallab) — Stefan Sechelmann's Java research library for discrete conformal geometry (TU Berlin, ~850 commits, v1.0.0 2018). The algorithmic foundation is his dissertation:
+
+> Stefan Sechelmann — *Variational Methods for Discrete Surface Parameterization: Applications and Implementation*, TU Berlin 2016.
+> DOI: [10.14279/depositonce-5415](https://depositonce.tu-berlin.de/items/8e2988b2-d991-45b5-aad5-9fb7988f3b2f) · CC BY-SA 4.0
+
+**The long-term goal is a CGAL package** — a submission to the CGAL library that brings discrete conformal maps (hyper-ideal, spherical, Euclidean) to the CGAL ecosystem using `CGAL::Surface_mesh` as the underlying halfedge data structure, with a traits-class design compatible with arbitrary CGAL-conforming mesh types.
+
+The project has four distinct phase blocks (updated 2026-05-22):
+- **Phase 1–7 (done, v0.7.0):** Direct port of the Java library algorithms to C++.
+- **Phase 8a MVP + 8b-Lite (done, v0.9.0):** CGAL public-API surface for all five DCE models via `<CGAL/Discrete_*.h>`.  Phase 8a.2 (generic FaceGraph), 8c (manuals), 8d (CGAL-test-format), 8e (YAML pipeline) deferred on-demand.
+- **Phase 9a + 9b (done, v0.9.0):** Two new functionals (CP-Euclidean port, Inversive-Distance research), two new Newton solvers, block-FD HyperIdeal Hessian.
+- **Phase 9b-analytic + 9c (planned):** Full analytic HyperIdeal Hessian via Schläfli identity (research, see `doc/roadmap/research-track.md`); 4g-polygon fundamental domain for genus g > 1 (mixed port + research).
+- **Phase 10+ (research):** Holomorphic differentials, Siegel period matrix Ω ∈ H_g, full uniformization for genus g ≥ 2.
+
+## Language
+
+**All code, comments, documentation, commit messages, and test descriptions must be in English.** The project is intended for international collaboration and CGAL submission. Existing German-language comments in older files should be replaced with English when editing those files.
+
+## Build commands
+
+All source lives under `code/`. Three build modes:
+
+```bash
+# Mode 1 — fast tests, no CGAL, no Boost, no display (CI default)
+cmake -S code -B build
+cmake --build build --target conformallab_tests -j$(nproc)
+ctest --test-dir build --output-on-failure
+
+# Mode 2 — CGAL tests, headless (CI full, requires Boost headers only)
+#   macOS: brew install boost    Linux: apt install libboost-dev
+cmake -S code -B build -DWITH_CGAL_TESTS=ON
+cmake --build build --target conformallab_cgal_tests -j$(nproc)
+ctest --test-dir build -R "^cgal\." --output-on-failure
+
+# Mode 3 — full local build: CLI app + viewer + examples (requires Wayland/X11)
+cmake -S code -B build -DWITH_CGAL=ON
+cmake --build build -j$(nproc)
+```
+
+`-DWITH_CGAL=ON` automatically enables `-DWITH_VIEWER=ON`, which pulls in GLFW and requires `wayland-scanner`. Never use this in headless CI.
+
+### Running a single test
+
+```bash
+# By GTest suite/test name
+./build/conformallab_cgal_tests --gtest_filter="NewtonSolver*"
+./build/conformallab_tests       --gtest_filter="Clausen*"
+
+# By CTest regex (prefix "cgal." for all CGAL tests)
+ctest --test-dir build -R "cgal.NewtonSolver" --output-on-failure
+```
+
+### Rebuilding the CI Docker image
+
+```bash
+docker buildx build \
+  --platform linux/arm64 \
+  -f .gitea/docker/Dockerfile.ci-cpp \
+  -t git.eulernest.eu/conformallab/ci-cpp:latest \
+  --push \
+  .gitea/docker/
+```
+
+## Architecture
+
+### Everything is header-only
+
+All algorithms live in `code/include/*.hpp`. There is no compiled library. The three CMake targets (`conformallab_tests`, `conformallab_cgal_tests`, `conformallab_core`) compile headers directly from their `.cpp` entry points. To add a new algorithm: create a `.hpp` in `code/include/`, add a test in `code/tests/cgal/`, and register the test file in `code/tests/cgal/CMakeLists.txt`.
+
+### Central type: `ConformalMesh`
+
+`conformal_mesh.hpp` defines the core type:
+```cpp
+using ConformalMesh = CGAL::Surface_mesh<Point3>;  // CGAL::Simple_cartesian<double>
+```
+
+This replaces the Java `CoHDS` (half-edge data structure) and its intrusive `CoVertex`/`CoEdge`/`CoFace` types. Data is attached via named CGAL property maps instead of intrusive fields:
+
+| Property map name | Type | Meaning |
+|---|---|---|
+| `"v:lambda"` | `double` per vertex | log scale factor (conformal variable uᵢ) |
+| `"v:theta"` | `double` per vertex | target cone angle Θᵥ |
+| `"v:idx"` | `int` per vertex | solver DOF index; `-1` = pinned/boundary |
+| `"e:alpha"` | `double` per edge | intersection angle αᵢⱼ (hyperbolic only) |
+| `"f:type"` | `int` per face | geometry type (0=Euclidean, 1=Hyperbolic, 2=Spherical) |
+
+`CGAL_DISABLE_GMP` and `CGAL_DISABLE_MPFR` are defined for all CGAL targets — the library deliberately uses `Simple_cartesian<double>` (floating-point, no exact arithmetic) because conformal geometry does not require exact predicates.
+
+### The five DCE models
+
+Each model has its own Maps struct that bundles all property maps, plus a functional, optional Hessian, Newton solver, and (since v0.9.0) a CGAL public-API entry function:
+
+| Model | Space | DOFs | Maps struct | Key headers | Newton function | CGAL entry |
+|---|---|---|---|---|---|---|
+| Euclidean | ℝ² | vertex | `EuclideanMaps` | `euclidean_functional.hpp`, `euclidean_hessian.hpp` | `newton_euclidean()` | `discrete_conformal_map_euclidean()` |
+| Spherical | S² | vertex | `SphericalMaps` | `spherical_functional.hpp`, `spherical_hessian.hpp` | `newton_spherical()` | `discrete_conformal_map_spherical()` |
+| Hyper-ideal | H² (Poincaré disk) | vertex + edge | `HyperIdealMaps` | `hyper_ideal_functional.hpp`, `hyper_ideal_hessian.hpp` (block-FD, Phase 9b) | `newton_hyper_ideal()` | `discrete_conformal_map_hyper_ideal()` |
+| CP-Euclidean (BPS 2010) | face-based circle packing | **face** | `CPEuclideanMaps` | `cp_euclidean_functional.hpp` | `newton_cp_euclidean()` | `discrete_circle_packing_euclidean()` |
+| Inversive-Distance (Luo 2004) | vertex-based circle packing | vertex | `InversiveDistanceMaps` | `inversive_distance_functional.hpp` | `newton_inversive_distance()` | `discrete_inversive_distance_map()` |
+
+DOF-assignment patterns:
+- **Vertex-only models** (Euclidean, Spherical, Inversive-Distance): pin one vertex manually (`maps.v_idx[first_vertex] = -1`) then assign sequential indices.  The CGAL public entries do this automatically with the "natural-theta" trick (so calling them with no arguments returns x = 0 as the equilibrium).
+- **HyperIdeal**: `assign_all_dof_indices(mesh, maps)` assigns vertex + edge DOFs automatically.
+- **CP-Euclidean**: face-based — `assign_cp_euclidean_face_dof_indices(mesh, maps, pinned_face)` pins one face and indexes the rest.
+
+### The full pipeline
+
+```
+load_mesh()                          → ConformalMesh  (OFF/OBJ/PLY)
+setup_*_maps(mesh)                   → *Maps  (property maps created, all zero)
+compute_*_lambda0_from_mesh(mesh, m) → λ° initialised from 3-D edge lengths
+DOF assignment                       → v_idx[v] set; -1 = pinned
+check_gauss_bonnet(mesh, maps)       → throws if Σ(2π−Θᵥ) ≠ 2π·χ(M)
+enforce_gauss_bonnet(mesh, maps)     → redistributes angle defect uniformly
+newton_*(mesh, x0, maps)             → NewtonResult{x*, iterations, converged}
+compute_cut_graph(mesh)              → CutGraph  (2g seam edges, tree-cotree)
+*_layout(mesh, x*, maps, &cg, &hol) → Layout2D/3D  + HolonomyData
+normalise_*(layout)                  → canonical position (PCA / Möbius / Rodrigues)
+compute_period_matrix(hol)           → PeriodData{τ∈ℍ}  (genus 1 flat torus)
+compute_fundamental_domain(hol)      → FundamentalDomain{vertices, generators}
+tiling_neighbourhood(layout, hol)    → vector of translated layout copies
+save_result_json/xml()               → serialised result
+```
+
+After `compute_*_lambda0_from_mesh()` the original vertex positions are no longer used — all subsequent computation is in log-length/scale-factor space.
+
+### Newton solver (`newton_solver.hpp`)
+
+Gradient sign convention differs across the five models:
+- **Euclidean / Spherical / Inversive-Distance:** `G_v = Θ_v − actual_angle_sum` (target minus actual).
+- **HyperIdeal:** `G_v = actual_angle_sum − Θ_v` (actual minus target).
+- **CP-Euclidean:** `G_f = φ_f − Σ_{h:face(h)=f} (p(θ*,Δρ) + θ*)` (face-based; see `cp_euclidean_functional.hpp` header for the full formula).
+
+Hessian sign and solver per model:
+- **Euclidean:** H is PSD (cotangent Laplacian) → `SimplicialLDLT(H)`.
+- **Spherical:** H is NSD (concave energy) → `SimplicialLDLT(−H)` (sign flip inside `newton_spherical`).
+- **HyperIdeal:** H is PSD (strictly convex) → `SimplicialLDLT(H)`.  Phase 9b uses a **block-FD Hessian** (per-face 6×6 local block, ~96× speed-up vs full FD on V=200).  Full analytic Hessian via the chain `(bᵢ, aₑ) → lᵢⱼ → ζ₁₃/ζ₁₄/ζ₁₅ → αᵢⱼ/βᵢ` is planned research — see `doc/roadmap/research-track.md` Phase 9b-analytic.
+- **CP-Euclidean:** analytic 2×2-per-edge `h_jk = sin θ / (cosh Δρ − cos θ)` (BPS 2010), strictly convex → `SimplicialLDLT(H)`.
+- **Inversive-Distance:** FD Hessian (inline in `newton_inversive_distance`).  Analytic via Glickenstein 2011 eq. (4.6) is planned research (Phase 9a.2-analytic).
+
+When `SimplicialLDLT` fails (rank-deficient H — gauge mode on a closed mesh without pinned vertex/face), the solver automatically retries with `Eigen::SparseQR` to find the minimum-norm step orthogonal to the null space.  Public API: `solve_linear_system(H, rhs, &used_fallback)`.
+
+### Layout and holonomy (`layout.hpp`)
+
+BFS-trilateration with a **priority min-heap on BFS depth** (`depth = max(depth[src], depth[tgt]) + 1`). Root face = largest 3-D area face. This minimises trilateration error accumulation compared to simple BFS.
+
+Key output fields:
+- `layout.uv[v.idx()]` — primary UV (first/shallowest BFS visit per vertex)
+- `layout.halfedge_uv[h.idx()]` — UV of `source(h)` as seen from `face(h)`; at seam halfedges the two opposite halfedges carry *different* UV values, enabling proper GPU texture atlasing without vertex duplication
+- `hol.translations[i]` — lattice generator ωᵢ ∈ ℂ (Euclidean/spherical)
+- `hol.mobius_maps[i]` — Möbius isometry Tᵢ ∈ SU(1,1) (hyperbolic, Poincaré disk)
+
+`MobiusMap` is defined in `layout.hpp`: T(z) = (az+b)/(cz+d). Key methods: `from_three()` (fit to 3 point correspondences via 3×3 complex linear system), `compose()`, `inverse()`, `apply(Vector2d)`.
+
+### Key mathematical reference for each header
+
+| Header | Java original | Key reference |
+|---|---|---|
+| `hyper_ideal_geometry.hpp` | `HyperIdealGeometry.java` | Springborn (2020) — ζ₁₃/ζ₁₄/ζ₁₅ functions |
+| `euclidean_hessian.hpp` | `EuclideanHessian.java` | Pinkall & Polthier (1993) — cotangent Laplacian |
+| `spherical_hessian.hpp` | `SphericalHessian.java` | ∂α/∂u from spherical law of cosines |
+| `cut_graph.hpp` | `CuttingUtility.java` | Erickson & Whittlesey (SODA 2005) — tree-cotree |
+| `period_matrix.hpp` | `PeriodMatrixUtility.java` | Sechelmann (2016) §4 — SL(2,ℤ) reduction |
+| `gauss_bonnet.hpp` | (distributed across Java) | Gauss–Bonnet: Σ(2π−Θᵥ) = 2π·χ(M) |
+
+### Java features not yet ported (Phase 9)
+
+The Java library under `de.varylab.discreteconformal` contains these items not yet in C++:
+
+| Java class | Planned C++ header | Phase |
+|---|---|---|
+| `InversiveDistanceFunctional` | `inversive_distance_functional.hpp` | 9a |
+| Analytic HyperIdeal Hessian | `hyper_ideal_hessian.hpp` (replace FD) | 9b |
+| 4g-polygon boundary walk in `FundamentalDomainUtility` | `fundamental_domain.hpp` (extend) | 9c |
+| `DiscreteHarmonicFormUtility` | Phase 10a prerequisite | 10 |
+| `DiscreteHolomorphicFormUtility` | Phase 10a | 10 |
+| `HomologyUtility`, `CanonicalBasisUtility` | Phase 10 | 10 |
+
+When porting a Java class, locate the original in `de.varylab.discreteconformal.*` at [github.com/varylab/conformallab](https://github.com/varylab/conformallab) and use it as the reference implementation.
+
+## Test design patterns
+
+### "Natural theta" — constructing a known equilibrium at x* = 0
+
+```cpp
+// Evaluate gradient at x=0; set target angles = actual angle sums → x*=0 by definition
+std::vector<double> x0(n_dofs, 0.0);
+auto G0 = euclidean_gradient(mesh, x0, maps);
+for (auto v : mesh.vertices())
+    if (maps.v_idx[v] >= 0)
+        maps.theta_v[v] -= G0[maps.v_idx[v]];  // shift so G(x=0) = 0
+```
+
+This is used in virtually every Newton convergence test — it avoids hardcoding specific angle values.
+
+### Gradient check pattern
+
+```cpp
+// Copy from any test_*_functional.cpp — GradientCheck_* test suite
+double eps = 1e-5;
+for (int i = 0; i < n; ++i) {
+    xp[i] += eps;  auto Gp = euclidean_gradient(mesh, xp, maps);
+    xm[i] -= eps;  auto Gm = euclidean_gradient(mesh, xm, maps);
+    double fd = (energy(xp) - energy(xm)) / (2*eps);
+    EXPECT_NEAR(G[i], fd, 1e-7);
+    xp[i] = xm[i] = x0[i];
+}
+```
+
+All new functionals must have a gradient-check test before being considered complete.
+
+### Halfedge traversal
+
+```cpp
+for (auto f : mesh.faces()) {
+    auto h0 = mesh.halfedge(f);          // canonical halfedge of face
+    auto h1 = mesh.next(h0);
+    auto h2 = mesh.next(h1);
+
+    Vertex_index v1 = mesh.source(h0);  // = mesh.target(h2)
+    Vertex_index v2 = mesh.source(h1);
+    Vertex_index v3 = mesh.source(h2);
+
+    // Angle at v3 is opposite to h0 (edge v1–v2)
+    // h_alpha[h0] = α₃,  h_alpha[h1] = α₁,  h_alpha[h2] = α₂
+
+    bool is_boundary = mesh.is_border(mesh.opposite(h0));
+}
+```
+
+### Attaching custom data to the mesh
+
+```cpp
+auto [my_map, created] = mesh.add_property_map<Vertex_index, double>("v:my_data", 0.0);
+my_map[v] = 3.14;
+```
+
+## CI pipeline
+
+Two jobs in `.gitea/workflows/cpp-tests.yml`:
+
+| Job | CMake flags | Deps | Triggers on |
+|---|---|---|---|
+| `test-fast` | *(none)* | Eigen + GTest only | all branches |
+| `test-cgal` | `-DWITH_CGAL_TESTS=ON` | + Boost | pull requests only |
+
+Runner: `eulernest` — self-hosted Raspberry Pi, ARM64, Ubuntu 22.04. Docker image: `git.eulernest.eu/conformallab/ci-cpp:latest`. `test-cgal` needs `test-fast` to pass first (`needs: test-fast`).
+
+Expected results: full test suite passing, 0 skipped, 0 failed.  The canonical counts live in `doc/api/tests.md` — do not hardcode them anywhere else (see [`doc/release-policy.md`](doc/release-policy.md)).
+
+## Release state
+
+Current release: **v0.9.0** (tag on `main`, released 2026-05-22).
+Phases 1–9a complete, Phase 8b-Lite CGAL API surface complete (all 5 DCE models reachable via `<CGAL/Discrete_*.h>`), Phase 9b block-FD HyperIdeal Hessian shipped (~96× speed-up).  Next planned milestones: Phase 9c (4g-polygon, genus g > 1) and Phase 9b-analytic (Schläfli identity).  See `doc/release-policy.md` for the version-tag policy and `doc/roadmap/phases.md` for the phase plan.
+
+## Phase 8 strategic decisions (2026-05-19)
+
+The CGAL-package architecture was frozen on 2026-05-19.  Full design:
+[`doc/api/cgal-package.md`](doc/api/cgal-package.md).  Key decisions:
+
+| Decision | Choice |
+|---|---|
+| Submission to upstream CGAL | **Pre-submission-ready, not bound.** 12+ months horizon. |
+| License | **MIT preserved** (no LGPL switch). |
+| Mesh-type flexibility | **Generic `FaceGraph + HalfedgeGraph`** in target design; MVP starts Surface_mesh-only. |
+| Parameter style | **Named Parameters** (`CGAL::parameters::...`). |
+| Default kernel | **`Simple_cartesian<double>`** (status quo). |
+| Backward compatibility | **Dual-layer wrapper** — `code/include/*.hpp` stays as implementation, `include/CGAL/*.h` is thin wrapper.  No algorithm duplication. |
+| Implementation strategy | **Hybrid MVP** — minimum Phase 8 (traits + one wrapper) first, then Phase 9 in full, then Phase 8 extensions only on concrete demand. |
+| Phase-8 MVP acceptance test | **Phase 9a (Inversive-Distance)** as the first new client of the new traits API. |
+
+### Implementation sequence (committed)
+
+```
+1. Phase 7.5  Doxygen + cleanup                         done ✅
+2. Phase 8 MVP — traits + one euclidean wrapper         3–5 days
+3. Phase 9a — Inversive-Distance against new traits     3–5 days
+4. Phase 9b — analytic HyperIdeal Hessian               1 week
+5. Phase 9c — 4g-polygon for genus g > 1                1 week
+   → port really complete, v0.9.0 release
+```
+
+Phase 8 extensions (8a.2 generic FaceGraph, 8c full Doxygen manuals, 8d
+CGAL-format tests, 8e YAML pipeline) are deferred to on-demand status —
+no speculative architecture for an uncertain submission.
+
+Root-level files added at v0.7.0:
+- `CITATION.cff` — machine-readable citation (Sechelmann 2016, Springborn 2020, Bobenko–Springborn 2004)
+- `CONTRIBUTING.md` — short root-level pointer to `doc/contributing.md`
+- `scripts/try_it.sh` — one-script quickstart: build → 209 tests → example run
+- CMake install target: `cmake --install build --prefix /usr/local` → headers land in `include/conformallab/`
+
+## Port-vs-research maintenance rule (2026-05-21 audit)
+
+Before claiming something "ports X from Java", **verify empirically**:
+
+```bash
+find /Users/tarikmoussa/Desktop/conformallab -iname "*X*"
+grep -r "ClassName" /Users/tarikmoussa/Desktop/conformallab/src
+```
+
+If zero matches, the work is **new research** — add it to
+`doc/roadmap/research-track.md` with primary literature citations,
+**not** to `doc/roadmap/java-parity.md`.
+
+The 2026-05-21 audit found four pre-existing mis-labels:
+
+| Item | Wrong claim | Reality |
+|---|---|---|
+| `InversiveDistanceFunctional` | "Java port (Luo 2004)" | No such Java class exists |
+| HyperIdeal Hessian (FD) | "Phase 4a" | Research — Java has `hasHessian()==false` |
+| HyperIdeal Hessian (analytic) | "Phase 9b port" | Research — derivation via Schläfli 1858 |
+| Tutorial framing | "ports `InversiveDistanceFunctional.java`" | Implementation from Luo 2004 + Glickenstein 2011 |
+
+All four are corrected as of this commit.  Future contributors must
+follow the empirical verification rule above before any new claim.
+
+## Documentation map
+
+24 documents across 6 categories. Read the relevant one before reasoning from scratch
+— do not hallucinate content that is already written down.
+
+### Mathematics & theory
+
+| Question | Document |
+|---|---|
+| What problem does this library solve mathematically? | `doc/math/discrete-conformal-theory.md` |
+| How do the three geometry modes differ (Euclidean/Spherical/HyperIdeal)? | `doc/math/geometry-modes.md` |
+| What analytic invariants can be used to validate correctness? | `doc/math/validation.md` |
+| What are the exact ctest commands with expected terminal output? | `doc/math/validation-protocol.md` |
+| What is the O() complexity and how does it scale with mesh size? | `doc/math/complexity.md` |
+| Which papers are referenced by which header? | `doc/math/references.md` |
+| How does conformallab++ compare to libigl, CGAL, geometry-central, pmp-library? | `doc/math/software-landscape.md` |
+| What is unique about conformallab++ (novelty, target audience)? | `doc/math/novelty-statement.md` |
+
+### Architecture & design
+
+| Question | Document |
+|---|---|
+| Full pipeline diagram and data-flow overview | `doc/architecture/overall_pipeline.md` |
+| Directory tree, build targets, file organisation | `doc/architecture/project-structure.md` |
+| Key architectural decisions and their rationale | `doc/architecture/design-decisions.md` |
+| Detailed comparison with geometry-central (CMU): overlap, adoption, scientific value | `doc/architecture/geometry-central-comparison.md` |
+| Phase 9a validation report (CP-Euclidean port + Luo-inversive-distance literature check) | `doc/architecture/phase-9a-validation.md` |
+
+### API & extension
+
+| Question | Document |
+|---|---|
+| All 24 public headers with descriptions | `doc/api/headers.md` |
+| Full pipeline API for all three geometries | `doc/api/pipeline.md` |
+| What does each processing unit require/provide (contracts)? | `doc/api/contracts.md` |
+| How to add a new functional / geometry mode / port from Java | `doc/api/extending.md` |
+| Per-suite breakdown and counts (single source of truth) | `doc/api/tests.md` |
+| Phase 8 CGAL package design + Declarative YAML pipeline spec | `doc/api/cgal-package.md` |
+
+### Concepts & specs
+
+| Question | Document |
+|---|---|
+| Declarative YAML pipeline: token vocabulary, 5 examples, validation algorithm | `doc/concepts/declarative-pipeline.md` |
+
+### Roadmap & porting
+
+| Question | Document |
+|---|---|
+| Phases 1–10 with status and sub-tasks | `doc/roadmap/phases.md` |
+| Which Java classes are ported, which are planned, which are skipped? | `doc/roadmap/java-parity.md` |
+| New research items (beyond Java) — citations, acceptance criteria | `doc/roadmap/research-track.md` |
+
+### Tutorials & onboarding
+
+| Question | Document |
+|---|---|
+| Build modes, single-test invocation, CLI, Docker image rebuild | `doc/getting-started.md` |
+| Step-by-step: port the Inversive Distance functional (Phase 9a template) | `doc/tutorials/add-inversive-distance.md` |
+| Language policy, test standards, release flow | `doc/contributing.md` |
+| Versioning rules + release process + single-source-of-truth list | `doc/release-policy.md` |
+
+### geometry-central context
+
+**geometry-central** (Keenan Crane, CMU) implements the same discrete conformal
+equivalence problem (Gillespie, Springborn & Crane, SIGGRAPH 2021) but uses
+Ptolemaic flips on intrinsic triangulations instead of Newton on the original mesh.
+It has no period matrix, holonomy, or spherical geometry mode.
+The shared mathematical core (Springborn 2020) means cross-validation is meaningful.
+Full analysis: `doc/architecture/geometry-central-comparison.md`.
+Optional adoption roadmap (GC-1/2/3): `doc/roadmap/phases.md` (Optional section).
+
+## Known quirks
+
+- **No GTEST_SKIP stubs remain** (since v0.9.0): the three stale HDS-port stub files were removed because the CGAL test suite covers the same functionality with real tests.  The pure-math `conformallab_tests` target now only contains active tests.
+- **Boost is header-only**: CGAL 6.x uses only Boost headers (`Boost.Config`, `Boost.Graph`). No compiled Boost libraries are needed. `find_package(Boost REQUIRED)` only locates the include path.
+- **`main` branch is protected** on `origin` (Gitea). Push to `dev`, then merge via pull request. Codeberg `main` can be pushed to directly.
+- **Both remotes must stay in sync**: `origin` = `git.eulernest.eu` (CI runs here), `codeberg` = `codeberg.org/TMoussa/ConformalLabpp` (public mirror). Push to both after every significant change.

CONTRIBUTING.md

@@ -0,0 +1,17 @@
+# Contributing to conformallab++
+
+See **[doc/contributing.md](doc/contributing.md)** for the full guide:
+
+- Git workflow (dev → PR → main)
+- CI pipeline (test-fast / test-cgal)
+- Test standards (gradient check, convergence test, registration)
+- Code style (C++17, header-only, namespace, property map naming)
+- Release process
+
+For the mathematical background of what's being implemented, see:
+
+- [doc/math/discrete-conformal-theory.md](doc/math/discrete-conformal-theory.md) — theory overview
+- [doc/math/validation.md](doc/math/validation.md) — how to validate the implementation
+- [doc/math/references.md](doc/math/references.md) — all referenced papers
+
+To add your own research, see [doc/api/extending.md](doc/api/extending.md).

Doxyfile

@@ -0,0 +1,166 @@
+# Doxyfile for conformallab++
+#
+# Phase 7.5 — minimal CGAL-style Doxygen configuration.
+# Only non-default values are set; Doxygen ≥ 1.9.5 supplies the rest.
+#
+# Usage:
+#   doxygen Doxyfile           # generates HTML into doc/doxygen/html/
+#   open doc/doxygen/html/index.html
+#
+# Or via CMake:
+#   cmake --build build --target doc
+
+# ── Project identity ─────────────────────────────────────────────────────────
+PROJECT_NAME           = "conformallab++"
+PROJECT_NUMBER         = 0.7.0
+PROJECT_BRIEF          = "Discrete conformal maps on triangle meshes — C++17 reimplementation of ConformalLab (TU Berlin)"
+PROJECT_LOGO           =
+OUTPUT_DIRECTORY       = doc/doxygen
+USE_MDFILE_AS_MAINPAGE = README.md
+
+# ── Input ────────────────────────────────────────────────────────────────────
+INPUT                  = README.md \
+                         CLAUDE.md \
+                         code/include \
+                         doc/api \
+                         doc/architecture \
+                         doc/math
+FILE_PATTERNS          = *.hpp *.h *.cpp *.md
+RECURSIVE              = YES
+EXCLUDE_PATTERNS       = */build*/* \
+                         */deps/* \
+                         */.git/* \
+                         */test-reports/* \
+                         *\ 2.hpp \
+                         *\ 2.h
+# Research-quality LaTeX notes use raw \sinh / \cosh / \frac / \beta /
+# \cdot / \partial / \zeta macros which are valid LaTeX but unknown to
+# Doxygen.  These files are intended to be read as PDF or in a LaTeX-
+# aware markdown viewer, not as Doxygen pages.  Excluding them removes
+# ~500 spurious "unknown command" warnings while keeping the .md files
+# discoverable on GitHub.
+EXCLUDE                = doc/math/hyperideal-hessian-derivation.md
+EXCLUDE_SYMBOLS        = Eigen::* boost::* std::*
+
+# Markdown filter: rewrites repo-relative links like [x](doc/api/tests.md)
+# into basename-only links [x](tests.md) so Doxygen's basename-indexed
+# \ref resolver can find them.  On-disk files are untouched (GitHub keeps
+# rendering them correctly).  See scripts/doxygen-md-filter.sh.
+FILTER_PATTERNS        = *.md=scripts/doxygen-md-filter.sh
+
+# ── Source browsing ──────────────────────────────────────────────────────────
+EXTRACT_ALL            = YES
+EXTRACT_PRIVATE        = NO
+EXTRACT_STATIC         = YES
+EXTRACT_LOCAL_CLASSES  = YES
+HIDE_UNDOC_MEMBERS     = NO
+SOURCE_BROWSER         = YES
+INLINE_SOURCES         = NO
+STRIP_CODE_COMMENTS    = NO
+REFERENCED_BY_RELATION = YES
+REFERENCES_RELATION    = YES
+REFERENCES_LINK_SOURCE = YES
+
+# ── Build options ────────────────────────────────────────────────────────────
+JAVADOC_AUTOBRIEF      = YES
+QT_AUTOBRIEF           = NO
+MARKDOWN_SUPPORT       = YES
+AUTOLINK_SUPPORT       = YES
+BUILTIN_STL_SUPPORT    = YES
+DISTRIBUTE_GROUP_DOC   = YES
+GROUP_NESTED_COMPOUNDS = YES
+SUBGROUPING            = YES
+INLINE_GROUPED_CLASSES = NO
+INLINE_SIMPLE_STRUCTS  = NO
+TYPEDEF_HIDES_STRUCT   = NO
+EXTENSION_MAPPING      = h=C++ hpp=C++
+
+# ── Warnings ─────────────────────────────────────────────────────────────────
+QUIET                  = NO
+WARNINGS               = YES
+WARN_IF_UNDOCUMENTED   = YES
+WARN_IF_DOC_ERROR      = YES
+WARN_IF_INCOMPLETE_DOC = YES
+WARN_NO_PARAMDOC       = NO
+WARN_AS_ERROR          = NO
+WARN_FORMAT            = "$file:$line: $text"
+WARN_LOGFILE           = doc/doxygen/doxygen-warnings.log
+
+# ── HTML output ──────────────────────────────────────────────────────────────
+GENERATE_HTML          = YES
+
+# MathJax — render LaTeX math in markdown ($...$ and $$...$$) and in
+# code-comment `\f$ ... \f$` blocks via MathJax in the generated HTML.
+# Required for the conformal-mapping math notation (\Theta, \omega, \tau,
+# \mathbb{H}, …) in doc/architecture/overall_pipeline.md and the
+# header docstrings.
+USE_MATHJAX            = YES
+MATHJAX_VERSION        = MathJax_3
+MATHJAX_FORMAT         = HTML-CSS
+MATHJAX_RELPATH        = https://cdn.jsdelivr.net/npm/mathjax@3/es5/
+HTML_OUTPUT            = html
+HTML_FILE_EXTENSION    = .html
+HTML_COLORSTYLE        = LIGHT
+HTML_COLORSTYLE_HUE    = 220
+HTML_COLORSTYLE_SAT    = 100
+HTML_COLORSTYLE_GAMMA  = 80
+# HTML_TIMESTAMP was removed in Doxygen 1.10; use TIMESTAMP=NO instead.
+TIMESTAMP              = NO
+HTML_DYNAMIC_SECTIONS  = YES
+GENERATE_TREEVIEW      = YES
+DISABLE_INDEX          = NO
+ENUM_VALUES_PER_LINE   = 1
+TREEVIEW_WIDTH         = 280
+EXT_LINKS_IN_WINDOW    = NO
+SEARCHENGINE           = YES
+SERVER_BASED_SEARCH    = NO
+
+# ── Disabled outputs (we only want HTML) ─────────────────────────────────────
+GENERATE_LATEX         = NO
+GENERATE_RTF           = NO
+GENERATE_MAN           = NO
+GENERATE_XML           = YES
+XML_OUTPUT             = xml
+XML_PROGRAMLISTING     = NO
+GENERATE_DOCBOOK       = NO
+GENERATE_AUTOGEN_DEF   = NO
+GENERATE_PERLMOD       = NO
+
+# ── Preprocessor ─────────────────────────────────────────────────────────────
+ENABLE_PREPROCESSING   = YES
+MACRO_EXPANSION        = YES
+EXPAND_ONLY_PREDEF     = YES
+SEARCH_INCLUDES        = YES
+INCLUDE_PATH           = code/include
+PREDEFINED             = CGAL_DISABLE_GMP \
+                         CGAL_DISABLE_MPFR \
+                         DOXYGEN_RUNNING
+
+# ── Diagrams ─────────────────────────────────────────────────────────────────
+HAVE_DOT               = NO
+CLASS_GRAPH            = YES
+COLLABORATION_GRAPH    = NO
+GROUP_GRAPHS           = YES
+INCLUDE_GRAPH          = NO
+INCLUDED_BY_GRAPH      = NO
+CALL_GRAPH             = NO
+CALLER_GRAPH           = NO
+
+# ── Aliases (CGAL-style) ─────────────────────────────────────────────────────
+ALIASES               += "concept{1}=\xrefitem concept \"Concept\" \"Concepts\" \1"
+ALIASES               += "models{1}=\xrefitem models \"Models\" \"Models\" \1"
+ALIASES               += "cgalRequires{1}=\par Requirements: \n\1"
+ALIASES               += "cgalParam{2}=\param \1 \2"
+# CGAL named-parameter block aliases — replicates the upstream
+# ${CGAL}/Documentation/doc/Documentation/Doxyfile_common conventions
+# so that \cgalParamNBegin{name} … \cgalParamNEnd blocks render as
+# nested HTML lists in our Doxygen output.
+ALIASES               += "cgalNamedParamsBegin=<dl class=\"params\"><dt>Optional named parameters</dt><dd><table class=\"params\">"
+ALIASES               += "cgalNamedParamsEnd=</table></dd></dl>"
+ALIASES               += "cgalParamNBegin{1}=<tr><td class=\"paramname\"><code>\1</code></td><td>"
+ALIASES               += "cgalParamNEnd=</td></tr>"
+ALIASES               += "cgalParamDescription{1}=<b>Description:</b> \1<br/>"
+ALIASES               += "cgalParamType{1}=<b>Type:</b> \1<br/>"
+ALIASES               += "cgalParamDefault{1}=<b>Default:</b> \1<br/>"
+ALIASES               += "cgalParamPrecondition{1}=<b>Precondition:</b> \1<br/>"
+ALIASES               += "cgalParamExtra{1}=<i>\1</i><br/>"

LICENSE

@@ -1,6 +1,6 @@
 MIT License
 
-Copyright (c) 2026 user2595
+Copyright (c) 2024–2026 Tarik Moussa <Tarik.moussa95@gmail.com>
 
 Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:
 

README.md

@@ -1,104 +1,175 @@
 # conformallab++
 
-conformallab++ is a modern C++ reimplementation of the [ConformalLab](https://github.com/sechel/conformallab) software by Stefan Sechelmann for experiments in discrete conformal geometry and related mesh transformations.
+[![CI](https://git.eulernest.eu/conformallab/ConformalLabpp/actions/workflows/cpp-tests.yml/badge.svg)](https://git.eulernest.eu/conformallab/ConformalLabpp/actions)
+[![License: MIT](https://img.shields.io/badge/License-MIT-yellow.svg)](LICENSE)
+[![DOI](https://img.shields.io/badge/doi-Sechelmann%202016-blue)](https://depositonce.tu-berlin.de/items/8e2988b2-d991-45b5-aad5-9fb7988f3b2f)
+[![API docs](https://img.shields.io/badge/API%20docs-Doxygen-orange)](https://tmoussa.codeberg.page/ConformalLabpp/)
 
-> **Status:** early prototype stage. API, file formats, and CLI are subject to change.
+C++17 reimplementation of [ConformalLab](https://github.com/varylab/conformallab) —
+Stefan Sechelmann's Java research library for discrete conformal geometry (TU Berlin).
+The long-term goal is a **CGAL package** for discrete conformal maps.
 
-## Features
+Algorithmic foundation:
+> Stefan Sechelmann — *Variational Methods for Discrete Surface Parameterization: Applications and Implementation*, TU Berlin 2016.
+> DOI: [10.14279/depositonce-5415](https://depositonce.tu-berlin.de/items/8e2988b2-d991-45b5-aad5-9fb7988f3b2f) · CC BY-SA 4.0 ·
+> [Java original](https://github.com/varylab/conformallab) · [sechel.de](https://sechel.de/)
 
-- Discrete conformal geometry utilities (Clausen function, hyper-ideal tetrahedra, surface curves)
-- Mesh I/O and conversion using CGAL — optional, only needed for the CLI app
-- Linear algebra routines with Eigen
-- Interactive mesh viewer using libigl / GLFW — optional
-- Lightweight CLI with CLI11 and JSON configuration
+**Status:** v0.9.0 — Phases 1–9a complete, Phase 8b-Lite CGAL API surface. Newton solvers for **five** DCE models (Euclidean / Spherical / HyperIdeal / CP-Euclidean / Inversive-Distance), priority-BFS layout in ℝ²/S²/Poincaré disk, Gauss–Bonnet, tree-cotree cut graph, Möbius holonomy, period matrix (genus 1), fundamental domain, halfedge_uv texture atlas, JSON/XML serialisation, CLI app.  Full test suite passing, 0 skipped — see [`doc/api/tests.md`](doc/api/tests.md) for the per-suite breakdown.
 
-## Build modes
+---
 
-The project uses three clearly separated CMake modes so you only pull in what you need.
-
-| Mode | CMake flag | What gets built |
-|------|-----------|-----------------|
-| **Tests only** (default, used in CI) | *(none)* | `conformallab_tests` · deps: Eigen + GTest |
-| **Viewer** | `-DWITH_VIEWER=ON` | `viewer` library · deps: libigl / GLFW / GLAD + Eigen |
-| **Full app** | `-DWITH_CGAL=ON` | `conformallab_core` CLI + viewer · deps: CGAL + libigl / GLFW / GLAD + Eigen |
-
-`-DWITH_CGAL=ON` automatically enables `WITH_VIEWER` because the CLI app uses the viewer library for mesh visualisation.
-
-External dependencies ship as tarballs in `code/deps/tarballs/` and are extracted lazily at CMake configure time — no internet access needed after cloning (GTest is the only exception: fetched from GitHub via FetchContent).
-
-## Prerequisites
-
-| Tool | Minimum version |
-|------|----------------|
-| C++ compiler (GCC or Clang) | C++17 |
-| CMake | 3.20 |
-
-No system-level libraries are required for the default tests-only build. CGAL and libigl are header-only and bundled in the repo.
-
-## Getting started
+## Quick start
 
 ```bash
-git clone https://codeberg.org/TMoussa/ConformalLabpp
-cd ConformalLabpp
-```
+git clone https://codeberg.org/TMoussa/ConformalLabpp && cd ConformalLabpp
 
-### Tests only (CI default)
-
-```bash
-cmake -S code -B build
-cmake --build build --target conformallab_tests -j$(nproc)
+# Fast tests — no system dependencies
+cmake -S code -B build && cmake --build build --target conformallab_tests -j$(nproc)
 ctest --test-dir build --output-on-failure
+
+# CGAL tests headless (apt install libboost-dev / brew install boost)
+cmake -S code -B build -DWITH_CGAL_TESTS=ON
+cmake --build build --target conformallab_cgal_tests -j$(nproc)
+ctest --test-dir build -R "^cgal\." --output-on-failure
+
+# Full build with CLI + viewer (requires Wayland/X11 dev headers)
+cmake -S code -B build -DWITH_CGAL=ON && cmake --build build -j$(nproc)
+./bin/conformallab_core -i input.off -g euclidean -o layout.off -j result.json
+
+# API documentation (requires doxygen: brew/apt install doxygen)
+cmake --build build --target doc
+open doc/doxygen/html/index.html
 ```
 
-### Full CLI app (CGAL + viewer)
+### Compile-time workflow modes
+
+The default build (PCH + Unity Build + Dense→Core trims) takes ~47 s
+clean for the full CGAL test target.  Five opt-in modes cover other
+iteration scenarios:
 
 ```bash
-cmake -S code -B build -DWITH_CGAL=ON
-cmake --build build -j$(nproc)
-./code/bin/conformallab_core --input data/off/example.off --show
+# Configure-only, no compile.  ~1 s configure, 0 s build — emits
+# compile_commands.json for IDE / clangd; skips the GTest fetch.
+cmake -S code -B build -DBUILD_TESTING=OFF
+
+# Header smoke check: per-public-header isolated compile.  ~12 s full,
+# ~0.1 s after touching one header.  "Does my refactor still parse?"
+cmake -S code -B build -DBUILD_TESTING=OFF -DCONFORMALLAB_HEADERS_CHECK=ON
+cmake --build build --target headers_check
+
+# Dev iteration: PCH on, Unity off.  Slower full build (~75 s) but
+# editing a single test rebuilds in ~16 s instead of ~46 s.
+cmake -S code -B build -DWITH_CGAL_TESTS=ON -DCONFORMALLAB_DEV_BUILD=ON
+
+# Fast CI tests: -O0 -g for the test executables only (library /
+# install targets keep -O3).  Linux + g++ typically ~40 % faster
+# build at the cost of 5–15× slower test RUN.  Neutral on macOS.
+cmake -S code -B build -DWITH_CGAL_TESTS=ON -DCONFORMALLAB_FAST_TEST_BUILD=ON
+
+# Pristine measurement: disable both performance levers, e.g. for
+# scripts/quality/coverage.sh that needs every TU compiled fresh.
+cmake -S code -B build -DWITH_CGAL_TESTS=ON \
+    -DCONFORMALLAB_USE_PCH=OFF -DCMAKE_UNITY_BUILD=OFF
 ```
 
-### Viewer only (no CGAL)
+ccache is detected automatically when present on `PATH`; disable with
+`-DCONFORMALLAB_USE_CCACHE=OFF`.  Full mode matrix + measurements in
+[`doc/architecture/compile-time.md`](doc/architecture/compile-time.md).
 
-```bash
-cmake -S code -B build -DWITH_VIEWER=ON
-cmake --build build --target viewer -j$(nproc)
+---
+
+## Minimal usage
+
+```cpp
+#include "conformal_mesh.hpp"
+#include "mesh_io.hpp"
+#include "euclidean_functional.hpp"
+#include "gauss_bonnet.hpp"
+#include "newton_solver.hpp"
+#include "layout.hpp"
+
+using namespace conformallab;
+
+ConformalMesh mesh = load_mesh("input.off");
+EuclideanMaps maps = setup_euclidean_maps(mesh);
+compute_euclidean_lambda0_from_mesh(mesh, maps);
+
+// Assign DOFs — pin first vertex (gauge fix)
+auto vit = mesh.vertices().begin();
+maps.v_idx[*vit++] = -1;
+int idx = 0;
+for (; vit != mesh.vertices().end(); ++vit) maps.v_idx[*vit] = idx++;
+
+// Natural equilibrium target: x* = 0 by construction
+std::vector<double> x0(idx, 0.0);
+auto G0 = euclidean_gradient(mesh, x0, maps);
+for (auto v : mesh.vertices())
+    if (maps.v_idx[v] >= 0) maps.theta_v[v] -= G0[maps.v_idx[v]];
+
+check_gauss_bonnet(mesh, maps);
+NewtonResult res = newton_euclidean(mesh, x0, maps);
+Layout2D layout = euclidean_layout(mesh, res.x, maps);
 ```
 
-## Project structure
+---
 
-```
-code/
-├── include/               # Public headers (Clausen, hyper-ideal, mesh utils, …)
-├── src/
-│   ├── apps/v0/           # conformallab_core CLI app (requires WITH_CGAL)
-│   └── viewer/            # simple_viewer (requires WITH_VIEWER)
-├── tests/                 # GTest unit tests (always built)
-└── deps/
-    ├── tarballs/          # Bundled dependency archives
-    ├── eigen-3.4.0/       # Header-only linear algebra (always extracted)
-    ├── CGAL-6.1.1/        # Header-only geometry (extracted with WITH_CGAL)
-    ├── libigl-2.6.0/      # Header-only viewer toolkit (extracted with WITH_VIEWER)
-    ├── glfw-3.4/          # Windowing (extracted with WITH_VIEWER)
-    ├── libigl-glad/       # OpenGL loader (extracted with WITH_VIEWER)
-    └── single_includes/   # CLI11, json.hpp
-```
+## Documentation
 
-## CI
+| | |
+|---|---|
+| **API reference (Doxygen HTML)** — every public class, function and named-parameter helper | https://tmoussa.codeberg.page/ConformalLabpp/ |
+| **Getting started** — build modes, single-test invocation, CLI, Docker | [doc/getting-started.md](doc/getting-started.md) |
+| **Pipeline API** — all three geometries, holonomy, serialisation | [doc/api/pipeline.md](doc/api/pipeline.md) |
+| **Public headers** — all public headers with descriptions | [doc/api/headers.md](doc/api/headers.md) |
+| **Test suites** — per-suite breakdown and counts (single source of truth) | [doc/api/tests.md](doc/api/tests.md) |
+| **Extending** — new functionals, geometry modes, porting from Java | [doc/api/extending.md](doc/api/extending.md) |
+| **Processing unit contracts** — preconditions / provides table | [doc/api/contracts.md](doc/api/contracts.md) |
+| **CGAL package design** — Phase 8 target, YAML pipeline | [doc/api/cgal-package.md](doc/api/cgal-package.md) |
+| **Architecture & pipeline diagram** | [doc/architecture/overall_pipeline.md](doc/architecture/overall_pipeline.md) |
+| **geometry-central comparison** — shared core, demarcation, adoption candidates, scientific added value | [doc/architecture/geometry-central-comparison.md](doc/architecture/geometry-central-comparison.md) |
+| **Design decisions** — key architectural choices + rationale | [doc/architecture/design-decisions.md](doc/architecture/design-decisions.md) |
+| **Project structure** — directory tree + build targets | [doc/architecture/project-structure.md](doc/architecture/project-structure.md) |
+| **Discrete conformal theory** — mathematical background for collaborators | [doc/math/discrete-conformal-theory.md](doc/math/discrete-conformal-theory.md) |
+| **Validation** — known analytic results + how to verify them | [doc/math/validation.md](doc/math/validation.md) |
+| **Validation protocol** — concrete commands with expected outputs | [doc/math/validation-protocol.md](doc/math/validation-protocol.md) |
+| **Tutorial: add a new functional** — step-by-step Inversive-Distance port | [doc/tutorials/add-inversive-distance.md](doc/tutorials/add-inversive-distance.md) |
+| **Declarative YAML pipeline** — concept, token vocabulary, 5 examples | [doc/concepts/declarative-pipeline.md](doc/concepts/declarative-pipeline.md) |
+| **Geometry modes** — Euclidean / Spherical / HyperIdeal comparison | [doc/math/geometry-modes.md](doc/math/geometry-modes.md) |
+| **References** — all papers by module | [doc/math/references.md](doc/math/references.md) |
+| **Software landscape** — how conformallab++ relates to libigl, CGAL, geometry-central | [doc/math/software-landscape.md](doc/math/software-landscape.md) |
+| **Novelty statement** — unique features, target audience, what this is not | [doc/math/novelty-statement.md](doc/math/novelty-statement.md) |
+| **Complexity & scalability** — O() analysis, measured timings on real meshes, HyperIdeal bottleneck | [doc/math/complexity.md](doc/math/complexity.md) |
+| **Roadmap** — Phases 1–10 | [doc/roadmap/phases.md](doc/roadmap/phases.md) |
+| **Java parity table** — what is ported, what is planned | [doc/roadmap/java-parity.md](doc/roadmap/java-parity.md) |
+| **Contributing** — language policy, test standards, release flow | [doc/contributing.md](doc/contributing.md) |
+| **Claude Code context** | [CLAUDE.md](CLAUDE.md) |
 
-Tests run automatically on push to `main`, `dev`, and `claude/**` branches via a self-hosted Gitea Actions runner (`eulernest`, ARM64 Raspberry Pi). The pipeline uses a minimal Docker image (`git.eulernest.eu/conformallab/ci-cpp:latest`) with cmake, g++, git, and Node.js 20 pre-installed.
+---
 
-The Dockerfile for the CI image lives in `.gitea/docker/Dockerfile.ci-cpp`. Build and push it once whenever the image needs updating:
+## Citing
 
-```bash
-docker buildx build \
-  --platform linux/arm64 \
-  -f .gitea/docker/Dockerfile.ci-cpp \
-  -t git.eulernest.eu/conformallab/ci-cpp:latest \
-  --push \
-  .gitea/docker/
-```
+If you use conformallab++ in your research, please cite it using the metadata
+in [`CITATION.cff`](CITATION.cff).  GitHub and Codeberg show a "Cite this repository"
+button that generates BibTeX and APA automatically.
+
+The primary algorithmic source is:
+
+> Stefan Sechelmann — *Variational Methods for Discrete Surface Parameterization:
+> Applications and Implementation*, TU Berlin 2016.
+> DOI: [10.14279/depositonce-5415](https://depositonce.tu-berlin.de/items/8e2988b2-d991-45b5-aad5-9fb7988f3b2f)
+
+---
+
+## Bugs & questions
+
+- **Bug reports / feature requests:** [Gitea Issues](https://git.eulernest.eu/conformallab/ConformalLabpp/issues)
+- **Code mirror (read-only):** [Codeberg](https://codeberg.org/TMoussa/ConformalLabpp)
+- **Contact:** Tarik Moussa · Tarik.moussa95@gmail.com
+
+---
 
 ## License
 
-conformallab++ is released under the MIT License (see [LICENSE](LICENSE)).
+conformallab++ is released under the MIT License (see [LICENSE](LICENSE)).  
+Copyright © 2024–2026 Tarik Moussa.  
+The dissertation (Sechelmann 2016) is CC BY-SA 4.0.

code/.gitignore

@@ -13,6 +13,7 @@ deps/*
 !deps/tarballs
 !deps/single_includes/
 !deps/CMakeLists.txt
+!deps/THIRD-PARTY-LICENSES.md
 
 # macOS iCloud Drive duplicates ("file 2.cpp", "file 2.hpp", …)
 * 2.*
@@ -40,6 +41,8 @@ Thumbs.db
 *.lo
 *.o
 *.obj
+# Exception: mesh data files in code/data/ are not compiled objects
+!data/**/*.obj
 
 # Precompiled Headers
 *.gch

code/CMakeLists.txt

@@ -7,24 +7,43 @@ message(STATUS "Configuring ${PROJECT_NAME}...")
 
 # ── Build modes ────────────────────────────────────────────────────────────────
 #
-#   Default (CI / tests-only): only Eigen + GTest are required.
+#   Default (CI fast / pure-math tests):
+#     cmake -S code -B build
+#     Only Eigen + GTest required.  36 non-CGAL tests.
 #
-#   -DWITH_CGAL=ON   builds the conformallab_core CLI app (needs CGAL).
-#                    Automatically enables WITH_VIEWER because the app uses
-#                    the viewer library for mesh visualisation.
+#   -DWITH_CGAL_TESTS=ON   CGAL test suite only — no viewer, no CLI app.
+#     cmake -S code -B build -DWITH_CGAL_TESTS=ON
+#     Requires: system Boost headers (apt install libboost-dev).
+#     Builds:   conformallab_cgal_tests  (158 tests).
+#     Does NOT require wayland-scanner, GLFW, libigl or a display.
+#     Use this in headless CI.
 #
-#   -DWITH_VIEWER=ON builds the viewer library standalone (libigl/GLFW/GLAD).
+#   -DWITH_CGAL=ON         Full build: CLI app + viewer + examples + CGAL tests.
+#     cmake -S code -B build -DWITH_CGAL=ON
+#     Requires: Boost + Wayland/X11 dev headers (wayland-scanner, libx11-dev …).
+#     Automatically enables WITH_VIEWER.
+#     Use this for local development with the interactive viewer.
+#
+#   -DWITH_VIEWER=ON        Viewer library only (libigl / GLFW / GLAD).
 #
 # ──────────────────────────────────────────────────────────────────────────────
-option(WITH_CGAL   "Build conformallab_core app (requires CGAL + Viewer)" OFF)
-option(WITH_VIEWER "Build viewer library (libigl / GLFW / GLAD)"          OFF)
+option(WITH_CGAL_TESTS "Build CGAL test suite without viewer/CLI (headless CI)" OFF)
+option(WITH_CGAL       "Build conformallab_core CLI app + viewer + CGAL tests"  OFF)
+option(WITH_VIEWER     "Build viewer library (libigl / GLFW / GLAD)"            OFF)
 
-# The CLI app always needs the viewer; enable it implicitly.
+# WITH_CGAL_TESTS is a strict subset of WITH_CGAL — no viewer, no CLI.
+# WITH_CGAL (full build) implies WITH_VIEWER.
 if(WITH_CGAL AND NOT WITH_VIEWER)
     message(STATUS "WITH_CGAL implies WITH_VIEWER – enabling automatically.")
     set(WITH_VIEWER ON CACHE BOOL "" FORCE)
 endif()
 
+# Propagate Boost requirement for both CGAL modes + headers_check (which
+# also compiles CGAL headers, hence needs Boost::graph_traits).
+if(WITH_CGAL OR WITH_CGAL_TESTS OR CONFORMALLAB_HEADERS_CHECK)
+    find_package(Boost REQUIRED)
+endif()
+
 # ── Standard settings ──────────────────────────────────────────────────────────
 set(CMAKE_CXX_STANDARD 17)
 set(CMAKE_CXX_STANDARD_REQUIRED ON)
@@ -36,9 +55,80 @@ if(NOT CMAKE_BUILD_TYPE)
   set(CMAKE_BUILD_TYPE "Release" CACHE STRING "Build type" FORCE)
 endif()
 
+# ── ccache integration (lever D) ───────────────────────────────────────────────
+#
+# Detect `ccache` on the host and prepend it to the compile + link launchers.
+# Effect: a second clean rebuild of an unchanged tree drops from ~55 s wall
+# to ~5 s (cache hits everywhere).  Costs nothing when ccache is absent.
+# Disable explicitly with `-DCONFORMALLAB_USE_CCACHE=OFF` if you want pristine
+# from-scratch measurements (e.g. when re-running scripts/quality/coverage.sh).
+option(CONFORMALLAB_USE_CCACHE
+    "Use ccache as compiler/linker launcher when present." ON)
+if(CONFORMALLAB_USE_CCACHE)
+    find_program(CCACHE_PROGRAM ccache)
+    if(CCACHE_PROGRAM)
+        set(CMAKE_C_COMPILER_LAUNCHER   "${CCACHE_PROGRAM}")
+        set(CMAKE_CXX_COMPILER_LAUNCHER "${CCACHE_PROGRAM}")
+        message(STATUS "ccache: enabled (${CCACHE_PROGRAM})")
+    endif()
+endif()
+
+# ── Dev-iteration build mode (lever C, opt-in) ─────────────────────────────────
+#
+# Turns off Unity Build target-wide.  With Unity Build OFF and PCH still ON,
+# editing a single test file rebuilds only that one TU + relinks (≈12 s on
+# Apple M1) instead of rebuilding its entire 4-file unity batch (~46 s).
+#
+# Trade-off: a clean full rebuild gets ~20 % slower (66 s vs 55 s) because
+# each TU re-pays the per-TU CGAL parse cost despite PCH.  Recommended for
+# trial-and-error workflows; recommended OFF when measuring CI build time.
+option(CONFORMALLAB_DEV_BUILD
+    "Dev iteration mode: PCH stays on, Unity Build is forced off." OFF)
+if(CONFORMALLAB_DEV_BUILD)
+    set(CMAKE_UNITY_BUILD OFF CACHE BOOL "" FORCE)
+    message(STATUS "CONFORMALLAB_DEV_BUILD active — Unity Build forced OFF.")
+endif()
+
+# ── Fast test-build mode (lever #10, opt-in for CI-PR loops) ───────────────────
+#
+# Compile the test targets with `-O0 -g` instead of the default `-O3`.
+# The Eigen + CGAL templates dominate the BACKEND (CodeGen + Opt) phase
+# of every TU at ~55 % of wall time (~9.3 s of a 17 s TU per
+# `clang -ftime-trace`).  Dropping to `-O0` collapses that phase to
+# <2 s and yields ~40 % faster full rebuilds.  The downside is that
+# the resulting binaries are 2-5× slower to RUN — fine for "does it
+# compile + do all 259 unit tests pass?" CI loops, NOT fine for any
+# scalability or benchmark workload.
+#
+# Library/installable code is never affected; only the test
+# executables compiled into build-*/ pick this flag up.
+option(CONFORMALLAB_FAST_TEST_BUILD
+    "Compile test executables with -O0 -g for faster CI / dev loops." OFF)
+if(CONFORMALLAB_FAST_TEST_BUILD)
+    message(STATUS "CONFORMALLAB_FAST_TEST_BUILD active — tests compile at -O0 -g.")
+endif()
+
 if(CMAKE_CXX_COMPILER_ID MATCHES "Clang|GNU")
+  # ─── Compiler-warning policy ─────────────────────────────────────────────
+  # `-Wall -Wextra -Wpedantic` is the project default for first-party code.
+  # Vendored deps under code/deps/ get a separate, looser policy (handled
+  # via per-target SYSTEM include marking when they are pulled in).
+  #
+  # `CONFORMALLAB_WARNINGS_AS_ERRORS=ON` flips on `-Werror` — used in CI's
+  # promotion-track and by `scripts/quality/sanitizers.sh` to make sure no
+  # new warning class slips in unannounced.  Off by default so regular
+  # builds on slightly older toolchains aren't broken by a new GCC's
+  # added warning.
+  option(CONFORMALLAB_WARNINGS_AS_ERRORS
+    "Treat compiler warnings as errors (-Werror)." OFF)
+
   add_compile_options(-Wall -Wextra -Wpedantic)
 
+  if(CONFORMALLAB_WARNINGS_AS_ERRORS)
+    add_compile_options(-Werror)
+    message(STATUS "Warnings-as-errors mode active (-Werror).")
+  endif()
+
   # AddressSanitizer only in Debug (gtest_discover_tests runs the binary at
   # configure time and hangs with ASan enabled).
   if(CMAKE_BUILD_TYPE STREQUAL "Debug" AND NOT BUILD_TESTING)
@@ -47,20 +137,29 @@ if(CMAKE_CXX_COMPILER_ID MATCHES "Clang|GNU")
   endif()
 endif()
 
-# ── GTest (always – tests are always built) ────────────────────────────────────
+# ── GTest (only when tests are enabled) ────────────────────────────────────────
+#
+# CMake's standard `BUILD_TESTING` option (defaults ON via include(CTest))
+# gates the entire test subtree below.  Pass `-DBUILD_TESTING=OFF` for a
+# configure-only / IDE-syntax-check workflow that needs `compile_commands.json`
+# but does NOT need to download GTest, build any test binary, or spend the
+# ~11 s on the fast-test target.
 include(FetchContent)
-include(CTest)
-enable_testing()
+include(CTest)               # also defines BUILD_TESTING (default ON)
 
-FetchContent_Declare(
-    googletest
-    GIT_REPOSITORY https://github.com/google/googletest.git
-    GIT_TAG        v1.14.0
-)
-set(gtest_force_shared_crt ON CACHE BOOL "" FORCE)
-set(INSTALL_GTEST          OFF CACHE BOOL "" FORCE)
-set(BUILD_GMOCK            OFF CACHE BOOL "" FORCE)
-FetchContent_MakeAvailable(googletest)
+if(BUILD_TESTING)
+    enable_testing()
+
+    FetchContent_Declare(
+        googletest
+        GIT_REPOSITORY https://github.com/google/googletest.git
+        GIT_TAG        v1.14.0
+    )
+    set(gtest_force_shared_crt ON CACHE BOOL "" FORCE)
+    set(INSTALL_GTEST          OFF CACHE BOOL "" FORCE)
+    set(BUILD_GMOCK            OFF CACHE BOOL "" FORCE)
+    FetchContent_MakeAvailable(googletest)
+endif()
 
 # ── External deps (lazy tarball extraction) ────────────────────────────────────
 add_subdirectory(deps)
@@ -85,11 +184,6 @@ endif()
 
 # ── Core CLI app (optional, requires CGAL + Viewer) ───────────────────────────
 if(WITH_CGAL)
-    # CGAL 6.x still needs Boost.Config headers unconditionally.
-    # Install via: brew install boost  (macOS)
-    #              apt install libboost-dev  (Debian/Ubuntu)
-    find_package(Boost REQUIRED)
-
     add_executable(${PROJECT_NAME} src/apps/v0/conformallab_cli.cpp)
     target_include_directories(${PROJECT_NAME} SYSTEM PRIVATE
         ${CMAKE_CURRENT_SOURCE_DIR}/deps/single_includes
@@ -106,5 +200,105 @@ if(WITH_CGAL)
         RUNTIME_OUTPUT_DIRECTORY ${CMAKE_CURRENT_SOURCE_DIR}/bin)
 endif()
 
-# ── Tests (always) ────────────────────────────────────────────────────────────
-add_subdirectory(tests)
+# ── Example programs (require WITH_CGAL; viewer example also needs WITH_VIEWER) ─
+if(WITH_CGAL)
+    add_subdirectory(examples)
+endif()
+
+# ── Tests (gated on BUILD_TESTING) ────────────────────────────────────────────
+#
+# Default ON (CTest convention).  Pass `-DBUILD_TESTING=OFF` to skip the
+# entire test subtree — configure-only / IDE-syntax-check workflow.
+if(BUILD_TESTING)
+    add_subdirectory(tests)
+endif()
+
+# ── headers_check target (lever A, opt-in) ────────────────────────────────────
+#
+# Lightweight per-header smoke-compile target.  For each public CGAL umbrella
+# header, emit one minimal TU `#include <…>\nint main() {}` and compile it
+# in isolation.  Cost: ~6 s per header on a cold build, ~6 s if a single
+# header changed (only the touched header's smoke TU rebuilds).
+#
+# Use case: "did my Phase-N refactor break the public API surface?" without
+# waiting 55 s for the full CGAL test build.  Decoupled from BUILD_TESTING
+# because it does not include any test framework; depends only on the
+# library headers themselves.
+#
+# Build with:  cmake --build build --target headers_check
+# Or enable as part of the default target list with -DCONFORMALLAB_HEADERS_CHECK=ON.
+option(CONFORMALLAB_HEADERS_CHECK
+    "Build the headers_check smoke target (per-header isolated compile)." OFF)
+
+if(CONFORMALLAB_HEADERS_CHECK OR DEFINED ENV{CI})
+    set(_hc_dir "${CMAKE_BINARY_DIR}/headers_check_stubs")
+    file(MAKE_DIRECTORY "${_hc_dir}")
+    set(_hc_headers
+        "CGAL/Discrete_conformal_map.h"
+        "CGAL/Discrete_circle_packing.h"
+        "CGAL/Discrete_inversive_distance.h"
+        "CGAL/Conformal_layout.h"
+        "CGAL/Conformal_map_traits.h"
+        "CGAL/Conformal_map/internal/parameters.h"
+    )
+    set(_hc_targets "")
+    foreach(_hdr IN LISTS _hc_headers)
+        string(REPLACE "/" "__" _slug "${_hdr}")
+        string(REPLACE "." "_"  _slug "${_slug}")
+        set(_stub "${_hc_dir}/${_slug}.cpp")
+        file(WRITE "${_stub}"
+"// Auto-generated by CMake at configure time; do not edit.
+// Smoke-compile sentinel for ${_hdr}.
+#include <${_hdr}>
+int main() { return 0; }
+")
+        add_executable(hc_${_slug} EXCLUDE_FROM_ALL "${_stub}")
+        target_include_directories(hc_${_slug} SYSTEM PRIVATE
+            ${CMAKE_CURRENT_SOURCE_DIR}/deps/eigen-3.4.0
+            ${CMAKE_CURRENT_SOURCE_DIR}/deps/CGAL-6.1.1/include
+            ${CMAKE_CURRENT_SOURCE_DIR}/deps/single_includes
+            ${Boost_INCLUDE_DIRS})
+        target_include_directories(hc_${_slug} PRIVATE
+            ${CMAKE_CURRENT_SOURCE_DIR}/include)
+        target_compile_definitions(hc_${_slug} PRIVATE
+            CGAL_DISABLE_GMP CGAL_DISABLE_MPFR)
+        list(APPEND _hc_targets hc_${_slug})
+    endforeach()
+    add_custom_target(headers_check DEPENDS ${_hc_targets})
+endif()
+
+# ── Install target (header-only library) ──────────────────────────────────────
+# Installs all public headers to <prefix>/include/conformallab/
+# Usage from another CMake project:
+#   cmake --install build --prefix /usr/local
+#   target_include_directories(myapp PRIVATE /usr/local/include/conformallab)
+include(GNUInstallDirs)
+install(DIRECTORY ${CMAKE_CURRENT_SOURCE_DIR}/include/
+        DESTINATION ${CMAKE_INSTALL_INCLUDEDIR}/conformallab
+        FILES_MATCHING PATTERN "*.hpp"
+        PATTERN "* 2.*" EXCLUDE)   # exclude macOS Finder duplicates
+install(FILES ${CMAKE_CURRENT_SOURCE_DIR}/../LICENSE
+              ${CMAKE_CURRENT_SOURCE_DIR}/../CITATION.cff
+        DESTINATION ${CMAKE_INSTALL_DATADIR}/conformallab)
+
+# ── Doxygen documentation target (Phase 7.5) ──────────────────────────────────
+# Generates HTML API documentation into doc/doxygen/html/.
+# Usage:
+#   cmake --build build --target doc
+#   open doc/doxygen/html/index.html
+#
+# Optional dependency: install Doxygen via `brew install doxygen` (macOS) or
+# `apt install doxygen graphviz` (Linux). The target is silently disabled
+# if Doxygen is not found.
+find_package(Doxygen QUIET)
+if(DOXYGEN_FOUND)
+    set(DOXYGEN_PROJECT_ROOT ${CMAKE_CURRENT_SOURCE_DIR}/..)
+    add_custom_target(doc
+        COMMAND ${DOXYGEN_EXECUTABLE} ${DOXYGEN_PROJECT_ROOT}/Doxyfile
+        WORKING_DIRECTORY ${DOXYGEN_PROJECT_ROOT}
+        COMMENT "Generating API documentation with Doxygen"
+        VERBATIM)
+    message(STATUS "Doxygen found: target 'doc' available (cmake --build build --target doc)")
+else()
+    message(STATUS "Doxygen not found — 'doc' target unavailable (install: brew/apt install doxygen)")
+endif()

code/data/obj/cathead.obj

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+v -0.106755 -0.496734 0.692812
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code/data/obj/tetraflat.obj

@@ -0,0 +1,16 @@
+# 
+# Wavefront OBJ file
+# Converted by the DEEP Exploration Deep Exploration 5 5.0.3.1555 Release
+# Right Hemisphere, LTD
+# http://www.righthemisphere.com/
+# 
+# object sgc 1
+g sgc_1
+v 0.00000 0.00000 0.00000
+v -1.29492 0.95275 -0.28653
+v 1.14390 0.47528 1.06408
+v 0.15103 -1.42803 -0.77755
+# 4 verticies
+f  1 2 3
+f  2 1 4
+f  1 3 4

code/data/off/torus_4x4.off

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code/data/off/torus_8x8.off

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code/data/off/torus_hex_6x6.off

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code/deps/CMakeLists.txt

@@ -5,9 +5,10 @@
 #
 # Which deps are extracted depends on the active build mode:
 #
-#   (default)      Eigen only          → tests-only build
-#   WITH_VIEWER=ON + Eigen, libigl, libigl-glad, glfw
-#   WITH_CGAL=ON   + Eigen, CGAL       (WITH_VIEWER is implied by WITH_CGAL)
+#   (default)           Eigen only
+#   WITH_CGAL_TESTS=ON  + Eigen, CGAL           (headless CI — no viewer)
+#   WITH_VIEWER=ON      + Eigen, libigl, libigl-glad, glfw
+#   WITH_CGAL=ON        + Eigen, CGAL, libigl, libigl-glad, glfw
 #
 
 function(setup_dependency NAME SUBDIR)
@@ -51,6 +52,7 @@ if(WITH_VIEWER)
 endif()
 
 # ── CGAL mode ─────────────────────────────────────────────────────────────────
-if(WITH_CGAL)
+# Both WITH_CGAL_TESTS (headless CI) and WITH_CGAL (full build) need CGAL headers.
+if(WITH_CGAL OR WITH_CGAL_TESTS)
     setup_dependency("CGAL-6.1.1" "include")
 endif()

code/deps/THIRD-PARTY-LICENSES.md

@@ -0,0 +1,98 @@
+# Third-party licenses
+
+This directory contains source code from external projects that
+conformallab++ vendors at fixed versions for build reproducibility.
+Each project is governed by its own license; this file enumerates them
+so downstream packagers, distributors, and reviewers can audit
+compatibility without crawling each upstream tarball.
+
+> **Why vendored at all?**  conformallab++ is header-only and ships
+> nothing it does not author except the optional CLI binary
+> (`-DWITH_CGAL=ON`).  Vendoring guarantees that the CGAL / Eigen /
+> Boost API surface every contributor sees is identical, removing
+> "works on my machine because I have CGAL 6.0 not 5.6" failure modes
+> during early review.  Downstream packagers replacing the vendored
+> trees with system installs is supported and is the recommended path
+> for distribution-level packaging (see `doc/architecture/dependencies.md`).
+
+## conformallab++ itself
+
+| Item | License | Notes |
+|---|---|---|
+| `code/include/**`, `code/src/**`, `code/tests/**`, `scripts/**`, `doc/**` | **MIT** (see `LICENSE` at repo root) | Every C++ source file carries `SPDX-License-Identifier: MIT`; CI gate `scripts/quality/license-headers.sh` enforces this. |
+
+## Vendored dependencies
+
+The table below lists each tree under `code/deps/`, its upstream
+license, the SPDX identifier, and any compatibility note relevant to
+shipping conformallab++ as MIT.
+
+| Directory | Upstream project | Version | License (SPDX) | Compatibility with MIT distribution | Notes |
+|---|---|---|---|---|---|
+| `CGAL-6.1.1/` | [CGAL](https://www.cgal.org) | 6.1.1 | **LGPL-3.0-or-later** (most headers) + **GPL-3.0-or-later** (a small subset — see CGAL's per-header `\cgal_license{...}` macro) | Header-only consumption is compatible; we ONLY include LGPL'd parts (`Surface_mesh`, `Polygon_mesh_processing`, BGL adapters, kernels). | conformallab++ does not include any of the GPL-only CGAL packages (e.g. `Triangulation_3` parts, certain mesh-3 internals). The `\cgal_license` macro is checked at compile time and would fail the build if a GPL-only header were transitively pulled in. Commercial licenses are available from GeometryFactory for users who can't accept (L)GPL. |
+| `eigen-3.4.0/` | [Eigen](https://eigen.tuxfamily.org) | 3.4.0 | **MPL-2.0** for almost everything, **LGPL-2.1-or-later** for a few legacy files (e.g. `Eigen/src/Core/util/NonMPL2.h` gates these) | MPL-2.0 is permissive enough for MIT; the LGPL files are NOT pulled in by `<Eigen/Dense>` / `<Eigen/Sparse>` (the only Eigen headers conformallab++ includes). | We define no preprocessor flag that activates the non-MPL2 code paths. The default Eigen build is pure MPL-2.0. |
+| `libigl-2.6.0/` | [libigl](https://libigl.github.io) | 2.6.0 | **MPL-2.0** | Compatible with MIT distribution. | Only the viewer subsystem under `code/src/viewer/` uses libigl, and only when `-DWITH_VIEWER=ON`. The library headers and the CGAL wrapper headers do not depend on libigl. |
+| `libigl-glad/` | [Glad](https://glad.dav1d.de/) (the generated OpenGL loader libigl ships) | bundled with libigl 2.6.0 | **MIT** (the generator's output is licensed permissively; the loader code itself is in the public domain via the original Khronos headers) | Compatible. | Built only with `-DWITH_VIEWER=ON`. |
+| `glfw-3.4/` | [GLFW](https://www.glfw.org) | 3.4 | **zlib/libpng** | Permissive; compatible with MIT. | Built only with `-DWITH_VIEWER=ON`. See `code/deps/glfw-3.4/LICENSE.md` for the verbatim text. |
+| `single_includes/json.hpp` | [nlohmann/json](https://github.com/nlohmann/json) | 3.x (header-only single-include) | **MIT** | Identical to ours. | The file itself carries the SPDX header `MIT`; see `code/deps/single_includes/json.hpp` first lines. |
+| `tarballs/` | (build-artefact cache) | — | n/a | n/a | This directory just caches the downloaded source tarballs to avoid re-downloading on every clean build. The tarballs are bit-for-bit identical to the upstream releases. |
+
+## Auto-fetched (not vendored)
+
+These are pulled by CMake `FetchContent` at configure time. They are
+**not** redistributed by conformallab++; the user's CMake fetches them
+during build. We list them anyway for transparency.
+
+| Item | Upstream | Version | License | Fetched by |
+|---|---|---|---|---|
+| **GoogleTest** | https://github.com/google/googletest | v1.14.0 | **BSD-3-Clause** | `code/CMakeLists.txt` (test target only) |
+
+## System dependencies (required at build time, not redistributed)
+
+| Item | Where it lives | License | Purpose |
+|---|---|---|---|
+| **Boost** (header-only subset) | system package (`apt install libboost-dev`, etc.) | **Boost Software License 1.0** | Required by CGAL's BGL adapters (only when `WITH_CGAL=ON` or `WITH_CGAL_TESTS=ON`). |
+| **C++17 standard library** | the compiler's libstdc++ / libc++ / msvc | LGPL-3.0 with exception / Apache 2.0 with LLVM exception / MSVC redist | normal compiler runtime. |
+
+## Summary for downstream packagers
+
+If you are packaging conformallab++ for a distribution, the practical
+license matrix is:
+
+```
+   binary you ship (CLI app, -DWITH_CGAL=ON)
+   ├── conformallab++  (MIT)
+   ├── CGAL            (LGPL-3.0-or-later — comply with §4 LGPL: source
+   │                    of CGAL must be obtainable or shipped)
+   ├── Eigen           (MPL-2.0 — comply with §3 MPL: any modifications
+   │                    must be released under MPL-2.0)
+   ├── libigl          (MPL-2.0 — same as Eigen)
+   ├── GLFW            (zlib/libpng — acknowledgement in product docs)
+   ├── Glad            (MIT — preserve copyright notice)
+   └── Boost (headers) (BSL-1.0 — preserve copyright notice)
+
+   header-only consumer (just #include our headers)
+   ├── conformallab++  (MIT)
+   ├── Eigen           (MPL-2.0 transitively)
+   ├── CGAL            (LGPL-3.0-or-later transitively)
+   └── Boost (headers) (BSL-1.0 transitively, only if you include any
+                        CGAL/* header)
+```
+
+The header-only consumer typically doesn't trigger LGPL §4 obligations
+because LGPL §3 explicitly permits use of LGPL'd material as
+"templates, inline functions, macros" by an "Application" without
+imposing copyleft on the Application — which is exactly the
+header-only consumption pattern.
+
+If you have specific compliance questions, the upstream license texts
+are authoritative; this file is a navigational aid.
+
+## How this file is maintained
+
+* Updated whenever a `code/deps/` tree is added, removed, or version-bumped.
+* Cross-referenced by `doc/architecture/dependencies.md`.
+* There is no CI gate that auto-verifies the SPDX entries against
+  upstream — that would require either an SBOM tool (e.g. `syft`,
+  `tern`) or a manual audit. The current policy is "review on
+  dep-tree change", logged in the commit message of the bump.

code/examples/CMakeLists.txt

@@ -0,0 +1,56 @@
+# examples/CMakeLists.txt
+#
+# Example programs for conformallab++.
+# All examples require -DWITH_CGAL=ON.
+# example_viewer additionally requires -DWITH_VIEWER=ON.
+#
+# Run after building:
+#   ./build/examples/example_euclidean   [input.off] [output.off]
+#   ./build/examples/example_hyper_ideal [input.off] [output.off]
+#   ./build/examples/example_viewer      [input.off]         (requires WITH_VIEWER)
+
+# ── Shared include paths for all examples ─────────────────────────────────────
+set(EXAMPLE_INCLUDES
+    ${CMAKE_SOURCE_DIR}/deps/eigen-3.4.0
+    ${CMAKE_SOURCE_DIR}/deps/CGAL-6.1.1/include
+    ${CMAKE_SOURCE_DIR}/deps/single_includes
+    ${Boost_INCLUDE_DIRS}
+)
+set(EXAMPLE_PRIVATE_INCLUDES
+    ${CMAKE_SOURCE_DIR}/include
+)
+set(EXAMPLE_DEFS
+    CGAL_DISABLE_GMP
+    CGAL_DISABLE_MPFR
+)
+
+# ── example_euclidean ─────────────────────────────────────────────────────────
+add_executable(example_euclidean example_euclidean.cpp)
+target_include_directories(example_euclidean SYSTEM PRIVATE ${EXAMPLE_INCLUDES})
+target_include_directories(example_euclidean PRIVATE        ${EXAMPLE_PRIVATE_INCLUDES})
+target_compile_definitions(example_euclidean PRIVATE        ${EXAMPLE_DEFS})
+
+# ── example_hyper_ideal ───────────────────────────────────────────────────────
+add_executable(example_hyper_ideal example_hyper_ideal.cpp)
+target_include_directories(example_hyper_ideal SYSTEM PRIVATE ${EXAMPLE_INCLUDES})
+target_include_directories(example_hyper_ideal PRIVATE        ${EXAMPLE_PRIVATE_INCLUDES})
+target_compile_definitions(example_hyper_ideal PRIVATE        ${EXAMPLE_DEFS})
+
+# ── example_layout ───────────────────────────────────────────────────────────
+add_executable(example_layout example_layout.cpp)
+target_include_directories(example_layout SYSTEM PRIVATE ${EXAMPLE_INCLUDES})
+target_include_directories(example_layout PRIVATE        ${EXAMPLE_PRIVATE_INCLUDES})
+target_compile_definitions(example_layout PRIVATE        ${EXAMPLE_DEFS})
+
+# ── example_viewer (requires WITH_VIEWER) ─────────────────────────────────────
+if(WITH_VIEWER)
+    add_executable(example_viewer example_viewer.cpp)
+    target_include_directories(example_viewer SYSTEM PRIVATE
+        ${EXAMPLE_INCLUDES}
+        ${CMAKE_SOURCE_DIR}/deps/libigl-2.6.0/include
+        ${CMAKE_SOURCE_DIR}/deps/libigl-glad/include
+    )
+    target_include_directories(example_viewer PRIVATE ${EXAMPLE_PRIVATE_INCLUDES})
+    target_compile_definitions(example_viewer PRIVATE ${EXAMPLE_DEFS})
+    target_link_libraries(example_viewer PRIVATE viewer)
+endif()

code/examples/example_euclidean.cpp

@@ -0,0 +1,117 @@
+// example_euclidean.cpp
+//
+// conformallab++ — Euclidean discrete conformal map  (headless example)
+//
+// This program demonstrates the full library pipeline for the EUCLIDEAN
+// discrete conformal functional:
+//
+//   1. Load a triangle mesh from an OFF file
+//   2. Set up the Euclidean functional maps
+//   3. Pin one vertex (gauge fix for open surfaces)
+//   4. Set target angles via "natural equilibrium" (x* = x_input)
+//   5. Solve with Newton + backtracking line search
+//   6. Print per-vertex conformal factors u_i = x[v_idx[v]]
+//   7. Save the result mesh (same geometry, solver state printed)
+//
+// Build (requires -DWITH_CGAL=ON):
+//   cmake -S code -B build -DWITH_CGAL=ON
+//   cmake --build build --target example_euclidean
+//   ./build/examples/example_euclidean  [input.off]  [output.off]
+//
+// If no input file is given the built-in make_quad_strip() mesh is used.
+
+#include "conformal_mesh.hpp"
+#include "mesh_builder.hpp"
+#include "mesh_io.hpp"
+#include "euclidean_functional.hpp"
+#include "newton_solver.hpp"
+#include <iostream>
+#include <string>
+#include <vector>
+
+using namespace conformallab;
+
+int main(int argc, char* argv[])
+{
+    // ── Step 1: obtain mesh ───────────────────────────────────────────────
+    ConformalMesh mesh;
+    std::string input_path  = (argc > 1) ? argv[1] : "";
+    std::string output_path = (argc > 2) ? argv[2] : "/tmp/conformallab_euclidean_out.off";
+
+    if (input_path.empty()) {
+        std::cout << "[example_euclidean] No input file given — using make_quad_strip().\n";
+        mesh = make_quad_strip();
+    } else {
+        std::cout << "[example_euclidean] Loading mesh from: " << input_path << "\n";
+        try { mesh = load_mesh(input_path); }
+        catch (const std::exception& e) {
+            std::cerr << "Error loading mesh: " << e.what() << "\n";
+            return 1;
+        }
+    }
+
+    std::cout << "[example_euclidean] Mesh: "
+              << mesh.number_of_vertices() << " vertices, "
+              << mesh.number_of_faces()    << " faces.\n";
+
+    // ── Step 2: set up functional maps ────────────────────────────────────
+    auto maps = setup_euclidean_maps(mesh);
+    compute_euclidean_lambda0_from_mesh(mesh, maps);
+
+    // ── Step 3: pin the first vertex (gauge fix) ──────────────────────────
+    auto vit = mesh.vertices().begin();
+    Vertex_index v_pinned = *vit++;
+    maps.v_idx[v_pinned] = -1;   // pinned: u[v_pinned] = 0 (fixed)
+    int idx = 0;
+    for (; vit != mesh.vertices().end(); ++vit)
+        maps.v_idx[*vit] = idx++;
+    const int n = idx;
+
+    std::cout << "[example_euclidean] DOFs: " << n << " (1 vertex pinned).\n";
+
+    // ── Step 4: natural equilibrium — set theta_v = actual angle sum at x=0 ─
+    //    After this step x* = 0 is the equilibrium (no deformation).
+    //    In a real application you would set theta_v = desired angle (e.g. 2π
+    //    for flat disks, or the cone angles for a cone metric).
+    {
+        std::vector<double> x0(static_cast<std::size_t>(n), 0.0);
+        auto G0 = euclidean_gradient(mesh, x0, maps);
+        for (auto v : mesh.vertices()) {
+            int iv = maps.v_idx[v];
+            if (iv >= 0) maps.theta_v[v] -= G0[static_cast<std::size_t>(iv)];
+        }
+    }
+
+    // ── Step 5: solve from a small perturbation to demonstrate Newton ─────
+    std::vector<double> x0(static_cast<std::size_t>(n), -0.05);
+    std::cout << "[example_euclidean] Solving Newton system…\n";
+    auto result = newton_euclidean(mesh, x0, maps, /*tol=*/1e-9, /*max_iter=*/100);
+
+    // ── Step 6: report ────────────────────────────────────────────────────
+    if (result.converged) {
+        std::cout << "[example_euclidean] Converged in " << result.iterations
+                  << " iterations.  ||G||_inf = " << result.grad_inf_norm << "\n";
+    } else {
+        std::cout << "[example_euclidean] Did NOT converge after " << result.iterations
+                  << " iterations.  ||G||_inf = " << result.grad_inf_norm << "\n";
+    }
+
+    std::cout << "[example_euclidean] Per-vertex conformal factors u_i:\n";
+    for (auto v : mesh.vertices()) {
+        int iv = maps.v_idx[v];
+        double u = (iv >= 0) ? result.x[static_cast<std::size_t>(iv)] : 0.0;
+        std::cout << "  v" << v << "  u = " << u;
+        if (iv < 0) std::cout << "  (pinned)";
+        std::cout << "\n";
+    }
+
+    // ── Step 7: write output mesh ─────────────────────────────────────────
+    try {
+        save_mesh(output_path, mesh);
+        std::cout << "[example_euclidean] Mesh saved to: " << output_path << "\n";
+    } catch (const std::exception& e) {
+        std::cerr << "Warning: could not write output: " << e.what() << "\n";
+    }
+
+    return result.converged ? 0 : 1;
+}

code/examples/example_hyper_ideal.cpp

@@ -0,0 +1,147 @@
+// example_hyper_ideal.cpp
+//
+// conformallab++ — Hyper-ideal discrete conformal map  (headless example)
+//
+// Demonstrates the full library pipeline for the HYPER-IDEAL discrete conformal
+// functional (Springborn 2020).  The hyper-ideal functional operates in
+// hyperbolic geometry: vertices have "horoball radii" (DOF b_i) and edges have
+// "intersection lengths" (DOF a_e).  The energy is strictly convex, so Newton
+// converges globally from any valid starting point.
+//
+// Pipeline:
+//   1. Load (or synthesise) a triangle mesh
+//   2. Set up HyperIdeal maps + assign all vertex and edge DOFs
+//   3. Choose equilibrium base point (b=1.0, a=0.5) and set natural targets
+//   4. Perturb and solve with Newton
+//   5. Print DOF values at equilibrium
+//   6. Save result mesh
+//
+// Build (requires -DWITH_CGAL=ON):
+//   cmake -S code -B build -DWITH_CGAL=ON
+//   cmake --build build --target example_hyper_ideal
+//   ./build/examples/example_hyper_ideal  [input.off]  [output.off]
+
+#include "conformal_mesh.hpp"
+#include "mesh_builder.hpp"
+#include "mesh_io.hpp"
+#include "hyper_ideal_functional.hpp"
+#include "newton_solver.hpp"
+#include <iostream>
+#include <string>
+#include <vector>
+#include <cmath>
+
+using namespace conformallab;
+
+int main(int argc, char* argv[])
+{
+    // ── Step 1: obtain mesh ───────────────────────────────────────────────
+    ConformalMesh mesh;
+    std::string input_path  = (argc > 1) ? argv[1] : "";
+    std::string output_path = (argc > 2) ? argv[2] : "/tmp/conformallab_hyper_ideal_out.off";
+
+    if (input_path.empty()) {
+        std::cout << "[example_hyper_ideal] No input file — using make_triangle().\n";
+        mesh = make_triangle();
+    } else {
+        std::cout << "[example_hyper_ideal] Loading mesh from: " << input_path << "\n";
+        try { mesh = load_mesh(input_path); }
+        catch (const std::exception& e) {
+            std::cerr << "Error loading mesh: " << e.what() << "\n";
+            return 1;
+        }
+    }
+
+    std::cout << "[example_hyper_ideal] Mesh: "
+              << mesh.number_of_vertices() << " vertices, "
+              << mesh.number_of_faces()    << " faces.\n";
+
+    // ── Step 2: set up functional maps ────────────────────────────────────
+    auto maps = setup_hyper_ideal_maps(mesh);
+    int n = assign_all_dof_indices(mesh, maps);
+
+    std::cout << "[example_hyper_ideal] DOFs: " << n
+              << " (" << mesh.number_of_vertices() << " vertex + "
+              << mesh.number_of_edges()            << " edge).\n";
+
+    // ── Step 3: choose equilibrium base point and set natural targets ─────
+    //
+    //    x = 0 is degenerate for the HyperIdeal functional (log-space).
+    //    We pick a valid base point (b_i = b_base, a_e = a_base), evaluate
+    //    the gradient there, and absorb it into the target angles so that
+    //    G(xbase) = 0.  This makes xbase the equilibrium x*.
+    //
+    //    In a real application you would set theta_v / theta_e to the desired
+    //    hyperbolic angle targets (e.g. from a reference mesh).
+    const double b_base = 1.0;   // horoball radii at equilibrium
+    const double a_base = 0.5;   // edge-length DOFs at equilibrium
+
+    const auto sz = static_cast<std::size_t>(n);
+    std::vector<double> xbase(sz, 0.0);
+    for (auto v : mesh.vertices()) {
+        int iv = maps.v_idx[v];
+        if (iv >= 0) xbase[static_cast<std::size_t>(iv)] = b_base;
+    }
+    for (auto e : mesh.edges()) {
+        int ie = maps.e_idx[e];
+        if (ie >= 0) xbase[static_cast<std::size_t>(ie)] = a_base;
+    }
+
+    // G = Σβ − theta_target; absorb G(xbase) into targets so G(xbase) = 0
+    auto G0 = evaluate_hyper_ideal(mesh, xbase, maps, false).gradient;
+    for (auto v : mesh.vertices()) {
+        int iv = maps.v_idx[v];
+        if (iv >= 0) maps.theta_v[v] += G0[static_cast<std::size_t>(iv)];
+    }
+    for (auto e : mesh.edges()) {
+        int ie = maps.e_idx[e];
+        if (ie >= 0) maps.theta_e[e] += G0[static_cast<std::size_t>(ie)];
+    }
+
+    // ── Step 4: perturb and solve ─────────────────────────────────────────
+    const double perturb = 0.25;
+    std::vector<double> x0 = xbase;
+    for (auto& v : x0) v += perturb;
+
+    double g_start = 0.0;
+    for (double v : G0) g_start = std::max(g_start, std::abs(v));
+    std::cout << "[example_hyper_ideal] Starting Newton from perturbation +" << perturb
+              << " (G at xbase = " << g_start << ").\n";
+
+    auto result = newton_hyper_ideal(mesh, x0, maps, /*tol=*/1e-9, /*max_iter=*/200);
+
+    // ── Step 5: report ────────────────────────────────────────────────────
+    if (result.converged) {
+        std::cout << "[example_hyper_ideal] Converged in " << result.iterations
+                  << " iterations.  ||G||_inf = " << result.grad_inf_norm << "\n";
+    } else {
+        std::cout << "[example_hyper_ideal] Did NOT converge after " << result.iterations
+                  << " iterations.  ||G||_inf = " << result.grad_inf_norm << "\n";
+    }
+
+    std::cout << "[example_hyper_ideal] DOF values at equilibrium:\n";
+    for (auto v : mesh.vertices()) {
+        int iv = maps.v_idx[v];
+        if (iv < 0) continue;
+        std::cout << "  v" << v
+                  << "  b = " << result.x[static_cast<std::size_t>(iv)]
+                  << "  (expected " << b_base << ")\n";
+    }
+    for (auto e : mesh.edges()) {
+        int ie = maps.e_idx[e];
+        if (ie < 0) continue;
+        std::cout << "  e" << e
+                  << "  a = " << result.x[static_cast<std::size_t>(ie)]
+                  << "  (expected " << a_base << ")\n";
+    }
+
+    // ── Step 6: write output mesh ─────────────────────────────────────────
+    try {
+        save_mesh(output_path, mesh);
+        std::cout << "[example_hyper_ideal] Mesh saved to: " << output_path << "\n";
+    } catch (const std::exception& e) {
+        std::cerr << "Warning: could not write output: " << e.what() << "\n";
+    }
+
+    return result.converged ? 0 : 1;
+}

code/examples/example_layout.cpp

@@ -0,0 +1,168 @@
+// example_layout.cpp
+//
+// Phase 5 example — Euclidean conformal layout with JSON/XML serialisation.
+//
+// Demonstrates the full pipeline that a library user would run:
+//
+//   1. Build (or load) a mesh.
+//   2. Set up Euclidean conformal maps + compute λ° from vertex positions.
+//   3. Choose natural target angles (→ x* = 0 is the equilibrium).
+//   4. Run Newton's method.
+//   5. Unfold the mesh in the plane (euclidean_layout).
+//   6. Save the layout as an OFF file for inspection in MeshLab/Blender.
+//   7. Serialise the solver result and UV coordinates to JSON and XML.
+//   8. Reload from JSON and verify the DOF vector is recovered.
+//
+// Build (from code/):
+//   cmake -S . -B build -DWITH_CGAL=ON && cmake --build build -t example_layout
+//
+// Run:
+//   ./build/examples/example_layout
+//   ./build/examples/example_layout input.off layout.off result.json result.xml
+
+#include "conformal_mesh.hpp"
+#include "mesh_builder.hpp"
+#include "mesh_io.hpp"
+#include "euclidean_functional.hpp"
+#include "newton_solver.hpp"
+#include "layout.hpp"
+#include "serialization.hpp"
+
+#include <iostream>
+#include <iomanip>
+#include <vector>
+#include <string>
+#include <cmath>
+
+using namespace conformallab;
+
+// ── Helper: natural target angles so x* = 0 is the equilibrium ───────────────
+static void set_natural_theta(ConformalMesh& mesh, EuclideanMaps& maps, int n)
+{
+    std::vector<double> x0(static_cast<std::size_t>(n), 0.0);
+    auto G = euclidean_gradient(mesh, x0, maps);
+    for (auto v : mesh.vertices()) {
+        int iv = maps.v_idx[v];
+        if (iv < 0) continue;
+        maps.theta_v[v] -= G[static_cast<std::size_t>(iv)];
+    }
+}
+
+// ── Helper: pin vertex 0, assign DOF indices 0..n-1 to the rest ──────────────
+static int pin_first(ConformalMesh& mesh, EuclideanMaps& maps)
+{
+    auto vit = mesh.vertices().begin();
+    maps.v_idx[*vit++] = -1;   // pinned
+    int idx = 0;
+    for (; vit != mesh.vertices().end(); ++vit)
+        maps.v_idx[*vit] = idx++;
+    return idx;
+}
+
+// ── Main ─────────────────────────────────────────────────────────────────────
+
+int main(int argc, char* argv[])
+{
+    // ── Step 1: mesh ──────────────────────────────────────────────────────────
+    ConformalMesh mesh;
+    std::string out_off  = "layout.off";
+    std::string out_json = "result.json";
+    std::string out_xml  = "result.xml";
+
+    if (argc >= 2) {
+        // User provided an input mesh
+        if (!read_mesh(argv[1], mesh) || mesh.is_empty()) {
+            std::cerr << "Cannot load " << argv[1] << "\n";
+            return 1;
+        }
+        if (argc >= 3) out_off  = argv[2];
+        if (argc >= 4) out_json = argv[3];
+        if (argc >= 5) out_xml  = argv[4];
+    } else {
+        // Use built-in quad-strip (6 vertices, 4 triangles)
+        mesh = make_quad_strip();
+        std::cout << "Using built-in quad-strip mesh (no arguments given).\n";
+    }
+
+    std::cout << "Mesh: " << mesh.number_of_vertices() << " vertices, "
+              << mesh.number_of_faces() << " faces\n";
+
+    // ── Step 2: maps + λ° ────────────────────────────────────────────────────
+    auto maps = setup_euclidean_maps(mesh);
+    compute_euclidean_lambda0_from_mesh(mesh, maps);
+
+    // ── Step 3: DOF assignment + natural targets ──────────────────────────────
+    int n = pin_first(mesh, maps);
+    std::cout << "DOFs: " << n << " (vertex 0 pinned)\n";
+
+    set_natural_theta(mesh, maps, n);
+
+    // ── Step 4: Newton ────────────────────────────────────────────────────────
+    std::vector<double> x0(static_cast<std::size_t>(n), 0.0);
+    auto res = newton_euclidean(mesh, x0, maps);
+
+    std::cout << std::boolalpha
+              << "Newton converged: " << res.converged
+              << "  iterations: "     << res.iterations
+              << "  |grad|_inf: "
+              << std::scientific << std::setprecision(3) << res.grad_inf_norm << "\n";
+
+    // Print scale factors u_v
+    std::cout << "Conformal factors u_v:\n";
+    for (auto v : mesh.vertices()) {
+        int iv = maps.v_idx[v];
+        double u = (iv >= 0) ? res.x[static_cast<std::size_t>(iv)] : 0.0;
+        std::cout << "  v" << v.idx() << ": u = " << std::fixed << std::setprecision(8) << u << "\n";
+    }
+
+    // ── Step 5: Layout ────────────────────────────────────────────────────────
+    auto layout = euclidean_layout(mesh, res.x, maps);
+
+    std::cout << "Layout success: " << layout.success
+              << "  has_seam: " << layout.has_seam << "\n";
+
+    if (layout.success) {
+        std::cout << "UV coordinates:\n";
+        for (auto v : mesh.vertices()) {
+            auto& p = layout.uv[v.idx()];
+            std::cout << "  v" << v.idx()
+                      << ": (" << std::fixed << std::setprecision(6)
+                      << p.x() << ", " << p.y() << ")\n";
+        }
+    }
+
+    // ── Step 6: Save layout OFF ───────────────────────────────────────────────
+    save_layout_off(out_off, mesh, layout);
+    std::cout << "Layout saved → " << out_off << "\n";
+
+    // ── Step 7: Serialise JSON + XML ──────────────────────────────────────────
+    save_result_json(out_json, res, "euclidean",
+                     static_cast<int>(mesh.number_of_vertices()),
+                     static_cast<int>(mesh.number_of_faces()),
+                     &layout);
+    std::cout << "JSON   saved → " << out_json << "\n";
+
+    save_result_xml(out_xml, res, "euclidean",
+                    static_cast<int>(mesh.number_of_vertices()),
+                    static_cast<int>(mesh.number_of_faces()),
+                    &layout);
+    std::cout << "XML    saved → " << out_xml << "\n";
+
+    // ── Step 8: JSON round-trip verification ──────────────────────────────────
+    NewtonResult res2;
+    std::string  geom;
+    Layout2D     layout2;
+    auto x2 = load_result_json(out_json, &res2, &geom, &layout2);
+
+    std::cout << "Round-trip: geometry=\"" << geom << "\"  "
+              << "DOFs=" << x2.size() << "  "
+              << "converged=" << res2.converged << "\n";
+
+    // Verify the DOF vector survived the round-trip
+    bool ok = (x2.size() == res.x.size());
+    for (std::size_t i = 0; i < x2.size() && ok; ++i)
+        ok = (std::abs(x2[i] - res.x[i]) < 1e-10);
+    std::cout << "DOF vector round-trip: " << (ok ? "OK ✓" : "MISMATCH ✗") << "\n";
+
+    return res.converged ? 0 : 1;
+}

code/examples/example_viewer.cpp

@@ -0,0 +1,150 @@
+// example_viewer.cpp
+//
+// conformallab++ — Interactive viewer example  (requires -DWITH_VIEWER=ON)
+//
+// This example demonstrates the full end-to-end pipeline WITH visual output:
+//
+//   1. Load (or synthesise) a mesh
+//   2. Compute the Euclidean discrete conformal map (Newton solver)
+//   3. Display the result in an interactive libigl / GLFW window
+//      • Left pane:  input mesh, coloured by per-vertex conformal factor u_i
+//      • Right pane: a flat parameterisation (future, placeholder)
+//
+// The viewer is split into two data sets using libigl's multi-mesh API so
+// users can inspect geometry and solution simultaneously.
+//
+// Build (requires -DWITH_CGAL=ON -DWITH_VIEWER=ON):
+//   cmake -S code -B build -DWITH_CGAL=ON -DWITH_VIEWER=ON
+//   cmake --build build --target example_viewer
+//   ./build/examples/example_viewer  [input.off]
+//
+// Navigation (libigl default):
+//   Mouse drag   — rotate
+//   Scroll       — zoom
+//   C            — toggle camera mode (trackball / 2D)
+//   Z / X / Y    — snap to axis-aligned view
+//   Q / Esc      — quit
+
+#include "conformal_mesh.hpp"
+#include "mesh_builder.hpp"
+#include "mesh_io.hpp"
+#include "euclidean_functional.hpp"
+#include "mesh_utils.hpp"
+#include "newton_solver.hpp"
+#include <igl/opengl/glfw/Viewer.h>
+#include <Eigen/Dense>
+#include <iostream>
+#include <string>
+#include <vector>
+#include <cmath>
+#include <algorithm>
+
+using namespace conformallab;
+
+// ── Jet colour map: scalar → RGB ─────────────────────────────────────────────
+static Eigen::RowVector3d jet(double t)
+{
+    t = std::max(0.0, std::min(1.0, t));
+    double r = std::clamp(1.5 - std::abs(4.0 * t - 3.0), 0.0, 1.0);
+    double g = std::clamp(1.5 - std::abs(4.0 * t - 2.0), 0.0, 1.0);
+    double b = std::clamp(1.5 - std::abs(4.0 * t - 1.0), 0.0, 1.0);
+    return {r, g, b};
+}
+
+int main(int argc, char* argv[])
+{
+    // ── Step 1: load or synthesise mesh ───────────────────────────────────
+    ConformalMesh mesh;
+    std::string input_path = (argc > 1) ? argv[1] : "";
+
+    if (input_path.empty()) {
+        std::cout << "[example_viewer] No input file — using make_quad_strip().\n";
+        mesh = make_quad_strip();
+    } else {
+        std::cout << "[example_viewer] Loading: " << input_path << "\n";
+        try { mesh = load_mesh(input_path); }
+        catch (const std::exception& e) {
+            std::cerr << "Error: " << e.what() << "\n";
+            return 1;
+        }
+    }
+
+    std::cout << "[example_viewer] Mesh: "
+              << mesh.number_of_vertices() << " vertices, "
+              << mesh.number_of_faces()    << " faces.\n";
+
+    // ── Step 2: solve the Euclidean discrete conformal map ────────────────
+    auto maps = setup_euclidean_maps(mesh);
+    compute_euclidean_lambda0_from_mesh(mesh, maps);
+
+    // Pin first vertex
+    auto vit = mesh.vertices().begin();
+    maps.v_idx[*vit++] = -1;
+    int idx = 0;
+    for (; vit != mesh.vertices().end(); ++vit)
+        maps.v_idx[*vit] = idx++;
+    const int n = idx;
+
+    // Natural equilibrium
+    {
+        std::vector<double> x0(static_cast<std::size_t>(n), 0.0);
+        auto G0 = euclidean_gradient(mesh, x0, maps);
+        for (auto v : mesh.vertices()) {
+            int iv = maps.v_idx[v];
+            if (iv >= 0) maps.theta_v[v] -= G0[static_cast<std::size_t>(iv)];
+        }
+    }
+
+    // Perturb and solve
+    std::vector<double> x0(static_cast<std::size_t>(n), -0.08);
+    auto result = newton_euclidean(mesh, x0, maps, 1e-9, 200);
+
+    if (result.converged)
+        std::cout << "[example_viewer] Converged in " << result.iterations << " iterations.\n";
+    else
+        std::cout << "[example_viewer] Warning: did not converge fully "
+                     "(||G||_inf = " << result.grad_inf_norm << ").\n";
+
+    // ── Step 3: build Eigen V / F for libigl ─────────────────────────────
+    using Kernel = CGAL::Simple_cartesian<double>;
+    Eigen::MatrixXd V;
+    Eigen::MatrixXi F;
+    mesh_utils::cgal_to_eigen<Kernel>(mesh, V, F);
+
+    // Per-vertex colour: conformal factor u_i, mapped via jet palette
+    const int nv = static_cast<int>(V.rows());
+    Eigen::MatrixXd C(nv, 3);
+
+    // Collect all u values to normalise
+    std::vector<double> u_all(static_cast<std::size_t>(nv), 0.0);
+    for (auto v : mesh.vertices()) {
+        int iv = maps.v_idx[v];
+        if (iv >= 0)
+            u_all[static_cast<std::size_t>(v.idx())] = result.x[static_cast<std::size_t>(iv)];
+    }
+    double u_min = *std::min_element(u_all.begin(), u_all.end());
+    double u_max = *std::max_element(u_all.begin(), u_all.end());
+    double u_range = (u_max > u_min) ? (u_max - u_min) : 1.0;
+
+    for (int vi = 0; vi < nv; ++vi) {
+        double t = (u_all[static_cast<std::size_t>(vi)] - u_min) / u_range;
+        C.row(vi) = jet(t);
+    }
+
+    // ── Step 4: launch interactive viewer ─────────────────────────────────
+    igl::opengl::glfw::Viewer viewer;
+    viewer.data().set_mesh(V, F);
+    viewer.data().set_colors(C);
+    viewer.data().show_lines   = true;
+    viewer.data().show_overlay = true;
+
+    // Status text overlay
+    viewer.data().add_label(
+        Eigen::Vector3d(V.col(0).mean(), V.col(1).mean(), V.col(2).maxCoeff()),
+        "Euclidean conformal factor u_i (jet: blue=min, red=max)");
+
+    std::cout << "[example_viewer] Launching viewer.  Press Q or Esc to quit.\n";
+    viewer.launch();
+
+    return 0;
+}

code/include/CGAL/Conformal_layout.h

@@ -0,0 +1,111 @@
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+//
+// Package: conformallab++ / Discrete_conformal_map (Phase 8b-Lite, 2026-05-21)
+
+/*!
+\file CGAL/Conformal_layout.h
+\ingroup PkgConformalMapRef
+
+Thin CGAL-style wrapper around the legacy `euclidean_layout()`,
+`spherical_layout()` and `hyper_ideal_layout()` functions defined in
+`code/include/layout.hpp`.
+
+This header lets a CGAL-side caller go directly from a `*Maps` bundle
+and the Newton-converged DOF vector to a `Layout2D` / `Layout3D` result,
+without needing to include the legacy header explicitly.
+*/
+
+#ifndef CGAL_CONFORMAL_LAYOUT_H
+#define CGAL_CONFORMAL_LAYOUT_H
+
+#include <CGAL/Conformal_map/internal/parameters.h>
+#include <CGAL/Named_function_parameters.h>
+#include <CGAL/boost/graph/named_params_helper.h>
+
+#include "../layout.hpp"
+
+namespace CGAL {
+
+// ── Re-exported layout types ─────────────────────────────────────────────────
+//
+// `Layout2D`, `Layout3D` and `HolonomyData` are defined in
+// `conformallab::layout` (see `code/include/layout.hpp`).  We re-export
+// them here so users of the CGAL API don't need to know the legacy
+// namespace.
+
+using ::conformallab::Layout2D;
+using ::conformallab::Layout3D;
+using ::conformallab::HolonomyData;
+using ::conformallab::CutGraph;
+
+// ── Wrapper functions ────────────────────────────────────────────────────────
+
+/*!
+\ingroup PkgConformalMapRef
+
+Compute the planar Euclidean layout of `mesh` from a converged DOF
+vector `x` and a `EuclideanMaps` bundle.  Optional named parameters:
+
+* `cut_graph` (pointer-to `CutGraph`, default `nullptr`) — supply a
+  pre-computed cut graph to get a globally consistent layout on closed
+  meshes.
+* `holonomy_data` (pointer-to `HolonomyData`, default `nullptr`) — if
+  non-null, the wrapper records translation/rotation holonomies around
+  each cut edge.
+* `normalise` (bool, default `false`) — apply the canonical PCA
+  centroid + major-axis normalisation.
+
+\returns A `Layout2D` with `uv[v]` per vertex.
+*/
+template <typename TriangleMesh,
+          typename CGAL_NP_TEMPLATE_PARAMETERS>
+Layout2D euclidean_layout(
+    TriangleMesh&                              mesh,
+    const std::vector<double>&                 x,
+    const ::conformallab::EuclideanMaps&       maps,
+    const CGAL_NP_CLASS& = parameters::default_values())
+{
+    // No CGAL-side named-parameter overrides needed for Phase 8b-Lite:
+    // forward straight to the legacy implementation with sensible
+    // defaults.  Richer parameter support (cut/holonomy/normalise via
+    // named params) is on the post-1.0 wishlist; the legacy API can be
+    // called directly in the meantime.
+    return ::conformallab::euclidean_layout(mesh, x, maps);
+}
+
+/*!
+\ingroup PkgConformalMapRef
+
+Compute the spherical layout of `mesh` (points on S² ⊂ ℝ³).
+*/
+template <typename TriangleMesh,
+          typename CGAL_NP_TEMPLATE_PARAMETERS>
+Layout3D spherical_layout(
+    TriangleMesh&                              mesh,
+    const std::vector<double>&                 x,
+    const ::conformallab::SphericalMaps&       maps,
+    const CGAL_NP_CLASS& = parameters::default_values())
+{
+    return ::conformallab::spherical_layout(mesh, x, maps);
+}
+
+/*!
+\ingroup PkgConformalMapRef
+
+Compute the hyperbolic layout of `mesh` (Poincaré disk model).
+*/
+template <typename TriangleMesh,
+          typename CGAL_NP_TEMPLATE_PARAMETERS>
+Layout2D hyper_ideal_layout(
+    TriangleMesh&                              mesh,
+    const std::vector<double>&                 x,
+    const ::conformallab::HyperIdealMaps&      maps,
+    const CGAL_NP_CLASS& = parameters::default_values())
+{
+    return ::conformallab::hyper_ideal_layout(mesh, x, maps);
+}
+
+}  // namespace CGAL
+
+#endif  // CGAL_CONFORMAL_LAYOUT_H

code/include/CGAL/Conformal_map/doxygen_groups.h

@@ -0,0 +1,47 @@
+// Copyright (c) 2024-2026 Tarik Moussa.
+/*! \file CGAL/Conformal_map/doxygen_groups.h
+    \brief Doxygen `\defgroup` registrations for the Conformal_map package.
+*/
+// SPDX-License-Identifier: MIT
+//
+// This header contains only Doxygen \defgroup commands.  It is included
+// nowhere in the build; its sole purpose is to register the package's
+// Doxygen group hierarchy so that `@ingroup Pkg...` references in the
+// other public headers resolve cleanly.
+
+#ifndef CGAL_CONFORMAL_MAP_DOXYGEN_GROUPS_H
+#define CGAL_CONFORMAL_MAP_DOXYGEN_GROUPS_H
+
+/*!
+\defgroup PkgConformalMap CGAL Discrete Conformal Map package
+\brief Discrete conformal maps on triangulated surfaces — five DCE models
+(Euclidean, Spherical, Hyper-Ideal, Circle-Packing Euclidean,
+Inversive-Distance).
+
+This package provides the C++ implementation of the variational discrete
+conformal equivalence solvers from Springborn 2020, Bobenko/Pinkall/Springborn
+2010, and Luo 2004, together with a CGAL-style named-parameter API.
+
+See `doc/api/cgal-package.md` for the full design rationale.
+*/
+
+/*!
+\defgroup PkgConformalMapRef Reference manual
+\ingroup PkgConformalMap
+\brief Public C++ API: entry functions, traits, layout helpers.
+*/
+
+/*!
+\defgroup PkgConformalMapConcepts Concepts
+\ingroup PkgConformalMap
+\brief C++ concepts and traits classes consumed by the entry functions.
+*/
+
+/*!
+\defgroup PkgConformalMapNamedParameters Named function parameters
+\ingroup PkgConformalMap
+\brief Package-specific named-parameter helpers in `CGAL::parameters::*`,
+plus the pipe-operator chaining convention.
+*/
+
+#endif  // CGAL_CONFORMAL_MAP_DOXYGEN_GROUPS_H

code/include/CGAL/Conformal_map/doxygen_namespaces.h

@@ -0,0 +1,91 @@
+// Copyright (c) 2024-2026 Tarik Moussa.
+/*! \file CGAL/Conformal_map/doxygen_namespaces.h
+    \brief Doxygen `\namespace` documentation blocks for the project namespaces.
+*/
+// SPDX-License-Identifier: MIT
+//
+// Doxygen namespace documentation only — no declarations.  Centralised
+// here so that each namespace gets a single, consistent description in
+// the generated HTML, regardless of which header is parsed first.
+
+#ifndef CGAL_CONFORMAL_MAP_DOXYGEN_NAMESPACES_H
+#define CGAL_CONFORMAL_MAP_DOXYGEN_NAMESPACES_H
+
+/*!
+\namespace CGAL
+\brief Root namespace of the CGAL library; conformallab++ adds its
+public entry points (`discrete_conformal_map_*`, `Conformal_map_traits`,
+…) directly into this namespace, matching CGAL package conventions.
+*/
+
+/*!
+\namespace CGAL::Conformal_map
+\brief Implementation-detail namespace for the Discrete Conformal Map
+package.  Users normally do not need to enter this namespace; all
+public entry points are re-exported into `CGAL::`.
+*/
+
+/*!
+\namespace CGAL::Conformal_map::internal_np
+\brief Tag types backing the package-local named-function parameters
+(`vertex_curvature_map_t`, `gradient_tolerance_t`, `output_uv_map_t`, …).
+Users invoke them via the helpers in `CGAL::parameters::*`.
+*/
+
+/*!
+\namespace CGAL::parameters
+\brief CGAL named-function-parameter helpers — both upstream CGAL's and
+the conformallab++ package extensions (`vertex_curvature_map(...)`,
+`gradient_tolerance(...)`, `output_uv_map(...)`, `normalise_layout(...)`).
+Also home of the pipe-operator chaining convention; see
+`doc/tutorials/add-output-uv-map.md` §3.4.
+*/
+
+/*!
+\namespace conformallab
+\brief Core math/algorithm namespace of conformallab++.  Holds the five
+DCE functionals (Euclidean / Spherical / HyperIdeal / CP-Euclidean /
+Inversive-Distance), the Newton solver, layout helpers, mesh-property
+typedefs, and serialisation utilities.  Lives under
+`code/include/*.hpp` and is consumed both by the standalone CLI and
+by the thin CGAL wrappers under `CGAL::`.
+*/
+
+/*!
+\namespace conformallab::detail
+\brief Implementation-private helpers for the `conformallab` namespace.
+Not part of the stable public API.
+*/
+
+/*!
+\namespace conformallab::cp_detail
+\brief Implementation-private helpers for the Circle-Packing Euclidean
+functional (see `cp_euclidean_functional.hpp`).
+*/
+
+/*!
+\namespace conformallab::id_detail
+\brief Implementation-private helpers for the Inversive-Distance
+functional (see `inversive_distance_functional.hpp`).
+*/
+
+/*!
+\namespace conformallab::detail_xml
+\brief Implementation-private XML helpers for the (de)serialisation
+layer (see `serialization.hpp`).
+*/
+
+/*!
+\namespace mesh_utils
+\brief Small, opinion-free mesh utilities (loaders, validators,
+property-map registration) used by both the standalone tools and the
+CGAL wrappers.
+*/
+
+/*!
+\namespace viewer_utils
+\brief libigl-based interactive viewer helpers; built only when
+`WITH_VIEWER=ON`.  Not part of the headless / CGAL public surface.
+*/
+
+#endif  // CGAL_CONFORMAL_MAP_DOXYGEN_NAMESPACES_H

code/include/CGAL/Conformal_map/internal/parameters.h

@@ -0,0 +1,237 @@
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+//
+// Package: conformallab++ / Discrete_conformal_map (Phase 8 MVP, 2026-05-19)
+
+/*!
+\file CGAL/Conformal_map/internal/parameters.h
+\internal
+\ingroup PkgConformalMapRef
+
+Named-parameter tag definitions specific to the Discrete_conformal_map
+package.  These tags extend the CGAL named-parameter mechanism
+(see `<CGAL/Named_function_parameters.h>`).
+
+Usage from a user perspective is in `CGAL::parameters::*`; the tags
+themselves live in `CGAL::Conformal_map::internal_np`.
+
+This is an internal header — users should not include it directly.
+*/
+
+#ifndef CGAL_CONFORMAL_MAP_INTERNAL_PARAMETERS_H
+#define CGAL_CONFORMAL_MAP_INTERNAL_PARAMETERS_H
+
+#include <CGAL/Named_function_parameters.h>
+
+namespace CGAL {
+namespace Conformal_map {
+
+/// \internal
+/// Parameter tags for the conformal-map package.  Each tag is an
+/// `enum` whose name ends in `_t` and a value whose name does not.
+/// The pattern follows CGAL convention so that the existing
+/// `choose_parameter` / `get_parameter` machinery works directly.
+namespace internal_np {
+
+// ─── Target curvature (Θᵥ) ──────────────────────────────────────────────────
+/// Property-map: vertex_descriptor → FT (target cone angle Θᵥ in radians).
+/// Default: 2π at every interior vertex, π at every boundary vertex.
+enum vertex_curvature_map_t { vertex_curvature_map };
+
+// ─── Newton solver tolerances ───────────────────────────────────────────────
+/// Convergence threshold for the Newton solver: ‖G(u)‖∞ < tol.
+/// Type: FT.  Default: 1e-10.
+enum gradient_tolerance_t { gradient_tolerance };
+
+/// Maximum number of Newton iterations.
+/// Type: int.  Default: 200.
+/// (Reuses the CGAL `number_of_iterations` tag where appropriate; this
+/// alias is provided for vocabulary continuity within the package.)
+enum max_iterations_t { max_iterations };
+
+// ─── DOF / gauge fixing ─────────────────────────────────────────────────────
+/// Property-map: vertex_descriptor → bool.  `true` ⇒ vertex is pinned
+/// (u_v = 0, removed from the Newton DOF vector).
+/// Default: first vertex is pinned, all others are variable.
+enum fixed_vertex_map_t { fixed_vertex_map };
+
+// ─── Layout output (Phase 8b-Lite extension) ────────────────────────────────
+/// Property-map: vertex_descriptor → 2-D / 3-D coordinate.  If provided,
+/// the entry function calls the appropriate `*_layout()` after Newton
+/// convergence and writes the per-vertex coordinates into this map:
+///   - Euclidean / Hyper-ideal / Inversive-Distance: `Point_2` (UV in ℝ²)
+///   - Spherical: `Point_3` (point on S² ⊂ ℝ³)
+/// If the parameter is absent, no layout step is performed.  Callers
+/// who need finer control should run `*_layout()` directly on
+/// `result.x` plus the maps from `setup_*_maps()`.
+enum output_uv_map_t { output_uv_map };
+
+/// Boolean flag: if `true`, apply the canonical post-layout
+/// normalisation (`normalise_euclidean` PCA centroid + axis,
+/// `normalise_spherical` Rodrigues to north pole, …) before writing
+/// into `output_uv_map`.  Default: `false`.
+enum normalise_layout_t { normalise_layout };
+
+}  // namespace internal_np
+}  // namespace Conformal_map
+
+namespace parameters {
+
+/*!
+\addtogroup PkgConformalMapNamedParameters
+\{
+*/
+
+/// \name Discrete conformal map — package-specific named parameters
+/// \{
+
+/// `vertex_curvature_map(pmap)` — target cone angle Θᵥ per vertex.
+/// Type: model of `ReadablePropertyMap` with key = `vertex_descriptor`,
+/// value = `FT`.  If omitted, the package uses 2π at interior vertices
+/// and π at boundary vertices (the natural Gauss–Bonnet target for an
+/// open disk or closed flat surface).
+template <typename PropertyMap>
+auto vertex_curvature_map(const PropertyMap& pmap)
+{
+    return CGAL::Named_function_parameters<
+        PropertyMap,
+        Conformal_map::internal_np::vertex_curvature_map_t,
+        CGAL::internal_np::No_property
+    >(pmap);
+}
+
+/// `gradient_tolerance(eps)` — Newton stopping criterion ‖G‖∞ < eps.
+template <typename FT>
+auto gradient_tolerance(FT eps)
+{
+    return CGAL::Named_function_parameters<
+        FT,
+        Conformal_map::internal_np::gradient_tolerance_t,
+        CGAL::internal_np::No_property
+    >(eps);
+}
+
+/// `max_iterations(n)` — Newton iteration limit.
+inline auto max_iterations(int n)
+{
+    return CGAL::Named_function_parameters<
+        int,
+        Conformal_map::internal_np::max_iterations_t,
+        CGAL::internal_np::No_property
+    >(n);
+}
+
+/// `fixed_vertex_map(pmap)` — which vertices are pinned for gauge-fixing.
+/// Type: model of `ReadablePropertyMap` with key = `vertex_descriptor`,
+/// value = `bool`.  If omitted, the first vertex in the mesh is pinned
+/// (compatible with the existing legacy API).
+template <typename PropertyMap>
+auto fixed_vertex_map(const PropertyMap& pmap)
+{
+    return CGAL::Named_function_parameters<
+        PropertyMap,
+        Conformal_map::internal_np::fixed_vertex_map_t,
+        CGAL::internal_np::No_property
+    >(pmap);
+}
+
+/// `output_uv_map(pmap)` — write the per-vertex layout coordinates
+/// into `pmap` after Newton converges.
+///
+/// Type: model of `WritablePropertyMap` with key = `vertex_descriptor`
+/// and value either `Point_2` (Euclidean / Hyper-ideal / Inversive-
+/// Distance entries) or `Point_3` (Spherical entry).
+///
+/// Implementation: the entry function runs the appropriate
+/// `*_layout()` from `code/include/layout.hpp` after Newton, then
+/// writes one coordinate per vertex into `pmap`.  If omitted, no
+/// layout is performed.
+template <typename PropertyMap>
+auto output_uv_map(const PropertyMap& pmap)
+{
+    return CGAL::Named_function_parameters<
+        PropertyMap,
+        Conformal_map::internal_np::output_uv_map_t,
+        CGAL::internal_np::No_property
+    >(pmap);
+}
+
+/// `normalise_layout(flag)` — apply the canonical post-layout
+/// normalisation (PCA centroid for Euclidean; north-pole alignment
+/// for Spherical; Möbius centring for Hyper-ideal).  Default: `false`.
+/// Only meaningful in combination with `output_uv_map`.
+inline auto normalise_layout(bool flag)
+{
+    return CGAL::Named_function_parameters<
+        bool,
+        Conformal_map::internal_np::normalise_layout_t,
+        CGAL::internal_np::No_property
+    >(flag);
+}
+
+/// \}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Pipe-operator chaining for the Discrete_conformal_map package
+//
+// CGAL's standard chaining syntax `a.b(...).c(...)` requires modifying the
+// CGAL upstream `parameters_interface.h` file, which we deliberately treat
+// as a read-only vendored dependency.  Instead, conformallab++ provides a
+// pipe-operator overload that achieves the same effect from
+// left-to-right composition:
+//
+//     auto p = CGAL::parameters::gradient_tolerance(1e-12)
+//            | CGAL::parameters::max_iterations(500)
+//            | CGAL::parameters::output_uv_map(uv);
+//     CGAL::discrete_conformal_map_euclidean(mesh, p);
+//
+// Semantics: `a | b` reads as "first apply a, then b".  The result is a
+// Named_function_parameters chain identical to what `.b()` chained onto
+// `a` would have produced, so the resulting object is accepted by every
+// entry function in the package.
+//
+// Implementation note: this operator is intentionally placed in the
+// CGAL::parameters namespace so it is found by ADL when the operands are
+// `Named_function_parameters` objects produced by the helpers above.  We
+// constrain it to no-base NPs only (i.e. the operands are fresh
+// single-parameter packs) to avoid colliding with any future CGAL
+// operator on the same type.
+// ════════════════════════════════════════════════════════════════════════════
+
+// Close the PkgConformalMapNamedParameters group block that was opened
+// above the helper functions (see \addtogroup at the top of this section).
+/// \}
+
+}  // namespace parameters
+
+/// Pipe-operator chaining for package-local named parameters.
+///
+/// `a | b` combines `a` and `b` into a single `Named_function_parameters`
+/// chain.  The right-hand side `b` must be a fresh single-parameter pack
+/// (i.e. its Base is `No_property`) — typically the direct return value
+/// of one of the helper functions in `CGAL::parameters::*`.  The
+/// left-hand side can be any chain length.
+///
+/// Lives in `namespace CGAL` (not `CGAL::parameters`) so ADL finds it
+/// when the operands are `CGAL::Named_function_parameters<...>` values.
+///
+/// Use as a workaround for the missing `a.b().c()` chaining syntax
+/// while CGAL upstream does not yet expose a per-package extension
+/// point for member-function chainers.
+template <typename T_a, typename Tag_a, typename Base_a,
+          typename T_b, typename Tag_b>
+auto operator|(const CGAL::Named_function_parameters<T_a, Tag_a, Base_a>& a,
+               const CGAL::Named_function_parameters<T_b, Tag_b, CGAL::internal_np::No_property>& b)
+{
+    // Re-build b as if it had been chained on top of a.
+    using LHS_NP   = CGAL::Named_function_parameters<T_a, Tag_a, Base_a>;
+    using Combined = CGAL::Named_function_parameters<T_b, Tag_b, LHS_NP>;
+    // Read b's value (Named_params_impl::v is the stored value).
+    using Impl_b = CGAL::internal_np::Named_params_impl<T_b, Tag_b, CGAL::internal_np::No_property>;
+    const auto& v_b = static_cast<const Impl_b&>(b).v;
+    return Combined(v_b, a);
+}
+
+}  // namespace CGAL
+
+#endif  // CGAL_CONFORMAL_MAP_INTERNAL_PARAMETERS_H

code/include/CGAL/Conformal_map_traits.h

@@ -0,0 +1,184 @@
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+//
+// Package: conformallab++ / Discrete_conformal_map (Phase 8 MVP, 2026-05-19)
+
+/*!
+\file CGAL/Conformal_map_traits.h
+\ingroup PkgConformalMapRef
+
+Defines the `ConformalMapTraits` concept and the default model
+`Default_conformal_map_traits<TriangleMesh, K>` for the package.
+
+The concept lists the types and property maps that the discrete-conformal
+algorithms require from any backing data structure.  By templatising the
+algorithms on this concept, the package can run on any CGAL halfedge
+mesh — `Surface_mesh`, `Polyhedron_3`, OpenMesh-adapter, pmp — without
+changes to the algorithm code.
+
+For Phase 8 MVP only the `Surface_mesh` specialisation is provided
+(specialisation 8a.1).  A generic `FaceGraph` specialisation is on the
+roadmap as 8a.2.
+
+\sa `CGAL::Discrete_conformal_map`
+\sa `CGAL::parameters::vertex_curvature_map`
+*/
+
+#ifndef CGAL_CONFORMAL_MAP_TRAITS_H
+#define CGAL_CONFORMAL_MAP_TRAITS_H
+
+#include <CGAL/Surface_mesh.h>
+#include <CGAL/Simple_cartesian.h>
+#include <boost/graph/graph_traits.hpp>
+
+namespace CGAL {
+
+// ════════════════════════════════════════════════════════════════════════════
+// \cgalConcept
+//
+// \concept ConformalMapTraits
+// \ingroup PkgConformalMapConcepts
+//
+// The concept `ConformalMapTraits` describes the requirements that any
+// Traits model must fulfil for the Discrete_conformal_map package.
+//
+// \cgalHasModelsBegin
+// \cgalHasModels{CGAL::Default_conformal_map_traits<TriangleMesh, K>}
+// \cgalHasModelsEnd
+//
+// \section RequiredTypes Required types
+//
+// | Type | Description |
+// |------|-------------|
+// | `Triangle_mesh` | A model of CGAL `FaceGraph` + `HalfedgeGraph`. |
+// | `Kernel`        | A CGAL kernel; defaults to `Simple_cartesian<double>`. |
+// | `FT`            | Field type used internally (typically `double`). |
+// | `Vertex_descriptor`   | `boost::graph_traits<Triangle_mesh>::vertex_descriptor`. |
+// | `Halfedge_descriptor` | analogously. |
+// | `Edge_descriptor`     | analogously. |
+// | `Face_descriptor`     | analogously. |
+//
+// \section RequiredProperties Required property-map accessors
+//
+// The Traits class is responsible for *locating* the property maps that
+// the algorithm reads from and writes to.  The semantics follow the
+// project conventions (see `doc/api/contracts.md` for the full table):
+//
+// | Property             | Key                     | Value | Access  | Used by |
+// |----------------------|-------------------------|-------|---------|---------|
+// | `vertex_points(m)`   | `Vertex_descriptor`     | `Point_3` | Read | input geometry |
+// | `theta_map(m)`       | `Vertex_descriptor`     | `FT`  | RW      | target cone angle Θᵥ |
+// | `vertex_index_map(m)`| `Vertex_descriptor`     | `int` | RW      | DOF index (−1 = pinned) |
+// | `lambda0_map(m)`     | `Edge_descriptor`       | `FT`  | RW      | base log-length λ°ᵢⱼ |
+//
+// Each accessor is a `static` member that returns the map; it must be
+// idempotent (calling twice yields the same map by name lookup).
+// ════════════════════════════════════════════════════════════════════════════
+
+
+// ════════════════════════════════════════════════════════════════════════════
+// Default_conformal_map_traits  —  primary template (undefined)
+// ════════════════════════════════════════════════════════════════════════════
+//
+/*!
+\ingroup PkgConformalMapConcepts
+\brief Primary `ConformalMapTraits` template — undefined, must be
+specialised per mesh type.  The MVP only ships the `Surface_mesh`
+specialisation below; further mesh types (Polyhedron_3, OpenMesh,
+pmp) are deferred to Phase 8a.2.
+*/
+template <typename TriangleMesh,
+          typename Kernel_ = CGAL::Simple_cartesian<double>>
+struct Default_conformal_map_traits;
+
+
+// ════════════════════════════════════════════════════════════════════════════
+// Specialisation: CGAL::Surface_mesh<K::Point_3>
+// ════════════════════════════════════════════════════════════════════════════
+
+/*!
+\ingroup PkgConformalMapRef
+
+Default traits for `CGAL::Surface_mesh`.  Wraps the property maps that
+the existing implementation (`code/include/euclidean_functional.hpp`)
+attaches to a Surface_mesh under the `"ev:idx"`, `"ev:theta"`,
+`"ee:lam0"` etc. names.
+
+This specialisation is the only one available in Phase 8 MVP.  It is
+selected automatically when `TriangleMesh = CGAL::Surface_mesh<...>`.
+
+\tparam K  Any CGAL kernel.  Defaults to `Simple_cartesian<double>`,
+           which is what `conformal_mesh.hpp` uses today.
+*/
+template <typename K>
+struct Default_conformal_map_traits<CGAL::Surface_mesh<typename K::Point_3>, K>
+{
+    /// The CGAL kernel parameter; defaults to `Simple_cartesian<double>`.
+    using Kernel        = K;
+    /// Field type used for all scalar conformal-map data (lengths, λ, Θ, …).
+    using FT            = typename K::FT;
+    /// 3-D point type used for vertex coordinates.
+    using Point_3       = typename K::Point_3;
+    /// The triangle-mesh type this specialisation targets.
+    using Triangle_mesh = CGAL::Surface_mesh<Point_3>;
+
+    /// Boost-graph vertex descriptor for `Triangle_mesh`.
+    using Vertex_descriptor   = typename boost::graph_traits<Triangle_mesh>::vertex_descriptor;
+    /// Boost-graph half-edge descriptor for `Triangle_mesh`.
+    using Halfedge_descriptor = typename boost::graph_traits<Triangle_mesh>::halfedge_descriptor;
+    /// Boost-graph edge descriptor for `Triangle_mesh`.
+    using Edge_descriptor     = typename boost::graph_traits<Triangle_mesh>::edge_descriptor;
+    /// Boost-graph face descriptor for `Triangle_mesh`.
+    using Face_descriptor     = typename boost::graph_traits<Triangle_mesh>::face_descriptor;
+
+    // Property-map types — match the names used by setup_euclidean_maps().
+
+    /// Property map vertex → `Point_3` (the mesh's geometric embedding).
+    using Vertex_point_map      = typename Triangle_mesh::template Property_map<Vertex_descriptor, Point_3>;
+    /// Property map vertex → target cone angle Θᵥ in radians (legacy name `ev:theta`).
+    using Theta_pmap            = typename Triangle_mesh::template Property_map<Vertex_descriptor, FT>;
+    /// Property map vertex → contiguous integer index (legacy name `ev:idx`).
+    using Vertex_index_pmap     = typename Triangle_mesh::template Property_map<Vertex_descriptor, int>;
+    /// Property map edge → log of original edge length λ⁰ (legacy name `ee:lam0`).
+    using Lambda0_pmap          = typename Triangle_mesh::template Property_map<Edge_descriptor, FT>;
+
+    // ─── Property-map accessors ───────────────────────────────────────────
+    //
+    // Each accessor returns a property map under its canonical legacy name.
+    // If no such map exists yet it is created with sensible defaults — so
+    // calling either `setup_euclidean_maps(m)` first or the accessor first
+    // is equivalent.
+
+    /// Return the built-in vertex-point map of `m` (the geometric embedding).
+    static Vertex_point_map vertex_points(Triangle_mesh& m) {
+        return m.points();
+    }
+
+    /// Return (or create with default 2π) the target-angle property map.
+    static Theta_pmap theta_map(Triangle_mesh& m) {
+        auto [pm, created] = m.template add_property_map<Vertex_descriptor, FT>(
+            "ev:theta", FT(2.0 * 3.141592653589793238));
+        (void)created;
+        return pm;
+    }
+
+    /// Return (or create with default −1) the vertex-index property map.
+    static Vertex_index_pmap vertex_index_map(Triangle_mesh& m) {
+        auto [pm, created] = m.template add_property_map<Vertex_descriptor, int>(
+            "ev:idx", -1);
+        (void)created;
+        return pm;
+    }
+
+    /// Return (or create with default 0) the λ⁰ (initial log-length) property map.
+    static Lambda0_pmap lambda0_map(Triangle_mesh& m) {
+        auto [pm, created] = m.template add_property_map<Edge_descriptor, FT>(
+            "ee:lam0", FT(0));
+        (void)created;
+        return pm;
+    }
+};
+
+}  // namespace CGAL
+
+#endif  // CGAL_CONFORMAL_MAP_TRAITS_H

code/include/CGAL/Discrete_circle_packing.h

@@ -0,0 +1,235 @@
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+//
+// Package: conformallab++ / Discrete_conformal_map (Phase 8b-Lite, 2026-05-21)
+
+/*!
+\file CGAL/Discrete_circle_packing.h
+\ingroup PkgConformalMapRef
+
+User-facing entry for the **face-based** circle-packing functional of
+Bobenko-Pinkall-Springborn 2010.  See `cp_euclidean_functional.hpp`
+for the underlying algorithm and `doc/architecture/phase-9a-validation.md`
+for the line-by-line mapping to the Java original
+`CPEuclideanFunctional.java`.
+
+This functional has a fundamentally different DOF structure to the
+classical Euclidean / Spherical / HyperIdeal modes — one log-radius
+`ρ_f` per **face** rather than one log-scale `u_v` per vertex.  We
+therefore expose it via a dedicated header with its own default-trait
+class (Strategy C of the Phase 8b architecture audit).
+*/
+
+#ifndef CGAL_DISCRETE_CIRCLE_PACKING_H
+#define CGAL_DISCRETE_CIRCLE_PACKING_H
+
+#include <CGAL/Conformal_map/internal/parameters.h>
+#include <CGAL/Kernel_traits.h>
+#include <CGAL/Named_function_parameters.h>
+#include <CGAL/boost/graph/named_params_helper.h>
+#include <CGAL/Surface_mesh.h>
+#include <CGAL/Simple_cartesian.h>
+#include <boost/graph/graph_traits.hpp>
+
+#include "../cp_euclidean_functional.hpp"
+#include "../newton_solver.hpp"
+
+#include <stdexcept>
+
+namespace CGAL {
+
+// ── Default traits for CP-Euclidean ───────────────────────────────────────────
+
+/*!
+\ingroup PkgConformalMapConcepts
+\brief Traits class for `discrete_circle_packing_euclidean()` —
+declares the kernel, mesh and property-map types used by the
+BPS-2010 face-based circle-packing functional.
+
+Primary template; specialise it for non-`Surface_mesh` triangle meshes.
+*/
+template <typename TriangleMesh,
+          typename Kernel_ = CGAL::Simple_cartesian<double>>
+struct Default_cp_euclidean_traits;
+
+/*!
+\ingroup PkgConformalMapConcepts
+\brief Specialisation for `CGAL::Surface_mesh<P>`; the only one shipped
+in Phase 8b-Lite.
+*/
+template <typename K>
+struct Default_cp_euclidean_traits<CGAL::Surface_mesh<typename K::Point_3>, K>
+{
+    /// CGAL kernel parameter (defaults to `Simple_cartesian<double>`).
+    using Kernel        = K;
+    /// Scalar field type used for all CP-Euclidean DOFs (`ρ_f`, `θ_e`, `φ_f`).
+    using FT            = typename K::FT;
+    /// 3-D point type (vertex coordinates).
+    using Point_3       = typename K::Point_3;
+    /// Triangle-mesh type this specialisation targets.
+    using Triangle_mesh = CGAL::Surface_mesh<Point_3>;
+
+    /// Boost-graph vertex descriptor for `Triangle_mesh`.
+    using Vertex_descriptor   = typename boost::graph_traits<Triangle_mesh>::vertex_descriptor;
+    /// Boost-graph half-edge descriptor for `Triangle_mesh`.
+    using Halfedge_descriptor = typename boost::graph_traits<Triangle_mesh>::halfedge_descriptor;
+    /// Boost-graph edge descriptor for `Triangle_mesh`.
+    using Edge_descriptor     = typename boost::graph_traits<Triangle_mesh>::edge_descriptor;
+    /// Boost-graph face descriptor for `Triangle_mesh`.
+    using Face_descriptor     = typename boost::graph_traits<Triangle_mesh>::face_descriptor;
+
+    // CP-Euclidean property maps — note the *face* DOF index map.
+
+    /// Property map face → contiguous integer DOF index (legacy `cf:idx`).
+    using Face_index_pmap = typename Triangle_mesh::template Property_map<Face_descriptor,   int>;
+    /// Property map edge → intersection angle θₑ (legacy `ce:theta`).
+    using Theta_e_pmap    = typename Triangle_mesh::template Property_map<Edge_descriptor,   FT>;
+    /// Property map face → target angle sum φ_f (legacy `cf:phi`).
+    using Phi_f_pmap      = typename Triangle_mesh::template Property_map<Face_descriptor,   FT>;
+};
+
+// ── Result type ───────────────────────────────────────────────────────────────
+
+/*!
+\ingroup PkgConformalMapRef
+
+Result of `discrete_circle_packing_euclidean`.  Carries face DOFs
+`ρ_f = log R_f` rather than the vertex DOFs of the classical modes.
+*/
+template <typename FT = double>
+struct Circle_packing_result
+{
+    /// Face DOFs `ρ_f = log R_f` (length = num_faces(mesh); pinned face = 0).
+    std::vector<FT> rho_per_face;
+
+    /// Newton iterations actually performed (≤ `max_iterations`).
+    int  iterations    = 0;
+    /// Final infinity-norm of the gradient (Newton stopping criterion).
+    FT   gradient_norm = FT(0);
+    /// `true` iff `gradient_norm < gradient_tolerance` at exit.
+    bool converged     = false;
+};
+
+// ── Entry function ────────────────────────────────────────────────────────────
+
+/*!
+\ingroup PkgConformalMapRef
+
+Compute the BPS-2010 face-based circle-packing of `mesh`.
+
+\tparam TriangleMesh  A `CGAL::Surface_mesh<P>`.
+\tparam NamedParameters  Optional CGAL named-parameter pack.
+
+\param mesh  Input triangle mesh.
+\param np    Named parameters (subset of those documented on
+             `discrete_conformal_map_euclidean`; the curvature-map
+             parameter `vertex_curvature_map` is **not** used in this
+             face-based mode — instead the per-face target angle sum
+             `φ_f` and per-edge intersection angle `θ_e` are set via
+             the property maps on `mesh` before this call, or left at
+             their defaults `φ_f = 2π`, `θ_e = π/2`).
+
+\returns A `Circle_packing_result<FT>` with `ρ_f` per face.
+
+\pre `mesh` is a triangle mesh.
+\pre `φ_f` and `θ_e` satisfy the BPS-2010 admissibility conditions
+     (Σ_f φ_f = 2π·χ + Σ_e (π − θ_e), see paper §6).
+*/
+template <typename TriangleMesh,
+          typename CGAL_NP_TEMPLATE_PARAMETERS>
+auto discrete_circle_packing_euclidean(
+    TriangleMesh& mesh,
+    const CGAL_NP_CLASS& np = parameters::default_values())
+{
+    using Point_type     = typename TriangleMesh::Point;
+    using Default_kernel = typename CGAL::Kernel_traits<Point_type>::Kernel;
+    using Default_traits = Default_cp_euclidean_traits<TriangleMesh, Default_kernel>;
+    using Traits         = typename internal_np::Lookup_named_param_def<
+                              internal_np::geom_traits_t,
+                              CGAL_NP_CLASS,
+                              Default_traits>::type;
+    using FT             = typename Traits::FT;
+
+    Circle_packing_result<FT> result;
+
+    auto maps = ::conformallab::setup_cp_euclidean_maps(mesh);
+
+    // Pin first face by default; `fixed_vertex_map` is reused here as the
+    // "fixed face" override hook (the parameter tag is generic enough).
+    // For a richer API, a dedicated `fixed_face_map` tag could be added.
+    auto it = mesh.faces().begin();
+    if (it == mesh.faces().end()) {
+        return result;   // empty mesh; trivial
+    }
+    const int n = ::conformallab::assign_cp_euclidean_face_dof_indices(mesh, maps, *it);
+
+    const FT  tol      = parameters::choose_parameter(
+        parameters::get_parameter(np, Conformal_map::internal_np::gradient_tolerance),
+        FT(1e-10));
+    const int max_iter = parameters::choose_parameter(
+        parameters::get_parameter(np, Conformal_map::internal_np::max_iterations),
+        200);
+
+    // Natural-phi default: shift φ_f so the gradient at ρ = 0 is zero.
+    std::vector<double> x0(static_cast<std::size_t>(n), 0.0);
+    auto G0 = ::conformallab::cp_euclidean_gradient(mesh, x0, maps);
+    for (auto f : mesh.faces()) {
+        int i = maps.f_idx[f];
+        if (i >= 0) maps.phi_f[f] -= G0[static_cast<std::size_t>(i)];
+    }
+
+    auto nr = ::conformallab::newton_cp_euclidean(mesh, x0, maps, tol, max_iter);
+
+    result.rho_per_face.assign(num_faces(mesh), FT(0));
+    for (auto f : mesh.faces()) {
+        int j = maps.f_idx[f];
+        if (j >= 0) result.rho_per_face[f.idx()] = nr.x[static_cast<std::size_t>(j)];
+    }
+    result.iterations    = nr.iterations;
+    result.gradient_norm = nr.grad_inf_norm;
+    result.converged     = nr.converged;
+
+    // ── output_uv_map (Phase 8b-Lite extension) ────────────────────────────
+    //
+    // The CP-Euclidean functional carries one DOF per *face* (the log of the
+    // face-circle radius `ρ_f = log R_f`), not per vertex.  A faithful
+    // layout therefore produces a circle packing in ℝ² — each face f is
+    // mapped to a circle of radius `R_f` at some centre `c_f`, with
+    // adjacent circles meeting at the prescribed intersection angle `θ_e`.
+    // That is a per-face output, not the per-vertex Point_2 that
+    // `output_uv_map` is typed for.
+    //
+    // For Phase 8b-Lite we deliberately don't fake it.  If the caller
+    // supplies `output_uv_map(pmap)` we throw `std::runtime_error` with a
+    // clear pointer to Phase 9c (BPS-2010 §6 face-based circle-packing
+    // layout, ~150 lines, on the porting roadmap).  Failing loudly is
+    // better than silently writing zeros.
+    //
+    // Users who want a UV-like coordinate today can:
+    //   1. Solve a Euclidean DCE on the same mesh (vertex DOFs),
+    //   2. Use `discrete_inversive_distance_map(... output_uv_map(pmap))`,
+    //   3. Or compute face-centre positions by hand from `result.rho_per_face`
+    //      + the per-edge `θ_e` values, plus a priority-BFS of their own.
+    {
+        auto uv_param = parameters::get_parameter(
+            np, Conformal_map::internal_np::output_uv_map);
+        constexpr bool has_uv = !std::is_same_v<
+            decltype(uv_param), internal_np::Param_not_found>;
+        if constexpr (has_uv) {
+            throw std::runtime_error(
+                "CGAL::discrete_circle_packing_euclidean: the "
+                "`output_uv_map(...)` named parameter is not yet supported "
+                "for face-based CP-Euclidean.  The faithful output is a "
+                "circle packing in the plane (per-face), not per-vertex "
+                "UVs.  Tracked as Phase 9c; "
+                "see doc/architecture/locked-vs-flexible.md and "
+                "doc/tutorials/add-output-uv-map.md.");
+        }
+    }
+
+    return result;
+}
+
+}  // namespace CGAL
+
+#endif  // CGAL_DISCRETE_CIRCLE_PACKING_H

code/include/CGAL/Discrete_conformal_map.h

@@ -0,0 +1,564 @@
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+//
+// Package: conformallab++ / Discrete_conformal_map (Phase 8 MVP, 2026-05-19)
+
+/*!
+\file CGAL/Discrete_conformal_map.h
+\ingroup PkgConformalMapRef
+
+User-facing entry point for the Discrete_conformal_map package.
+
+This header provides a single function — `discrete_conformal_map_euclidean`
+— that computes a Euclidean discrete-conformal flattening of an open or
+closed triangle mesh.  Spherical and hyperbolic variants are scheduled
+for Phase 8b.2 once the Euclidean pattern is validated by Phase 9a
+(Inversive-Distance functional).
+
+\section Example Simplest usage
+
+\code{.cpp}
+#include <CGAL/Simple_cartesian.h>
+#include <CGAL/Surface_mesh.h>
+#include <CGAL/Discrete_conformal_map.h>
+
+using K = CGAL::Simple_cartesian<double>;
+using Mesh = CGAL::Surface_mesh<K::Point_3>;
+
+int main() {
+    Mesh mesh = ...;  // load a triangle mesh
+    auto result = CGAL::discrete_conformal_map_euclidean(mesh);
+    if (!result.converged)
+        return 1;
+    // result.u_per_vertex[v] now holds the conformal scale factor at v.
+}
+\endcode
+
+\section NamedParams Tuning via named parameters
+
+\code{.cpp}
+auto result = CGAL::discrete_conformal_map_euclidean(
+    mesh,
+    CGAL::parameters::gradient_tolerance(1e-12)
+                     .max_iterations(500));
+\endcode
+
+\sa `CGAL::Default_conformal_map_traits`
+\sa `CGAL::parameters::vertex_curvature_map`
+*/
+
+#ifndef CGAL_DISCRETE_CONFORMAL_MAP_H
+#define CGAL_DISCRETE_CONFORMAL_MAP_H
+
+#include <CGAL/Conformal_map_traits.h>
+#include <CGAL/Conformal_map/internal/parameters.h>
+#include <CGAL/Kernel_traits.h>
+#include <CGAL/Named_function_parameters.h>
+#include <CGAL/boost/graph/named_params_helper.h>
+#include <CGAL/property_map.h>
+
+// Existing implementation headers (Layer 1 — unchanged).
+#include "../euclidean_functional.hpp"
+#include "../layout.hpp"
+#include "../spherical_functional.hpp"
+#include "../hyper_ideal_functional.hpp"
+#include "../gauss_bonnet.hpp"
+#include "../newton_solver.hpp"
+
+#include <vector>
+#include <unordered_map>
+
+namespace CGAL {
+
+// ════════════════════════════════════════════════════════════════════════════
+// Result type
+// ════════════════════════════════════════════════════════════════════════════
+
+/*!
+\ingroup PkgConformalMapRef
+
+Result of `discrete_conformal_map_euclidean`.  Carries the converged
+scale factors `u_v`, Newton diagnostics, and the convergence flag.
+*/
+template <typename FT = double>
+struct Conformal_map_result
+{
+    /// Conformal scale factor `u_v` per vertex (indexed by raw vertex index).
+    /// Length: `num_vertices(mesh)`.
+    std::vector<FT> u_per_vertex;
+
+    /// Number of Newton iterations performed.
+    int iterations = 0;
+
+    /// `‖G(u*)‖∞` at termination.
+    FT gradient_norm = FT(0);
+
+    /// `true` iff `gradient_norm < gradient_tolerance`.
+    bool converged = false;
+
+    /// `true` iff the linear solver used the SparseQR fallback at any
+    /// Newton step (gauge mode on closed mesh without pinned vertex).
+    bool sparse_qr_fallback_used = false;
+};
+
+
+// ════════════════════════════════════════════════════════════════════════════
+// discrete_conformal_map_euclidean  —  user-facing entry
+// ════════════════════════════════════════════════════════════════════════════
+
+/*!
+\ingroup PkgConformalMapRef
+
+Compute the Euclidean discrete-conformal map of `mesh`.
+
+This is the user-facing entry of Phase 8 MVP.  Internally it delegates
+to the existing implementation in `code/include/euclidean_functional.hpp`
+and `code/include/newton_solver.hpp` (Phase 1–7), so the algorithmic
+behaviour is identical to the legacy API; this function only changes
+the public façade.
+
+\tparam TriangleMesh  A `CGAL::Surface_mesh<P>` for some point type `P`.
+                      Other `FaceGraph` models are planned for Phase 8a.2.
+\tparam NamedParameters  Optional CGAL named-parameter pack.
+
+\param mesh  The input mesh (modified in place: property maps are attached).
+\param np    Named parameters:
+  \cgalParamNBegin{vertex_curvature_map}
+    \cgalParamDescription{Property map `vertex → FT` of target cone angles Θᵥ.}
+    \cgalParamDefault{2π at interior vertices, π at boundary vertices.}
+  \cgalParamNEnd
+  \cgalParamNBegin{gradient_tolerance}
+    \cgalParamDescription{Newton stops when `‖G(u)‖∞ < tol`.}
+    \cgalParamDefault{`1e-10`}
+  \cgalParamNEnd
+  \cgalParamNBegin{max_iterations}
+    \cgalParamDescription{Hard limit on Newton steps.}
+    \cgalParamDefault{`200`}
+  \cgalParamNEnd
+  \cgalParamNBegin{fixed_vertex_map}
+    \cgalParamDescription{Property map `vertex → bool`; `true` ⇒ pinned.}
+    \cgalParamDefault{The first vertex in `mesh.vertices()` is pinned.}
+  \cgalParamNEnd
+
+\returns A `Conformal_map_result<FT>` carrying `u_v` and Newton diagnostics.
+
+\pre `mesh` is a triangle mesh.
+\pre `mesh` satisfies the Gauss–Bonnet relation
+     `Σ(2π − Θᵥ) = 2π·χ(mesh)` for the chosen target curvature map.
+*/
+template <typename TriangleMesh,
+          typename CGAL_NP_TEMPLATE_PARAMETERS>
+auto discrete_conformal_map_euclidean(
+    TriangleMesh& mesh,
+    const CGAL_NP_CLASS& np = parameters::default_values())
+{
+    // ── Type plumbing ──────────────────────────────────────────────────────
+    //
+    // Deduce the kernel from the mesh's Point_3 type rather than hard-coding
+    // Simple_cartesian<double>.  This lets the wrapper work with any
+    // Surface_mesh<P> whose P is a CGAL kernel point.  The user can override
+    // the entire traits class via the `geom_traits(...)` named parameter
+    // (Phase 8b.2 extension; default below covers the common case).
+    using Point_type      = typename TriangleMesh::Point;
+    using Default_kernel  = typename CGAL::Kernel_traits<Point_type>::Kernel;
+    using Default_traits  = Default_conformal_map_traits<TriangleMesh, Default_kernel>;
+    using Traits          = typename internal_np::Lookup_named_param_def<
+                                internal_np::geom_traits_t,
+                                CGAL_NP_CLASS,
+                                Default_traits>::type;
+    using FT              = typename Traits::FT;
+
+    Conformal_map_result<FT> result;
+
+    // ── 1. Set up property maps (legacy layer) ─────────────────────────────
+    auto maps = ::conformallab::setup_euclidean_maps(mesh);
+    ::conformallab::compute_euclidean_lambda0_from_mesh(mesh, maps);
+
+    // ── 2. Target curvature: user-supplied or "natural-theta" default ─────
+    auto theta_param = parameters::get_parameter(
+        np, Conformal_map::internal_np::vertex_curvature_map);
+    constexpr bool has_theta = !std::is_same_v<
+        decltype(theta_param), internal_np::Param_not_found>;
+    if constexpr (has_theta) {
+        // User-provided Θ: copy into the property map and verify Gauss–Bonnet.
+        // Throws std::runtime_error if the user-supplied Θ violates GB.
+        for (auto v : mesh.vertices())
+            maps.theta_v[v] = get(theta_param, v);
+        ::conformallab::check_gauss_bonnet(mesh, maps);
+    }
+    // If no Θ is supplied, the default behaviour is "natural-theta": set Θ
+    // such that x = 0 is the natural equilibrium (the actual angle sums at
+    // x = 0 become the targets).  This matches the convention of the
+    // existing test suite and guarantees that the default invocation
+    // converges immediately for any well-formed triangle mesh.
+    // The actual Θ adjustment is done after DOF assignment (step 5b below).
+
+    // ── 3. Pin vertices: user map, or the first vertex by default ──────────
+    //
+    // The legacy v_idx property map has -1 as default (= pinned).  We must
+    // first mark every vertex as "free" (any non-negative sentinel), then
+    // pin the requested ones, then assign sequential DOF indices.
+    constexpr int FREE = 0;
+    for (auto v : mesh.vertices())
+        maps.v_idx[v] = FREE;
+
+    auto pin_param = parameters::get_parameter(
+        np, Conformal_map::internal_np::fixed_vertex_map);
+    constexpr bool has_pin = !std::is_same_v<
+        decltype(pin_param), internal_np::Param_not_found>;
+
+    bool any_pinned = false;
+    if constexpr (has_pin) {
+        for (auto v : mesh.vertices())
+            if (get(pin_param, v)) {
+                maps.v_idx[v] = -1;
+                any_pinned = true;
+            }
+    }
+    if (!any_pinned) {
+        auto it = mesh.vertices().begin();
+        if (it != mesh.vertices().end()) {
+            maps.v_idx[*it] = -1;
+            any_pinned = true;
+        }
+    }
+
+    // ── 4. Assign DOF indices 0..n−1 to non-pinned vertices ─────────────────
+    int idx = 0;
+    for (auto v : mesh.vertices())
+        if (maps.v_idx[v] != -1)
+            maps.v_idx[v] = idx++;
+
+    // ── 5. Read tolerances ─────────────────────────────────────────────────
+    const FT  tol      = parameters::choose_parameter(
+        parameters::get_parameter(np, Conformal_map::internal_np::gradient_tolerance),
+        FT(1e-10));
+    const int max_iter = parameters::choose_parameter(
+        parameters::get_parameter(np, Conformal_map::internal_np::max_iterations),
+        200);
+
+    // ── 5b. Natural-theta default: shift Θ so that x = 0 is the equilibrium
+    //
+    // Only applied when the user did NOT supply a vertex_curvature_map.
+    // The trick: evaluate G at x = 0, then subtract G_v from Θ_v.  After
+    // this shift the new G(0) is identically zero, so Newton starts at the
+    // optimum and immediately reports "converged".  This matches the
+    // contract of the existing test suite ("natural-theta" pattern).
+    std::vector<double> x0(static_cast<std::size_t>(idx), 0.0);
+    if constexpr (!has_theta) {
+        auto G0 = ::conformallab::euclidean_gradient(mesh, x0, maps);
+        for (auto v : mesh.vertices()) {
+            const int j = maps.v_idx[v];
+            if (j >= 0)
+                maps.theta_v[v] -= G0[static_cast<std::size_t>(j)];
+        }
+    }
+
+    // ── 6. Newton on x_0 = 0 ───────────────────────────────────────────────
+    auto nr = ::conformallab::newton_euclidean(mesh, x0, maps, tol, max_iter);
+
+    // ── 7. Pack result: u_v for every vertex, including pinned (u=0) ──────
+    result.u_per_vertex.assign(num_vertices(mesh), FT(0));
+    for (auto v : mesh.vertices()) {
+        const int j = maps.v_idx[v];
+        if (j >= 0)
+            result.u_per_vertex[v.idx()] = nr.x[static_cast<std::size_t>(j)];
+        // else: pinned ⇒ u_v stays 0
+    }
+    result.iterations    = nr.iterations;
+    result.gradient_norm = nr.grad_inf_norm;
+    result.converged     = nr.converged;
+
+    // ── 8. Optional layout step (Phase 8b-Lite extension) ──────────────────
+    //
+    // If the caller supplied `output_uv_map(pmap)`, run the priority-BFS
+    // trilateration on the converged x and write per-vertex `Point_2`
+    // coordinates into `pmap`.  Optional `normalise_layout(true)` applies
+    // the canonical PCA centroid + major-axis normalisation.
+    auto uv_param = parameters::get_parameter(
+        np, Conformal_map::internal_np::output_uv_map);
+    constexpr bool has_uv = !std::is_same_v<
+        decltype(uv_param), internal_np::Param_not_found>;
+    if constexpr (has_uv) {
+        if (nr.converged) {
+            auto layout = ::conformallab::euclidean_layout(mesh, nr.x, maps);
+
+            const bool do_norm = parameters::choose_parameter(
+                parameters::get_parameter(np, Conformal_map::internal_np::normalise_layout),
+                false);
+            if (do_norm) ::conformallab::normalise_euclidean(layout);
+
+            for (auto v : mesh.vertices()) {
+                const auto& uv = layout.uv[v.idx()];
+                put(uv_param, v,
+                    typename Traits::Kernel::Point_2(uv.x(), uv.y()));
+            }
+        }
+    }
+
+    return result;
+}
+
+
+// ════════════════════════════════════════════════════════════════════════════
+// discrete_conformal_map_spherical  —  Phase 8b-Lite
+// ════════════════════════════════════════════════════════════════════════════
+
+/*!
+\ingroup PkgConformalMapRef
+
+Compute the spherical discrete-conformal map of a closed genus-0 mesh.
+
+The spherical DCE energy is *concave*, so its Hessian is NSD at the
+optimum and `newton_spherical()` factorises −H internally (handled by
+the legacy implementation; no caller action required).  A gauge vertex
+is pinned automatically to remove the rotational mode.
+
+\tparam TriangleMesh  A `CGAL::Surface_mesh<P>` for some point type `P`.
+\tparam NamedParameters  Optional CGAL named-parameter pack.
+
+\param mesh  The input mesh (modified in place: property maps attached).
+\param np    Same named parameters as `discrete_conformal_map_euclidean`.
+
+\returns A `Conformal_map_result<FT>` carrying `u_v` per vertex and
+         Newton diagnostics.
+
+\pre `mesh` is a closed genus-0 triangle mesh.
+\pre The user-supplied or natural-theta Θ satisfies the spherical
+     Gauss–Bonnet relation `Σ(2π − Θᵥ) = 4π` (sphere).
+*/
+template <typename TriangleMesh,
+          typename CGAL_NP_TEMPLATE_PARAMETERS>
+auto discrete_conformal_map_spherical(
+    TriangleMesh& mesh,
+    const CGAL_NP_CLASS& np = parameters::default_values())
+{
+    using Point_type     = typename TriangleMesh::Point;
+    using Default_kernel = typename CGAL::Kernel_traits<Point_type>::Kernel;
+    using Default_traits = Default_conformal_map_traits<TriangleMesh, Default_kernel>;
+    using Traits         = typename internal_np::Lookup_named_param_def<
+                              internal_np::geom_traits_t,
+                              CGAL_NP_CLASS,
+                              Default_traits>::type;
+    using FT             = typename Traits::FT;
+
+    Conformal_map_result<FT> result;
+
+    auto maps = ::conformallab::setup_spherical_maps(mesh);
+    ::conformallab::compute_lambda0_from_mesh(mesh, maps);
+
+    auto theta_param = parameters::get_parameter(
+        np, Conformal_map::internal_np::vertex_curvature_map);
+    constexpr bool has_theta = !std::is_same_v<
+        decltype(theta_param), internal_np::Param_not_found>;
+    if constexpr (has_theta) {
+        for (auto v : mesh.vertices())
+            maps.theta_v[v] = get(theta_param, v);
+    }
+
+    // Pin one vertex (gauge fix) — user-supplied or first vertex.
+    constexpr int FREE = 0;
+    for (auto v : mesh.vertices()) maps.v_idx[v] = FREE;
+
+    auto pin_param = parameters::get_parameter(
+        np, Conformal_map::internal_np::fixed_vertex_map);
+    constexpr bool has_pin = !std::is_same_v<
+        decltype(pin_param), internal_np::Param_not_found>;
+
+    bool any_pinned = false;
+    if constexpr (has_pin) {
+        for (auto v : mesh.vertices())
+            if (get(pin_param, v)) { maps.v_idx[v] = -1; any_pinned = true; }
+    }
+    if (!any_pinned) {
+        auto it = mesh.vertices().begin();
+        if (it != mesh.vertices().end()) { maps.v_idx[*it] = -1; any_pinned = true; }
+    }
+
+    int idx = 0;
+    for (auto v : mesh.vertices())
+        if (maps.v_idx[v] != -1) maps.v_idx[v] = idx++;
+
+    const FT  tol      = parameters::choose_parameter(
+        parameters::get_parameter(np, Conformal_map::internal_np::gradient_tolerance),
+        FT(1e-10));
+    const int max_iter = parameters::choose_parameter(
+        parameters::get_parameter(np, Conformal_map::internal_np::max_iterations),
+        200);
+
+    // Natural-theta default for the spherical functional.
+    std::vector<double> x0(static_cast<std::size_t>(idx), 0.0);
+    if constexpr (!has_theta) {
+        auto G0 = ::conformallab::spherical_gradient(mesh, x0, maps);
+        for (auto v : mesh.vertices()) {
+            const int j = maps.v_idx[v];
+            if (j >= 0) maps.theta_v[v] -= G0[static_cast<std::size_t>(j)];
+        }
+    }
+
+    auto nr = ::conformallab::newton_spherical(mesh, x0, maps, tol, max_iter);
+
+    result.u_per_vertex.assign(num_vertices(mesh), FT(0));
+    for (auto v : mesh.vertices()) {
+        const int j = maps.v_idx[v];
+        if (j >= 0) result.u_per_vertex[v.idx()] = nr.x[static_cast<std::size_t>(j)];
+    }
+    result.iterations    = nr.iterations;
+    result.gradient_norm = nr.grad_inf_norm;
+    result.converged     = nr.converged;
+
+    // Optional 3-D layout step (point on S²)
+    auto uv_param = parameters::get_parameter(
+        np, Conformal_map::internal_np::output_uv_map);
+    constexpr bool has_uv = !std::is_same_v<
+        decltype(uv_param), internal_np::Param_not_found>;
+    if constexpr (has_uv) {
+        if (nr.converged) {
+            auto layout = ::conformallab::spherical_layout(mesh, nr.x, maps);
+            const bool do_norm = parameters::choose_parameter(
+                parameters::get_parameter(np, Conformal_map::internal_np::normalise_layout),
+                false);
+            if (do_norm) ::conformallab::normalise_spherical(layout);
+            for (auto v : mesh.vertices()) {
+                const auto& p = layout.pos[v.idx()];
+                put(uv_param, v,
+                    typename Traits::Kernel::Point_3(p.x(), p.y(), p.z()));
+            }
+        }
+    }
+
+    return result;
+}
+
+
+// ════════════════════════════════════════════════════════════════════════════
+// discrete_conformal_map_hyper_ideal  —  Phase 8b-Lite
+// ════════════════════════════════════════════════════════════════════════════
+
+/*!
+\ingroup PkgConformalMapRef
+
+Result of `discrete_conformal_map_hyper_ideal`.  Carries both vertex
+DOFs `b_v` and edge DOFs `a_e` (hyper-ideal triangles in H³).
+*/
+template <typename FT = double>
+struct Hyper_ideal_map_result
+{
+    /// Vertex DOFs `b_v` (length = num_vertices(mesh); pinned vertices = 0).
+    std::vector<FT> b_per_vertex;
+    /// Edge DOFs `a_e` (length = num_edges(mesh); pinned edges = 0).
+    std::vector<FT> a_per_edge;
+
+    /// Newton iterations actually performed (≤ `max_iterations`).
+    int  iterations    = 0;
+    /// Final infinity-norm of the gradient (Newton stopping criterion).
+    FT   gradient_norm = FT(0);
+    /// `true` iff `gradient_norm < gradient_tolerance` at exit.
+    bool converged     = false;
+};
+
+/*!
+\ingroup PkgConformalMapRef
+
+Compute the hyper-ideal discrete-conformal map of a triangle mesh
+(Springborn 2020 §4).
+
+\note  Phase 8b-Lite scope: vertex DOFs `b_v` are assigned automatically
+       to all vertices; edge DOFs `a_e` are similarly assigned.  The
+       block-FD Hessian (Phase 9b) is used internally — see
+       `newton_hyper_ideal` for the solver convention.
+*/
+template <typename TriangleMesh,
+          typename CGAL_NP_TEMPLATE_PARAMETERS>
+auto discrete_conformal_map_hyper_ideal(
+    TriangleMesh& mesh,
+    const CGAL_NP_CLASS& np = parameters::default_values())
+{
+    using Point_type     = typename TriangleMesh::Point;
+    using Default_kernel = typename CGAL::Kernel_traits<Point_type>::Kernel;
+    using Default_traits = Default_conformal_map_traits<TriangleMesh, Default_kernel>;
+    using Traits         = typename internal_np::Lookup_named_param_def<
+                              internal_np::geom_traits_t,
+                              CGAL_NP_CLASS,
+                              Default_traits>::type;
+    using FT             = typename Traits::FT;
+
+    Hyper_ideal_map_result<FT> result;
+
+    auto maps = ::conformallab::setup_hyper_ideal_maps(mesh);
+    // Hyper-ideal init does not derive from mesh geometry: the user's
+    // Θ_v and θ_e are the model inputs.  Defaults from setup are
+    // Θ_v = 2π, θ_e = π (orthogonal).
+
+    auto theta_param = parameters::get_parameter(
+        np, Conformal_map::internal_np::vertex_curvature_map);
+    constexpr bool has_theta = !std::is_same_v<
+        decltype(theta_param), internal_np::Param_not_found>;
+    if constexpr (has_theta) {
+        for (auto v : mesh.vertices())
+            maps.theta_v[v] = get(theta_param, v);
+    }
+
+    const int n = ::conformallab::assign_all_dof_indices(mesh, maps);
+
+    const FT  tol      = parameters::choose_parameter(
+        parameters::get_parameter(np, Conformal_map::internal_np::gradient_tolerance),
+        FT(1e-8));
+    const int max_iter = parameters::choose_parameter(
+        parameters::get_parameter(np, Conformal_map::internal_np::max_iterations),
+        200);
+
+    // Initial point: b_v = 1.0 (positive log-scale), a_e = 0.5 (moderate).
+    std::vector<double> x0(static_cast<std::size_t>(n), 0.0);
+    for (auto v : mesh.vertices()) {
+        int i = maps.v_idx[v];
+        if (i >= 0) x0[static_cast<std::size_t>(i)] = 1.0;
+    }
+    for (auto e : mesh.edges()) {
+        int i = maps.e_idx[e];
+        if (i >= 0) x0[static_cast<std::size_t>(i)] = 0.5;
+    }
+
+    auto nr = ::conformallab::newton_hyper_ideal(mesh, x0, maps, tol, max_iter);
+
+    result.b_per_vertex.assign(num_vertices(mesh), FT(0));
+    result.a_per_edge   .assign(num_edges(mesh),    FT(0));
+    for (auto v : mesh.vertices()) {
+        int j = maps.v_idx[v];
+        if (j >= 0) result.b_per_vertex[v.idx()] = nr.x[static_cast<std::size_t>(j)];
+    }
+    for (auto e : mesh.edges()) {
+        int j = maps.e_idx[e];
+        if (j >= 0) result.a_per_edge[e.idx()] = nr.x[static_cast<std::size_t>(j)];
+    }
+    result.iterations    = nr.iterations;
+    result.gradient_norm = nr.grad_inf_norm;
+    result.converged     = nr.converged;
+
+    // Optional Poincaré-disk layout (2-D in the unit disk).
+    auto uv_param = parameters::get_parameter(
+        np, Conformal_map::internal_np::output_uv_map);
+    constexpr bool has_uv = !std::is_same_v<
+        decltype(uv_param), internal_np::Param_not_found>;
+    if constexpr (has_uv) {
+        if (nr.converged) {
+            auto layout = ::conformallab::hyper_ideal_layout(mesh, nr.x, maps);
+            const bool do_norm = parameters::choose_parameter(
+                parameters::get_parameter(np, Conformal_map::internal_np::normalise_layout),
+                false);
+            if (do_norm) ::conformallab::normalise_hyperbolic(layout);
+            for (auto v : mesh.vertices()) {
+                const auto& uv = layout.uv[v.idx()];
+                put(uv_param, v,
+                    typename Traits::Kernel::Point_2(uv.x(), uv.y()));
+            }
+        }
+    }
+
+    (void)n;   // already-counted by maps; silence unused-var warnings if any
+    return result;
+}
+
+}  // namespace CGAL
+
+#endif  // CGAL_DISCRETE_CONFORMAL_MAP_H

code/include/CGAL/Discrete_inversive_distance.h

@@ -0,0 +1,263 @@
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+//
+// Package: conformallab++ / Discrete_conformal_map (Phase 8b-Lite, 2026-05-21)
+
+/*!
+\file CGAL/Discrete_inversive_distance.h
+\ingroup PkgConformalMapRef
+
+User-facing entry for the **vertex-based** inversive-distance circle-
+packing functional of Luo (2004), with the Bowers-Stephenson (2004)
+initialisation.  See `inversive_distance_functional.hpp` for the
+underlying algorithm and `doc/roadmap/research-track.md` (item 9a.2)
+for the research-track classification — this functional has **no Java
+original** (verified empirically), it is from-the-literature research.
+
+DOF structure
+─────────────
+* Per-vertex `u_i = log r_i` (compatible with the classical Euclidean
+  trait).
+* Per-edge constant `I_ij` computed once by Bowers-Stephenson from the
+  input mesh geometry (handled internally by
+  `compute_inversive_distance_init_from_mesh`).
+
+Because the per-edge constant has a different meaning from the
+Euclidean `λ°_e`, this entry has its own default-trait class
+`Default_inversive_distance_traits`.
+*/
+
+#ifndef CGAL_DISCRETE_INVERSIVE_DISTANCE_H
+#define CGAL_DISCRETE_INVERSIVE_DISTANCE_H
+
+#include <CGAL/Conformal_map/internal/parameters.h>
+#include <CGAL/Kernel_traits.h>
+#include <CGAL/Named_function_parameters.h>
+#include <CGAL/boost/graph/named_params_helper.h>
+#include <CGAL/Surface_mesh.h>
+#include <CGAL/Simple_cartesian.h>
+#include <boost/graph/graph_traits.hpp>
+
+#include <CGAL/Discrete_conformal_map.h>   // for Conformal_map_result<FT>
+
+#include "../inversive_distance_functional.hpp"
+#include "../newton_solver.hpp"
+
+namespace CGAL {
+
+// ── Default traits for Inversive-Distance ────────────────────────────────────
+
+/*!
+\ingroup PkgConformalMapConcepts
+\brief Traits class for `discrete_inversive_distance_map()` — declares
+the kernel, mesh and property-map types used by Luo's 2004 vertex-based
+inversive-distance circle packing.
+
+Primary template; specialise it for non-`Surface_mesh` triangle meshes.
+*/
+template <typename TriangleMesh,
+          typename Kernel_ = CGAL::Simple_cartesian<double>>
+struct Default_inversive_distance_traits;
+
+/*!
+\ingroup PkgConformalMapConcepts
+\brief Specialisation for `CGAL::Surface_mesh<P>`; the only one shipped
+in Phase 8b-Lite.
+*/
+template <typename K>
+struct Default_inversive_distance_traits<CGAL::Surface_mesh<typename K::Point_3>, K>
+{
+    /// CGAL kernel parameter (defaults to `Simple_cartesian<double>`).
+    using Kernel        = K;
+    /// Scalar field type used for all inversive-distance DOFs.
+    using FT            = typename K::FT;
+    /// 3-D point type (vertex coordinates).
+    using Point_3       = typename K::Point_3;
+    /// Triangle-mesh type this specialisation targets.
+    using Triangle_mesh = CGAL::Surface_mesh<Point_3>;
+
+    /// Boost-graph vertex descriptor for `Triangle_mesh`.
+    using Vertex_descriptor   = typename boost::graph_traits<Triangle_mesh>::vertex_descriptor;
+    /// Boost-graph edge descriptor for `Triangle_mesh`.
+    using Edge_descriptor     = typename boost::graph_traits<Triangle_mesh>::edge_descriptor;
+
+    // Inversive-distance specific property maps.
+
+    /// Property map vertex → contiguous integer DOF index (legacy `iv:idx`).
+    using Vertex_index_pmap = typename Triangle_mesh::template Property_map<Vertex_descriptor, int>;
+    /// Property map vertex → target cone angle Θᵥ in radians (legacy `iv:theta`).
+    using Theta_v_pmap      = typename Triangle_mesh::template Property_map<Vertex_descriptor, FT>;
+    /// Property map vertex → initial radius r⁰ᵥ (legacy `iv:r0`).
+    using R0_pmap           = typename Triangle_mesh::template Property_map<Vertex_descriptor, FT>;
+    /// Property map edge → inversive distance Iᵢⱼ (legacy `ie:I`).
+    using I_e_pmap          = typename Triangle_mesh::template Property_map<Edge_descriptor,   FT>;
+};
+
+// ── Entry function ────────────────────────────────────────────────────────────
+
+/*!
+\ingroup PkgConformalMapRef
+
+Compute the Luo-2004 vertex-based inversive-distance circle packing of `mesh`.
+
+The per-edge constant `I_ij` is computed once at the start from the input
+3-D geometry via the Bowers-Stephenson identity
+    `I_ij = (ℓ_ij² − r_i² − r_j²) / (2 r_i r_j)`,
+with `r_i^(0) = (1/3) min{ℓ_e : e adj v_i}` as the default initial radii.
+The user can override the initial radii by writing into the `r0`
+property map before calling this function.
+
+\tparam TriangleMesh  A `CGAL::Surface_mesh<P>`.
+\tparam NamedParameters  Optional CGAL named-parameter pack.
+
+\param mesh  Input triangle mesh.
+\param np    Named parameters:
+  - `vertex_curvature_map(pmap)` — per-vertex Θ_v target.
+  - `fixed_vertex_map(pmap)` — pinning override.
+  - `gradient_tolerance(ε)` — Newton stop.
+  - `max_iterations(n)` — Newton iteration cap.
+
+\returns A `Conformal_map_result<FT>` with `u_per_vertex[v] = log r_v`
+         (the converged log-radius at each vertex).
+
+\pre `mesh` is a triangle mesh with positive edge lengths.
+\pre The user-supplied or natural-theta Θ satisfies Gauss–Bonnet.
+
+\note  Convergence is sensitive to the initial point and to extreme
+       `I_ij` values.  For testing purposes the natural-theta default
+       (Θ_v shifted so that u = 0 is the equilibrium) always converges
+       in zero iterations.
+*/
+template <typename TriangleMesh,
+          typename CGAL_NP_TEMPLATE_PARAMETERS>
+auto discrete_inversive_distance_map(
+    TriangleMesh& mesh,
+    const CGAL_NP_CLASS& np = parameters::default_values())
+{
+    using Point_type     = typename TriangleMesh::Point;
+    using Default_kernel = typename CGAL::Kernel_traits<Point_type>::Kernel;
+    using Default_traits = Default_inversive_distance_traits<TriangleMesh, Default_kernel>;
+    using Traits         = typename internal_np::Lookup_named_param_def<
+                              internal_np::geom_traits_t,
+                              CGAL_NP_CLASS,
+                              Default_traits>::type;
+    using FT             = typename Traits::FT;
+
+    Conformal_map_result<FT> result;
+
+    auto maps = ::conformallab::setup_inversive_distance_maps(mesh);
+    ::conformallab::compute_inversive_distance_init_from_mesh(mesh, maps);
+
+    auto theta_param = parameters::get_parameter(
+        np, Conformal_map::internal_np::vertex_curvature_map);
+    constexpr bool has_theta = !std::is_same_v<
+        decltype(theta_param), internal_np::Param_not_found>;
+    if constexpr (has_theta) {
+        for (auto v : mesh.vertices())
+            maps.theta_v[v] = get(theta_param, v);
+    }
+
+    // Pin first vertex by default; user can override with fixed_vertex_map.
+    constexpr int FREE = 0;
+    for (auto v : mesh.vertices()) maps.v_idx[v] = FREE;
+
+    auto pin_param = parameters::get_parameter(
+        np, Conformal_map::internal_np::fixed_vertex_map);
+    constexpr bool has_pin = !std::is_same_v<
+        decltype(pin_param), internal_np::Param_not_found>;
+
+    bool any_pinned = false;
+    if constexpr (has_pin) {
+        for (auto v : mesh.vertices())
+            if (get(pin_param, v)) { maps.v_idx[v] = -1; any_pinned = true; }
+    }
+    if (!any_pinned) {
+        auto it = mesh.vertices().begin();
+        if (it != mesh.vertices().end()) { maps.v_idx[*it] = -1; any_pinned = true; }
+    }
+
+    int idx = 0;
+    for (auto v : mesh.vertices())
+        if (maps.v_idx[v] != -1) maps.v_idx[v] = idx++;
+
+    const FT  tol      = parameters::choose_parameter(
+        parameters::get_parameter(np, Conformal_map::internal_np::gradient_tolerance),
+        FT(1e-10));
+    const int max_iter = parameters::choose_parameter(
+        parameters::get_parameter(np, Conformal_map::internal_np::max_iterations),
+        200);
+
+    // Natural-theta default.
+    std::vector<double> x0(static_cast<std::size_t>(idx), 0.0);
+    if constexpr (!has_theta) {
+        auto G0 = ::conformallab::inversive_distance_gradient(mesh, x0, maps);
+        for (auto v : mesh.vertices()) {
+            const int j = maps.v_idx[v];
+            if (j >= 0) maps.theta_v[v] -= G0[static_cast<std::size_t>(j)];
+        }
+    }
+
+    auto nr = ::conformallab::newton_inversive_distance(mesh, x0, maps, tol, max_iter);
+
+    result.u_per_vertex.assign(num_vertices(mesh), FT(0));
+    for (auto v : mesh.vertices()) {
+        const int j = maps.v_idx[v];
+        if (j >= 0) result.u_per_vertex[v.idx()] = nr.x[static_cast<std::size_t>(j)];
+    }
+    result.iterations    = nr.iterations;
+    result.gradient_norm = nr.grad_inf_norm;
+    result.converged     = nr.converged;
+
+    // ── Optional layout step (Phase 8b-Lite extension) ─────────────────────
+    //
+    // If the caller supplied `output_uv_map(pmap)`, lay out the converged
+    // packing in ℝ² and write per-vertex `Point_2` coordinates into `pmap`.
+    //
+    // Method: the converged Inversive-Distance radii `r_i = exp(u_i)`
+    // together with the fixed per-edge `I_ij` constants determine effective
+    // Euclidean edge lengths via the Bowers-Stephenson identity
+    //   ℓᵢⱼ² = rᵢ² + rⱼ² + 2·Iᵢⱼ·rᵢ·rⱼ
+    // so we can populate a temporary `EuclideanMaps` whose `lambda0` carries
+    // `log(ℓᵢⱼ²)` per edge and then reuse `euclidean_layout(mesh, 0, eucl)`
+    // — the existing priority-BFS trilateration on the resulting triangle
+    // metric.  All vertex/edge DOF indices stay at −1 (pinned), so the empty
+    // DOF vector `0` produces lengths driven purely by `lambda0`.
+    auto uv_param = parameters::get_parameter(
+        np, Conformal_map::internal_np::output_uv_map);
+    constexpr bool has_uv = !std::is_same_v<
+        decltype(uv_param), internal_np::Param_not_found>;
+    if constexpr (has_uv) {
+        if (nr.converged) {
+            auto eucl = ::conformallab::setup_euclidean_maps(mesh);
+            for (auto e : mesh.edges()) {
+                auto h = mesh.halfedge(e);
+                const double u_i = result.u_per_vertex[mesh.source(h).idx()];
+                const double u_j = result.u_per_vertex[mesh.target(h).idx()];
+                const double I   = maps.I_e[e];
+                const double l2  = ::conformallab::id_detail::edge_length_squared(u_i, u_j, I);
+                eucl.lambda0[e]  = (l2 > 0.0) ? std::log(l2) : -30.0;
+            }
+            // Empty DOF vector: every vertex is pinned (idx=-1), so the
+            // layout depends purely on the lambda0 we just computed.
+            std::vector<double> zero;
+            auto layout = ::conformallab::euclidean_layout(mesh, zero, eucl);
+
+            const bool do_norm = parameters::choose_parameter(
+                parameters::get_parameter(np, Conformal_map::internal_np::normalise_layout),
+                false);
+            if (do_norm) ::conformallab::normalise_euclidean(layout);
+
+            for (auto v : mesh.vertices()) {
+                const auto& uv = layout.uv[v.idx()];
+                put(uv_param, v,
+                    typename Traits::Kernel::Point_2(uv.x(), uv.y()));
+            }
+        }
+    }
+
+    return result;
+}
+
+}  // namespace CGAL
+
+#endif  // CGAL_DISCRETE_INVERSIVE_DISTANCE_H

code/include/clausen.hpp

@@ -1,4 +1,7 @@
 #pragma once
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
 
 // Clausen integral, Lobachevsky function, and Im(Li2).
 // Ported from de.varylab.discreteconformal.functional.Clausen (Java).
@@ -128,9 +131,9 @@ inline int ncl5pi6() noexcept {
 
 } // namespace detail
 
-// Clausen's integral Cl2(x) = -integral_0^x log|2 sin(t/2)| dt.
-// High-precision Chebyshev implementation.
-// Corresponds to Java Clausen.clausen2().
+/// Clausen's integral `Cl₂(x) = −∫₀ˣ log|2 sin(t/2)| dt`,
+/// computed via a high-precision Chebyshev expansion.
+/// Same as Java `Clausen.clausen2()`.
 inline double clausen2(double x) noexcept {
     using namespace detail;
     constexpr double pi  = 3.14159265358979323846264338328;
@@ -154,8 +157,8 @@ inline double clausen2(double x) noexcept {
     return rh ? -f : f;
 }
 
-// Milnor's Lobachevsky function Л(x) = Cl2(2x)/2.
-// Corresponds to Java Clausen.Л().
+/// Milnor's Lobachevsky function `Л(x) = Cl₂(2x) / 2`.
+/// Same as Java `Clausen.Л()`.
 inline double Lobachevsky(double x) noexcept {
     constexpr double pi = 3.14159265358979323846264338328;
     x = std::fmod(x, pi);
@@ -163,8 +166,8 @@ inline double Lobachevsky(double x) noexcept {
     return clausen2(2.0 * x) / 2.0;
 }
 
-// Imaginary part of the dilogarithm Im(Li2(z)).
-// Corresponds to Java Clausen.ImLi2().
+/// Imaginary part of the dilogarithm `Im(Li₂(z))` for complex `z`.
+/// Same as Java `Clausen.ImLi2()`.
 inline double ImLi2(std::complex<double> z) noexcept {
     auto a = std::log(1.0 - std::conj(z));  // log(1 - conj(z))
     auto b = std::log(1.0 - z);             // log(1 - z)

code/include/conformal_mesh.hpp

@@ -0,0 +1,110 @@
+#pragma once
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
+// conformal_mesh.hpp
+//
+// Central mesh type for the discrete conformal mapping algorithms.
+// Replaces the Java CoHDS (de.varylab.discreteconformal.heds.CoHDS)
+// and its associated vertex/edge/face types (CoVertex, CoEdge, CoFace).
+//
+// Design
+// ------
+//   Java                         │  C++ (this file)
+//   ─────────────────────────────┼──────────────────────────────────────────
+//   CoHDS                        │  ConformalMesh   (CGAL::Surface_mesh)
+//   CoVertex / CoEdge / CoFace   │  Vertex_index / Edge_index / Face_index
+//   HyperIdealRadiusAdapter      │  property_map<Vertex_index, double>
+//   HalfedgeInterface adapters   │  named property maps ("v:lambda", …)
+//
+// Property-map naming convention
+// ───────────────────────────────
+//   "v:lambda"    per-vertex log scale factor  (conformal variable u_i)
+//   "v:theta"     per-vertex target cone angle
+//   "v:idx"       per-vertex solver DOF index  (-1 = pinned / boundary)
+//   "e:alpha"     per-edge intersection angle  (α_ij, hyperbolic geometry)
+//   "f:type"      per-face geometry type       (0=Euclidean, 1=Hyperbolic, 2=Spherical)
+//
+// All property maps are optional; add only what a given algorithm needs.
+//
+// Note on descriptor types (CGAL 6.x)
+// ────────────────────────────────────
+// CGAL::Surface_mesh exposes its index types as nested types:
+//   Surface_mesh::Vertex_index, ::Halfedge_index, ::Edge_index, ::Face_index
+// The BGL graph_traits aliases expose the same types as vertex_descriptor etc.,
+// but those live in boost::graph_traits<Surface_mesh>, not in Surface_mesh itself.
+// We use the Surface_mesh member names throughout for clarity.
+
+#include <CGAL/Simple_cartesian.h>
+#include <CGAL/Surface_mesh.h>
+#include <string>
+
+namespace conformallab {
+
+// ── Kernel ──────────────────────────────────────────────────────────────────
+// Simple double-precision Cartesian.  Conformal mapping algorithms never
+// need exact arithmetic — they operate on floating-point lengths and angles.
+
+/// CGAL kernel used by all conformallab algorithms (double precision).
+using Kernel = CGAL::Simple_cartesian<double>;
+/// 3-D point type (vertex coordinates).
+using Point3 = Kernel::Point_3;
+/// 2-D point type (UV-domain layout coordinates).
+using Point2 = Kernel::Point_2;
+
+// ── Mesh type ────────────────────────────────────────────────────────────────
+/// Triangle mesh carrying all conformal-map data as property maps.
+using ConformalMesh = CGAL::Surface_mesh<Point3>;
+
+// ── Index/descriptor aliases (CGAL 6.x naming) ───────────────────────────────
+/// Vertex descriptor of `ConformalMesh`.
+using Vertex_index   = ConformalMesh::Vertex_index;
+/// Half-edge descriptor of `ConformalMesh`.
+using Halfedge_index = ConformalMesh::Halfedge_index;
+/// Edge descriptor of `ConformalMesh`.
+using Edge_index     = ConformalMesh::Edge_index;
+/// Face descriptor of `ConformalMesh`.
+using Face_index     = ConformalMesh::Face_index;
+
+// ── Geometry type constant (replaces Java CoFace.type enum) ─────────────────
+/// Discrete geometry type that a face/mesh is interpreted in.
+/// Replaces the original Java `CoFace.type` enum.
+enum class GeometryType : int {
+    Euclidean  = 0, ///< Flat metric (ℝ²).
+    Hyperbolic = 1, ///< Hyperbolic metric (ℍ²).
+    Spherical  = 2  ///< Spherical metric (S²).
+};
+
+// ── Standard property-map bundles ────────────────────────────────────────────
+
+/// Register and return the vertex-side property maps used by all five
+/// DCE functionals: `v:lambda` (log conformal factor), `v:theta` (target
+/// cone angle), `v:idx` (contiguous integer index).
+inline auto add_vertex_properties(ConformalMesh& mesh)
+{
+    auto [lambda, ok1] = mesh.add_property_map<Vertex_index, double>("v:lambda", 0.0);
+    auto [theta,  ok2] = mesh.add_property_map<Vertex_index, double>("v:theta",  0.0);
+    auto [idx,    ok3] = mesh.add_property_map<Vertex_index, int>   ("v:idx",   -1);
+    (void)ok1; (void)ok2; (void)ok3;
+    return std::make_tuple(lambda, theta, idx);
+}
+
+/// Register and return the edge intersection-angle property `e:alpha`
+/// (used by the hyper-ideal functional).
+inline auto add_edge_properties(ConformalMesh& mesh)
+{
+    auto [alpha, ok] = mesh.add_property_map<Edge_index, double>("e:alpha", 0.0);
+    (void)ok;
+    return alpha;
+}
+
+/// Register and return the per-face geometry-type property `f:type`.
+inline auto add_face_properties(ConformalMesh& mesh)
+{
+    auto [ftype, ok] = mesh.add_property_map<Face_index, int>(
+        "f:type", static_cast<int>(GeometryType::Euclidean));
+    (void)ok;
+    return ftype;
+}
+
+} // namespace conformallab

code/include/constants.hpp

@@ -0,0 +1,20 @@
+#pragma once
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
+// constants.hpp
+//
+// Single source of truth for mathematical constants used throughout
+// conformallab++.  Include this header instead of defining π locally.
+//
+// All constants are in the conformallab namespace and are constexpr double.
+
+namespace conformallab {
+
+/// π to 30 significant digits (well beyond double precision of ~15-16 digits).
+constexpr double PI     = 3.14159265358979323846264338328;
+
+/// 2π (full turn).
+constexpr double TWO_PI = 2.0 * PI;
+
+} // namespace conformallab

code/include/cp_euclidean_functional.hpp

@@ -0,0 +1,390 @@
+#pragma once
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
+// cp_euclidean_functional.hpp
+//
+// Phase 9a.1 — Circle-Packing Euclidean functional (CP-Euclidean).
+//
+// Ported from de.varylab.discreteconformal.functional.CPEuclideanFunctional
+// (Java, 260 lines).  Mathematical reference:
+//   Bobenko, A. I., Pinkall, U. & Springborn, B. (2010)
+//   "Discrete conformal maps and ideal hyperbolic polyhedra"
+//   Geometry & Topology 14, 379-426.
+//
+// ┌──────────────────────────────────────────────────────────────────────────┐
+// │  FACE-based circle packing                                                │
+// │                                                                           │
+// │  Each face f of the mesh carries a circle of radius R_f.                  │
+// │  The variable is ρ_f = log R_f.                                           │
+// │  Adjacent face-circles intersect at a prescribed angle θ_e per edge.      │
+// │                                                                           │
+// │  This is the FACE-DUAL of the classical vertex-based Luo (2004)           │
+// │  inversive-distance circle packing implemented in                          │
+// │  inversive_distance_functional.hpp (Phase 9a.2).  The relation            │
+// │     I_ij  =  cos θ_e                                                       │
+// │  identifies the two parametrisations (Glickenstein 2011 §5).              │
+// │                                                                           │
+// │  DOFs                                                                     │
+// │    x[f_idx[f]] = ρ_f   (face-dual log-radius)                             │
+// │    f_idx[f] = −1 means f is pinned (ρ_f = 0, gauge fix)                  │
+// │                                                                           │
+// │  Constants                                                                │
+// │    θ_e per edge   intersection angle of the two face-circles              │
+// │    φ_f per face   target sum of corner-angles inside the face             │
+// │                                                                           │
+// │  Energy  (BPS-2010 §6)                                                    │
+// │    E(ρ) =  Σ_f  φ_f · ρ_f                                                  │
+// │         +  Σ_{(h,f=face(h)):                                              │
+// │              [ if opposite face exists ]                                   │
+// │                ½ p(θ*,Δρ)·Δρ + Λ(θ*+p) − θ*·ρ_left                       │
+// │              [ else (boundary halfedge) ]                                  │
+// │                −2 θ*·ρ_left                                                │
+// │            ]                                                              │
+// │                                                                           │
+// │    where    θ* = π − θ                                                    │
+// │             Δρ = ρ_right − ρ_left                                          │
+// │             p(θ*, Δρ) = 2·atan( tan(θ*/2) · tanh(Δρ/2) )                  │
+// │             Λ        = Clausen function (Lobachevsky)                     │
+// │                                                                           │
+// │  Gradient                                                                 │
+// │    Per face f:        +φ_f                                                │
+// │    Per interior hf:   −(p + θ*)  added to G[face(h)]                      │
+// │    Per boundary hf:   −2 θ*       added to G[face(h)]                     │
+// │                                                                           │
+// │  Hessian (analytic, BPS-2010 eq. 6.8; Java getHessian lines 127-166)     │
+// │    Per interior undirected edge e (connecting faces j and k):             │
+// │      h_jk = sin θ / (cosh Δρ − cos θ)                                     │
+// │      H[j,j] += h_jk,  H[k,k] += h_jk,  H[j,k] −= h_jk,  H[k,j] −= h_jk    │
+// │    Pinned faces contribute nothing (their row/col is removed).            │
+// └──────────────────────────────────────────────────────────────────────────┘
+//
+// Halfedge convention (matches Java's "leftFace / rightFace"):
+//   For a directed halfedge h in CGAL::Surface_mesh:
+//     mesh.face(h)             ≡ leftFace
+//     mesh.face(opposite(h))   ≡ rightFace  (may be null on boundary)
+//     mesh.is_border(h) == true iff h has no face (h points outward).
+//   Property-map name prefix: "cf:" (face) and "ce:" (edge).
+
+#include "conformal_mesh.hpp"
+#include "constants.hpp"
+#include "clausen.hpp"
+#include <Eigen/Sparse>
+#include <CGAL/boost/graph/iterator.h>
+#include <vector>
+#include <cmath>
+#include <cstdint>
+#include <iostream>
+
+namespace conformallab {
+
+// ── Property-map type aliases ────────────────────────────────────────────────
+
+/// Property map face → `int`    for the CP-Euclidean functional.
+using CPFMapI = ConformalMesh::Property_map<Face_index, int>;
+/// Property map face → `double` for the CP-Euclidean functional.
+using CPFMapD = ConformalMesh::Property_map<Face_index, double>;
+/// Property map edge → `double` for the CP-Euclidean functional.
+using CPEMapD = ConformalMesh::Property_map<Edge_index, double>;
+
+// ── Persistent map bundle ─────────────────────────────────────────────────────
+
+/// Bundle of the three property maps consumed by the CP-Euclidean
+/// (Bobenko-Pinkall-Springborn 2010) circle-packing functional.
+struct CPEuclideanMaps {
+    CPFMapI f_idx;     ///< DOF index per face (−1 = pinned)
+    CPEMapD theta_e;   ///< intersection angle per edge (default π/2 = orthogonal)
+    CPFMapD phi_f;     ///< target face-angle sum   (default 2π)
+};
+
+/// Attach the three CP-Euclidean property maps to `mesh` with default
+/// values and return their handles.
+///
+/// Defaults:
+/// * `theta_e[e] = π/2` for every edge — orthogonal circle packing
+///   (Koebe-Andreev-Thurston).
+/// * `phi_f[f] = 2π` for every face — flat target.
+/// * `f_idx[f] = -1` for every face — all faces pinned initially;
+///   call `assign_cp_euclidean_face_dof_indices()` next to assign
+///   DOF indices to all faces except one gauge-pinned face.
+///
+/// The maps are named with the `"cf:"` / `"ce:"` prefixes
+/// (cf = circle-packing-face, ce = circle-packing-edge) so they do
+/// not collide with the Euclidean / Spherical / HyperIdeal maps.
+///
+/// \param mesh  Input mesh.  Modified in place: three property maps are
+///              attached if not already present, otherwise the existing
+///              maps are returned unchanged (CGAL property-map idempotence).
+/// \returns     A bundle of all three property maps for caller use.
+inline CPEuclideanMaps setup_cp_euclidean_maps(ConformalMesh& mesh)
+{
+    CPEuclideanMaps m;
+    m.f_idx   = mesh.add_property_map<Face_index, int>   ("cf:idx",   -1     ).first;
+    m.theta_e = mesh.add_property_map<Edge_index, double>("ce:theta", PI / 2 ).first;
+    m.phi_f   = mesh.add_property_map<Face_index, double>("cf:phi",   TWO_PI ).first;
+    return m;
+}
+
+/// Assign sequential DOF indices `0..n-1` to all faces except `pinned`,
+/// which receives the sentinel `-1` (gauge-fixed face, `ρ_pinned = 0`).
+///
+/// Mirrors the Java CPEuclideanFunctional's "skip face index 0"
+/// convention from `evaluateEnergyAndGradient` (lines 184-185 of
+/// CPEuclideanFunctional.java).  The C++ port exposes the choice of
+/// pinned face explicitly rather than hard-coding it.
+///
+/// \param mesh    The mesh.  Read for face iteration only; not modified.
+/// \param m       Map bundle whose `f_idx` is overwritten.
+/// \param pinned  The face whose DOF is fixed at zero (the gauge).
+/// \returns       The number of free DOFs assigned (`num_faces(mesh) - 1`).
+inline int assign_cp_euclidean_face_dof_indices(ConformalMesh&  mesh,
+                                                  CPEuclideanMaps& m,
+                                                  Face_index       pinned)
+{
+    int idx = 0;
+    for (auto f : mesh.faces()) {
+        if (f == pinned) m.f_idx[f] = -1;
+        else             m.f_idx[f] = idx++;
+    }
+    return idx;
+}
+
+/// Convenience overload: pin the **first** face in `mesh.faces()` order.
+/// Use this when any face works as the gauge (typically true for
+/// closed mesh experiments).
+inline int assign_cp_euclidean_face_dof_indices(ConformalMesh&  mesh,
+                                                  CPEuclideanMaps& m)
+{
+    auto it = mesh.faces().begin();
+    if (it == mesh.faces().end()) return 0;
+    return assign_cp_euclidean_face_dof_indices(mesh, m, *it);
+}
+
+/// Count the free DOFs (faces with `f_idx >= 0`).
+/// Equivalent to `num_faces(mesh) - <number of pinned faces>`.
+inline int cp_euclidean_dimension(const ConformalMesh& mesh,
+                                    const CPEuclideanMaps& m)
+{
+    int dim = 0;
+    for (auto f : mesh.faces()) if (m.f_idx[f] >= 0) ++dim;
+    return dim;
+}
+
+// ── Internal helpers ──────────────────────────────────────────────────────────
+
+namespace cp_detail {
+
+// p(θ*, Δρ) = 2·atan( tan(θ*/2) · tanh(Δρ/2) )
+// Numerically stable form lifted directly from CPEuclideanFunctional.java
+// (private method `p`, lines 243-247).
+inline double p_function(double thStar, double dRho) noexcept
+{
+    const double e          = std::exp(dRho);
+    const double tanh_half  = (e - 1.0) / (e + 1.0);
+    return 2.0 * std::atan(std::tan(0.5 * thStar) * tanh_half);
+}
+
+// DOF reader: returns 0 for the pinned face (idx = −1).
+inline double dof_val(int idx, const std::vector<double>& x) noexcept
+{
+    return idx >= 0 ? x[static_cast<std::size_t>(idx)] : 0.0;
+}
+
+} // namespace cp_detail
+
+/// CP-Euclidean energy value at DOF vector `x` (ρ per face).
+/// Mirrors `evaluateEnergyAndGradient()` in the Java original (lines 170-240).
+inline double cp_euclidean_energy(const ConformalMesh&        mesh,
+                                    const std::vector<double>&  x,
+                                    const CPEuclideanMaps&      m)
+{
+    using cp_detail::dof_val;
+
+    double E = 0.0;
+
+    // Per-face linear term: + φ_f · ρ_f
+    // (The pinned face has f_idx = −1; its ρ is fixed at 0 so it contributes nothing.)
+    for (auto f : mesh.faces()) {
+        const int i = m.f_idx[f];
+        if (i < 0) continue;
+        E += m.phi_f[f] * x[static_cast<std::size_t>(i)];
+    }
+
+    // Per directed halfedge term.  Java iterates over `getEdges()` which in jtem
+    // yields one Edge per directed side; in CGAL we iterate halfedges directly.
+    for (auto h : mesh.halfedges()) {
+        if (mesh.is_border(h)) continue;   // h is in the outer "border" face → skip
+        const Face_index fL = mesh.face(h);
+        const Halfedge_index ho = mesh.opposite(h);
+        const Face_index fR = mesh.is_border(ho) ? Face_index() : mesh.face(ho);
+
+        const double th     = m.theta_e[mesh.edge(h)];
+        const double thStar = PI - th;
+        const double rho_L  = dof_val(m.f_idx[fL], x);
+
+        if (fR == Face_index()) {
+            // Boundary halfedge: only the left face exists.
+            E += -2.0 * thStar * rho_L;
+        } else {
+            const double rho_R = dof_val(m.f_idx[fR], x);
+            const double dRho  = rho_R - rho_L;
+            const double p     = cp_detail::p_function(thStar, dRho);
+            E += 0.5 * p * dRho;
+            E += clausen2(thStar + p);
+            E += -thStar * rho_L;
+        }
+    }
+
+    return E;
+}
+
+/// CP-Euclidean gradient `∂E/∂ρ_f` (per face DOF).  Interior term
+/// `−(p + θ*)`, boundary term `−2 θ*`; see `setup_cp_euclidean_maps`.
+inline std::vector<double> cp_euclidean_gradient(const ConformalMesh&        mesh,
+                                                   const std::vector<double>&  x,
+                                                   const CPEuclideanMaps&      m)
+{
+    using cp_detail::dof_val;
+
+    const int n = cp_euclidean_dimension(mesh, m);
+    std::vector<double> G(static_cast<std::size_t>(n), 0.0);
+
+    // Per-face linear term.
+    for (auto f : mesh.faces()) {
+        const int i = m.f_idx[f];
+        if (i < 0) continue;
+        G[static_cast<std::size_t>(i)] += m.phi_f[f];
+    }
+
+    // Per directed halfedge term.
+    for (auto h : mesh.halfedges()) {
+        if (mesh.is_border(h)) continue;
+        const Face_index fL = mesh.face(h);
+        const int        iL = m.f_idx[fL];
+        if (iL < 0) continue;   // pinned face: gradient component is forced to 0
+
+        const Halfedge_index ho = mesh.opposite(h);
+        const Face_index fR = mesh.is_border(ho) ? Face_index() : mesh.face(ho);
+
+        const double th     = m.theta_e[mesh.edge(h)];
+        const double thStar = PI - th;
+        const double rho_L  = dof_val(iL, x);
+
+        if (fR == Face_index()) {
+            G[static_cast<std::size_t>(iL)] -= 2.0 * thStar;
+        } else {
+            const double rho_R = dof_val(m.f_idx[fR], x);
+            const double dRho  = rho_R - rho_L;
+            const double p     = cp_detail::p_function(thStar, dRho);
+            G[static_cast<std::size_t>(iL)] -= (p + thStar);
+        }
+    }
+
+    return G;
+}
+
+/// Analytic CP-Euclidean Hessian, sparse.  Per interior edge `(j,k)`
+/// the contribution is `h_jk = sin θ / (cosh(Δρ) − cos θ)`, added to
+/// diagonals `H_jj`, `H_kk` and subtracted off-diagonals `H_jk = H_kj`.
+/// Pinned faces are excluded (DOF index −1).
+inline Eigen::SparseMatrix<double> cp_euclidean_hessian(const ConformalMesh&        mesh,
+                                                          const std::vector<double>&  x,
+                                                          const CPEuclideanMaps&      m)
+{
+    using cp_detail::dof_val;
+
+    const int n = cp_euclidean_dimension(mesh, m);
+    std::vector<Eigen::Triplet<double>> trips;
+    trips.reserve(static_cast<std::size_t>(4 * mesh.number_of_edges()));
+
+    for (auto e : mesh.edges()) {
+        const Halfedge_index h  = mesh.halfedge(e);
+        const Halfedge_index ho = mesh.opposite(h);
+        if (mesh.is_border(h) || mesh.is_border(ho)) continue;   // boundary edge
+
+        const int j = m.f_idx[mesh.face(h)];
+        const int k = m.f_idx[mesh.face(ho)];
+
+        const double rho_j = dof_val(j, x);
+        const double rho_k = dof_val(k, x);
+        const double dRho  = rho_k - rho_j;
+        const double th    = m.theta_e[e];
+        const double hjk   = std::sin(th) / (std::cosh(dRho) - std::cos(th));
+
+        if (j >= 0)              trips.emplace_back(j, j,  hjk);
+        if (k >= 0)              trips.emplace_back(k, k,  hjk);
+        if (j >= 0 && k >= 0) {
+            trips.emplace_back(j, k, -hjk);
+            trips.emplace_back(k, j, -hjk);
+        }
+    }
+
+    Eigen::SparseMatrix<double> H(n, n);
+    H.setFromTriplets(trips.begin(), trips.end());
+    return H;
+}
+
+/// FD gradient check for the CP-Euclidean functional.  Mirrors the
+/// Java `FunctionalTest`; default `eps = 1e-5`, `tol = 1e-6`.
+inline bool gradient_check_cp_euclidean(const ConformalMesh&        mesh,
+                                          const std::vector<double>&  x,
+                                          const CPEuclideanMaps&      m,
+                                          double eps = 1e-5,
+                                          double tol = 1e-6)
+{
+    auto G = cp_euclidean_gradient(mesh, x, m);
+    const std::size_t n = G.size();
+
+    for (std::size_t i = 0; i < n; ++i) {
+        std::vector<double> xp = x, xm = x;
+        xp[i] += eps;
+        xm[i] -= eps;
+        const double Ep = cp_euclidean_energy(mesh, xp, m);
+        const double Em = cp_euclidean_energy(mesh, xm, m);
+        const double fd = (Ep - Em) / (2.0 * eps);
+        if (std::abs(G[i] - fd) > tol) {
+            std::cerr << "[cp-euclidean] FD gradient mismatch at DOF " << i
+                      << ": analytic=" << G[i]
+                      << "  FD=" << fd
+                      << "  diff=" << (G[i] - fd) << "\n";
+            return false;
+        }
+    }
+    return true;
+}
+
+/// FD Hessian check for the CP-Euclidean functional.  Verifies analytic
+/// `H` column-by-column against `(G(x+εe_j) − G(x−εe_j)) / (2ε)`.
+inline bool hessian_check_cp_euclidean(const ConformalMesh&        mesh,
+                                         const std::vector<double>&  x,
+                                         const CPEuclideanMaps&      m,
+                                         double eps = 1e-5,
+                                         double tol = 1e-5)
+{
+    const auto H = cp_euclidean_hessian(mesh, x, m);
+    const int n = static_cast<int>(H.rows());
+
+    for (int j = 0; j < n; ++j) {
+        std::vector<double> xp = x, xm = x;
+        xp[static_cast<std::size_t>(j)] += eps;
+        xm[static_cast<std::size_t>(j)] -= eps;
+        auto Gp = cp_euclidean_gradient(mesh, xp, m);
+        auto Gm = cp_euclidean_gradient(mesh, xm, m);
+
+        for (int i = 0; i < n; ++i) {
+            double fd = (Gp[static_cast<std::size_t>(i)] - Gm[static_cast<std::size_t>(i)])
+                        / (2.0 * eps);
+            double an = H.coeff(i, j);
+            if (std::abs(an - fd) > tol) {
+                std::cerr << "[cp-euclidean] FD Hessian mismatch at ("
+                          << i << "," << j << "): analytic=" << an
+                          << "  FD=" << fd
+                          << "  diff=" << (an - fd) << "\n";
+                return false;
+            }
+        }
+    }
+    return true;
+}
+
+} // namespace conformallab

code/include/cut_graph.hpp

@@ -0,0 +1,151 @@
+#pragma once
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
+// cut_graph.hpp
+//
+// Phase 6 — Tree-cotree algorithm for computing a cut graph of a triangulated
+// surface.
+//
+// For a closed, orientable, genus-g surface:
+//   #vertices (V), #edges (E), #faces (F)
+//   Euler: V − E + F = 2 − 2g
+//   Primal spanning tree:  V − 1  edges
+//   Dual spanning tree:    F − 1  edges  (avoiding duals of tree edges)
+//   Remaining:             E − (V−1) − (F−1) = 2g  cut edges
+//
+// These 2g cut edges generate H₁(M, ℤ) ≅ ℤ^{2g}.
+// Cutting along them turns M into a topological disk.
+//
+// For open meshes (boundary present) the algorithm still works: the dual BFS
+// starts from a boundary-adjacent face, and boundary half-edges are skipped.
+// The number of cut edges will be E − (V−1) − (F−1) − B where B counts
+// boundary edges treated as dual tree edges.
+//
+// Usage:
+//   CutGraph cg = compute_cut_graph(mesh);
+//   // cg.cut_edge_flags[e.idx()] == true  →  treat edge as seam in BFS
+//   euclidean_layout(mesh, x, maps, &cg);    // layout with holonomy tracking
+
+#include "conformal_mesh.hpp"
+#include "gauss_bonnet.hpp"     // for euler_characteristic / genus
+#include <vector>
+#include <queue>
+#include <cstddef>
+
+namespace conformallab {
+
+// ─────────────────────────────────────────────────────────────────────────────
+// CutGraph
+// ─────────────────────────────────────────────────────────────────────────────
+
+/// Cut-graph result of the tree-cotree algorithm: the set of `2g` edges
+/// whose removal turns a closed genus-`g` surface into a topological disk.
+struct CutGraph {
+    /// cut_edge_flags[e.idx()] = true ↔ this edge is a cut edge.
+    /// Size = mesh.number_of_edges().
+    std::vector<bool> cut_edge_flags;
+
+    /// Indices of the 2g cut edges in order (size = 2g).
+    std::vector<std::size_t> cut_edge_indices;
+
+    /// Genus of the surface (0 for topological spheres and open patches).
+    int genus = 0;
+
+    /// `true` iff edge `e` is a cut edge of this graph.
+    bool is_cut(Edge_index e) const
+    {
+        return static_cast<std::size_t>(e.idx()) < cut_edge_flags.size()
+            && cut_edge_flags[static_cast<std::size_t>(e.idx())];
+    }
+};
+
+/// Compute the cut graph of `mesh` via the standard tree-cotree
+/// algorithm (Erickson–Whittlesey 2005): primal BFS spanning tree T,
+/// dual BFS spanning tree T* avoiding T-primals, then the `2g` cut
+/// edges are those in neither T nor T*.
+inline CutGraph compute_cut_graph(const ConformalMesh& mesh)
+{
+    const std::size_t nv = mesh.number_of_vertices();
+    const std::size_t ne = mesh.number_of_edges();
+    const std::size_t nf = mesh.number_of_faces();
+
+    CutGraph cg;
+    cg.cut_edge_flags.assign(ne, false);
+    cg.genus = conformallab::genus(mesh);
+
+    if (nv == 0 || nf == 0) return cg;
+
+    // ── Step 1: primal spanning tree via BFS from vertex 0 ───────────────────
+    std::vector<bool> tree_edge(ne, false);
+    std::vector<bool> v_visited(nv, false);
+
+    {
+        std::queue<Vertex_index> q;
+        auto v0 = *mesh.vertices().begin();
+        v_visited[v0.idx()] = true;
+        q.push(v0);
+        while (!q.empty()) {
+            Vertex_index v = q.front(); q.pop();
+            for (Halfedge_index h : CGAL::halfedges_around_target(v, mesh)) {
+                Vertex_index u = mesh.source(h);
+                if (!v_visited[static_cast<std::size_t>(u.idx())]) {
+                    v_visited[static_cast<std::size_t>(u.idx())] = true;
+                    tree_edge[static_cast<std::size_t>(mesh.edge(h).idx())] = true;
+                    q.push(u);
+                }
+            }
+        }
+    }
+
+    // ── Step 2: dual spanning tree via BFS from face 0 ───────────────────────
+    // Dual edge between face f and face f_adj crosses primal edge e.
+    // Include dual edge only if:
+    //   (a) e is not a primal tree edge (tree_edge[e] == false)
+    //   (b) h_adj is not a border halfedge
+    std::vector<bool> dual_tree_edge(ne, false);
+    std::vector<bool> f_visited(nf, false);
+
+    {
+        std::queue<Face_index> q;
+        auto f0 = *mesh.faces().begin();
+        f_visited[static_cast<std::size_t>(f0.idx())] = true;
+        q.push(f0);
+        while (!q.empty()) {
+            Face_index f = q.front(); q.pop();
+            for (Halfedge_index h :
+                 CGAL::halfedges_around_face(mesh.halfedge(f), mesh))
+            {
+                Halfedge_index h_opp = mesh.opposite(h);
+                if (mesh.is_border(h_opp)) continue;        // boundary edge
+                Face_index f_adj = mesh.face(h_opp);
+                if (f_visited[static_cast<std::size_t>(f_adj.idx())]) continue;
+
+                std::size_t eidx = static_cast<std::size_t>(mesh.edge(h).idx());
+                if (!tree_edge[eidx]) {
+                    // Use this dual edge in T*
+                    f_visited[static_cast<std::size_t>(f_adj.idx())] = true;
+                    dual_tree_edge[eidx] = true;
+                    q.push(f_adj);
+                }
+            }
+        }
+    }
+
+    // ── Step 3: cut edges = neither in T nor in T* nor on boundary ───────────
+    // Boundary edges are adjacent to the "outer face" and need no cutting —
+    // they are implicitly handled by the boundary itself.
+    for (Edge_index e : mesh.edges()) {
+        std::size_t idx = static_cast<std::size_t>(e.idx());
+        if (tree_edge[idx] || dual_tree_edge[idx]) continue;
+        // Skip boundary edges — they are not interior homological cycles.
+        Halfedge_index h = mesh.halfedge(e);
+        if (mesh.is_border(h) || mesh.is_border(mesh.opposite(h))) continue;
+        cg.cut_edge_flags[idx] = true;
+        cg.cut_edge_indices.push_back(idx);
+    }
+
+    return cg;
+}
+
+} // namespace conformallab

code/include/discrete_elliptic_utility.hpp

@@ -0,0 +1,48 @@
+#pragma once
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
+
+// Ported from de.varylab.discreteconformal.util.DiscreteEllipticUtility (Java).
+// Only the pure-math subset (no HDS required).
+
+#include <complex>
+#include <cmath>
+
+namespace conformallab {
+
+// Move tau into the fundamental domain of the modular group SL(2,Z):
+//   |Re(tau)| <= 0.5,  Im(tau) >= 0,  Re(tau) >= 0,  |tau| >= 1
+//
+// Algorithm: iteratively apply
+//   1. T-shift:   Re > 0.5 or Re < 0  →  Re -= sign(Re)
+//   2. Im-flip:   Im < 0              →  Im = -Im
+//   3. Re-flip:   Re < 0              →  Re = -Re
+//   4. S-invert:  |tau| < 1           →  tau = 1/tau
+//
+/// Normalise a complex modulus `τ` into the standard fundamental
+/// domain of an elliptic curve (`|τ| ≥ 1`, `0 ≤ Re τ ≤ ½`, `Im τ ≥ 0`).
+/// Same as Java `DiscreteEllipticUtility.normalizeModulus(Complex)`.
+inline std::complex<double> normalizeModulus(std::complex<double> tau) {
+    int maxIter = 100;
+    while (--maxIter > 0) {
+        double re = tau.real();
+        double im = tau.imag();
+        // exit when all conditions satisfied
+        if (std::abs(re) <= 0.5 && im >= 0.0 && re >= 0.0 && std::abs(tau) >= 1.0)
+            break;
+
+        if (std::abs(re) > 0.5)
+            re -= (re > 0.0 ? 1.0 : -1.0);   // signum shift
+        if (im < 0.0)
+            im = -im;
+        if (re < 0.0)
+            re = -re;
+        tau = std::complex<double>(re, im);
+        if (std::abs(tau) < 1.0)
+            tau = 1.0 / tau;                  // S-transformation: invert
+    }
+    return tau;
+}
+
+} // namespace conformallab

code/include/euclidean_functional.hpp

@@ -0,0 +1,345 @@
+#pragma once
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
+// euclidean_functional.hpp
+//
+// Energy and gradient of the Euclidean discrete conformal functional
+// (EuclideanCyclicFunctional) evaluated on a ConformalMesh.
+//
+// Ported from de.varylab.discreteconformal.functional.EuclideanCyclicFunctional.
+//
+// ┌──────────────────────────────────────────────────────────────────────────┐
+// │  DOFs                                                                    │
+// │    x[v_idx[v]]  = u_v   – conformal factor at vertex v                  │
+// │    x[e_idx[e]]  = λ_e   – edge log-length variable (optional)           │
+// │    -1 means "pinned" (u_v = 0 / λ_e = 0, only λ° contributes)          │
+// │                                                                          │
+// │  Effective log-length (always additive, unlike SphericalFunctional):    │
+// │    Λ̃_ij = λ°_ij + u_i + u_j + (x[e_idx[e]] if variable, else 0)       │
+// │                                                                          │
+// │  Side length:  l_ij = exp(Λ̃_ij / 2)                                    │
+// │                                                                          │
+// │  Gradient:                                                               │
+// │    ∂E/∂u_v  = Θ_v  −  Σ_{faces adj. v} α_v(face)                       │
+// │    ∂E/∂λ_e  = α_opp(face⁺) + α_opp(face⁻) − φ_e                       │
+// │                                                                          │
+// │  Energy:                                                                 │
+// │    Computed as the path integral E(x) = ∫₀¹ ⟨G(tx), x⟩ dt             │
+// │    using 10-point Gauss-Legendre quadrature (same as SphericalFunctional)│
+// │    This is E(0)=0 by construction and exact for conservative G.         │
+// └──────────────────────────────────────────────────────────────────────────┘
+//
+// Halfedge convention (identical to SphericalFunctional):
+//   h0 = mesh.halfedge(f),  h1 = next(h0),  h2 = next(h1)
+//   v1 = source(h0), v2 = source(h1), v3 = source(h2)
+//   h_alpha[h0] = α3  (angle at v3, opposite edge h0 = v1v2)
+//   h_alpha[h1] = α1  (angle at v1, opposite edge h1 = v2v3)
+//   h_alpha[h2] = α2  (angle at v2, opposite edge h2 = v3v1)
+//
+// Property-map name prefix: "ev:" (vertex) and "ee:" (edge).
+
+#include "conformal_mesh.hpp"
+#include "constants.hpp"
+#include "euclidean_geometry.hpp"
+#include <CGAL/boost/graph/iterator.h>
+#include <vector>
+#include <cmath>
+#include <cstdint>
+
+namespace conformallab {
+
+// ── Property-map type aliases ─────────────────────────────────────────────────
+
+/// Property map vertex → `double` for the Euclidean functional.
+using EuclVMapD = ConformalMesh::Property_map<Vertex_index, double>;
+/// Property map vertex → `int`    for the Euclidean functional.
+using EuclVMapI = ConformalMesh::Property_map<Vertex_index, int>;
+/// Property map edge   → `double` for the Euclidean functional.
+using EuclEMapD = ConformalMesh::Property_map<Edge_index,   double>;
+/// Property map edge   → `int`    for the Euclidean functional.
+using EuclEMapI = ConformalMesh::Property_map<Edge_index,   int>;
+
+// ── Persistent map bundle ─────────────────────────────────────────────────────
+
+/// Bundle of the five property maps consumed by the Euclidean functional.
+struct EuclideanMaps {
+    EuclVMapI v_idx;    ///< DOF index per vertex   (-1 = pinned / u_v = 0)
+    EuclEMapI e_idx;    ///< DOF index per edge      (-1 = no edge DOF)
+    EuclVMapD theta_v;  ///< target cone angle Θ_v   (default 2π)
+    EuclEMapD phi_e;    ///< target edge turn angle φ_e (default π)
+    EuclEMapD lambda0;  ///< base log-length λ°_e    (default 0.0)
+};
+
+/// Attach the five Euclidean property maps to `mesh` with sensible
+/// defaults and return their handles.
+///
+/// Defaults:
+/// * `v_idx[v] = -1`  (every vertex pinned; user must reassign before solving)
+/// * `e_idx[e] = -1`  (no edge DOFs by default; use `assign_euclidean_all_dof_indices` for cyclic functional)
+/// * `theta_v[v] = 2π`  (flat interior vertex target)
+/// * `phi_e[e] = π`     (interior edge turn angle target — flat surface)
+/// * `lambda0[e] = 0`   (placeholder; call `compute_euclidean_lambda0_from_mesh` next)
+///
+/// Map name prefix: `"ev:"` (vertex) and `"ee:"` (edge).
+inline EuclideanMaps setup_euclidean_maps(ConformalMesh& mesh)
+{
+    EuclideanMaps m;
+    m.v_idx  = mesh.add_property_map<Vertex_index, int>   ("ev:idx",   -1     ).first;
+    m.e_idx  = mesh.add_property_map<Edge_index,   int>   ("ee:idx",   -1     ).first;
+    m.theta_v= mesh.add_property_map<Vertex_index, double>("ev:theta", TWO_PI ).first;
+    m.phi_e  = mesh.add_property_map<Edge_index,   double>("ee:phi",   PI     ).first;
+    m.lambda0= mesh.add_property_map<Edge_index,   double>("ee:lam0",  0.0    ).first;
+    return m;
+}
+
+/// Assign sequential DOF indices `0..n-1` to all vertices.
+///
+/// **Note:** does NOT pin a gauge vertex.  For closed meshes the caller
+/// must set one `m.v_idx[v] = -1` either before or after this call to
+/// remove the rotational mode (the Newton solver's SparseQR fallback
+/// will otherwise pick a minimum-norm solution but at higher cost).
+inline int assign_euclidean_vertex_dof_indices(ConformalMesh& mesh, EuclideanMaps& m)
+{
+    int idx = 0;
+    for (auto v : mesh.vertices()) m.v_idx[v] = idx++;
+    return idx;
+}
+
+/// Assign DOF indices for all vertices AND all edges (vertex-DOFs first,
+/// then edge-DOFs).  Use this overload for the "cyclic" formulation that
+/// includes per-edge log-length DOFs (`λ_e`) on top of per-vertex scale
+/// factors (`u_v`).
+inline int assign_euclidean_all_dof_indices(ConformalMesh& mesh, EuclideanMaps& m)
+{
+    int idx = 0;
+    for (auto v : mesh.vertices()) m.v_idx[v] = idx++;
+    for (auto e : mesh.edges())    m.e_idx[e] = idx++;
+    return idx;
+}
+
+/// Count the free DOFs (vertices + edges with index `≥ 0`).
+inline int euclidean_dimension(const ConformalMesh& mesh, const EuclideanMaps& m)
+{
+    int dim = 0;
+    for (auto v : mesh.vertices()) if (m.v_idx[v] >= 0) ++dim;
+    for (auto e : mesh.edges())    if (m.e_idx[e] >= 0) ++dim;
+    return dim;
+}
+
+/// Set `lambda0` from mesh vertex positions:
+/// `λ°_e = 2·log(|p_i − p_j|)` (natural log of Euclidean edge length²).
+/// This gives `exp(Λ̃_ij / 2) = l_ij` at `x = 0`.
+inline void compute_euclidean_lambda0_from_mesh(ConformalMesh& mesh, EuclideanMaps& m)
+{
+    for (auto e : mesh.edges()) {
+        auto h  = mesh.halfedge(e);
+        auto p1 = mesh.point(mesh.source(h));
+        auto p2 = mesh.point(mesh.target(h));
+        double dx = p1.x() - p2.x();
+        double dy = p1.y() - p2.y();
+        double dz = p1.z() - p2.z();
+        double len = std::sqrt(dx*dx + dy*dy + dz*dz);
+        if (len > 1e-15)
+            m.lambda0[e] = 2.0 * std::log(len);
+        else
+            m.lambda0[e] = -30.0;   // degenerate edge
+    }
+}
+
+// ── Internal helpers ──────────────────────────────────────────────────────────
+
+/// Read DOF value from `x` for index `idx`; return 0 if pinned (idx < 0).
+static inline double eucl_dof_val(int idx, const std::vector<double>& x)
+{
+    return idx >= 0 ? x[static_cast<std::size_t>(idx)] : 0.0;
+}
+
+/// Convert a CGAL half-edge index to a plain `std::size_t` for vector indexing.
+static inline std::size_t eucl_hidx(Halfedge_index h)
+{
+    return static_cast<std::size_t>(static_cast<std::uint32_t>(h));
+}
+
+/// Compute the Euclidean-functional gradient G(x):
+///   * `G_v = Θ_v − Σ_faces α_v(face)`
+///   * `G_e = α_opp(face⁺) + α_opp(face⁻) − φ_e`
+///
+/// Same half-edge corner-angle storage convention as `spherical_gradient`.
+inline std::vector<double> euclidean_gradient(
+    ConformalMesh&             mesh,
+    const std::vector<double>& x,
+    const EuclideanMaps&       m)
+{
+    const int n = euclidean_dimension(mesh, m);
+    std::vector<double> G(static_cast<std::size_t>(n), 0.0);
+
+    // Per-halfedge corner-angle storage.
+    const std::size_t nh = mesh.number_of_halfedges();
+    std::vector<double> h_alpha(nh, 0.0);
+
+    // ── Pass 1: compute corner angles per face ────────────────────────────────
+    for (auto f : mesh.faces()) {
+        Halfedge_index h0 = mesh.halfedge(f);
+        Halfedge_index h1 = mesh.next(h0);
+        Halfedge_index h2 = mesh.next(h1);
+
+        Vertex_index v1 = mesh.source(h0);
+        Vertex_index v2 = mesh.source(h1);
+        Vertex_index v3 = mesh.source(h2);
+
+        Edge_index e12 = mesh.edge(h0);
+        Edge_index e23 = mesh.edge(h1);
+        Edge_index e31 = mesh.edge(h2);
+
+        // Effective log-lengths (always additive: u_i + u_j regardless of DOF status)
+        double u1 = eucl_dof_val(m.v_idx[v1], x);
+        double u2 = eucl_dof_val(m.v_idx[v2], x);
+        double u3 = eucl_dof_val(m.v_idx[v3], x);
+
+        double lam12 = m.lambda0[e12] + u1 + u2 + eucl_dof_val(m.e_idx[e12], x);
+        double lam23 = m.lambda0[e23] + u2 + u3 + eucl_dof_val(m.e_idx[e23], x);
+        double lam31 = m.lambda0[e31] + u3 + u1 + eucl_dof_val(m.e_idx[e31], x);
+
+        auto fa = euclidean_angles(lam12, lam23, lam31);
+        if (!fa.valid) continue;   // degenerate triangle: contributes 0
+
+        // h_alpha[h] = corner angle OPPOSITE to h's edge:
+        //   h0 (edge v1v2) → opposite corner at v3 → α3
+        //   h1 (edge v2v3) → opposite corner at v1 → α1
+        //   h2 (edge v3v1) → opposite corner at v2 → α2
+        h_alpha[eucl_hidx(h0)] = fa.alpha3;
+        h_alpha[eucl_hidx(h1)] = fa.alpha1;
+        h_alpha[eucl_hidx(h2)] = fa.alpha2;
+    }
+
+    // ── Pass 2: accumulate vertex gradient ───────────────────────────────────
+    // G_v = Θ_v − Σ h_alpha[prev(h)]  for each incoming non-border h to v.
+    for (auto v : mesh.vertices()) {
+        int iv = m.v_idx[v];
+        if (iv < 0) continue;
+        double sum_alpha = 0.0;
+        for (auto h : CGAL::halfedges_around_target(v, mesh)) {
+            if (mesh.is_border(h)) continue;
+            sum_alpha += h_alpha[eucl_hidx(mesh.prev(h))];
+        }
+        G[static_cast<std::size_t>(iv)] = m.theta_v[v] - sum_alpha;
+    }
+
+    // ── Pass 3: accumulate edge gradient ─────────────────────────────────────
+    // G_e = α_opp(f⁺) + α_opp(f⁻) − φ_e
+    // α_opp of edge e in face f = h_alpha[halfedge h of e pointing INTO f].
+    for (auto e : mesh.edges()) {
+        int ie = m.e_idx[e];
+        if (ie < 0) continue;
+        auto h  = mesh.halfedge(e);
+        auto ho = mesh.opposite(h);
+        double sum = -m.phi_e[e];
+        if (!mesh.is_border(h))  sum += h_alpha[eucl_hidx(h)];
+        if (!mesh.is_border(ho)) sum += h_alpha[eucl_hidx(ho)];
+        G[static_cast<std::size_t>(ie)] = sum;
+    }
+
+    return G;
+}
+
+/// Euclidean energy `E(x) = ∫₀¹ ⟨G(t·x), x⟩ dt`, evaluated with
+/// 10-point Gauss-Legendre quadrature (same as the Spherical functional).
+inline double euclidean_energy(
+    ConformalMesh&             mesh,
+    const std::vector<double>& x,
+    const EuclideanMaps&       m)
+{
+    static const double gl_s[10] = {
+        -0.9739065285171717,  -0.8650633666889845,
+        -0.6794095682990244,  -0.4333953941292472,
+        -0.1488743389816312,   0.1488743389816312,
+         0.4333953941292472,   0.6794095682990244,
+         0.8650633666889845,   0.9739065285171717
+    };
+    static const double gl_w[10] = {
+        0.0666713443086881,  0.1494513491505806,
+        0.2190863625159820,  0.2692667193099963,
+        0.2955242247147529,  0.2955242247147529,
+        0.2692667193099963,  0.2190863625159820,
+        0.1494513491505806,  0.0666713443086881
+    };
+
+    const std::size_t n = x.size();
+    double E = 0.0;
+
+    for (int k = 0; k < 10; ++k) {
+        double t  = (1.0 + gl_s[k]) * 0.5;
+        double wt = gl_w[k] * 0.5;
+
+        std::vector<double> tx(n);
+        for (std::size_t i = 0; i < n; ++i) tx[i] = t * x[i];
+
+        auto G = euclidean_gradient(mesh, tx, m);
+
+        double dot = 0.0;
+        for (std::size_t i = 0; i < n; ++i) dot += G[i] * x[i];
+        E += wt * dot;
+    }
+    return E;
+}
+
+// ── Full evaluation (energy + gradient) ──────────────────────────────────────
+
+/// Output of `evaluate_euclidean()` — energy plus optional gradient.
+struct EuclideanResult {
+    double              energy   = 0.0; ///< Functional value at input DOFs.
+    std::vector<double> gradient;       ///< Gradient ∇E (empty if not requested).
+};
+
+/// Evaluate the Euclidean functional at DOFs `x`.  Returns energy and
+/// gradient (toggle via `need_energy` / `need_gradient`).
+inline EuclideanResult evaluate_euclidean(
+    ConformalMesh&             mesh,
+    const std::vector<double>& x,
+    const EuclideanMaps&       m,
+    bool                       need_energy   = true,
+    bool                       need_gradient = true)
+{
+    EuclideanResult res;
+    if (need_gradient)
+        res.gradient = euclidean_gradient(mesh, x, m);
+    if (need_energy)
+        res.energy = euclidean_energy(mesh, x, m);
+    return res;
+}
+
+/// Finite-difference gradient check for the Euclidean functional
+/// (central differences).  Defaults `eps = 1e-5`, `tol = 1e-4`.
+inline bool gradient_check_euclidean(
+    ConformalMesh&             mesh,
+    const std::vector<double>& x0,
+    const EuclideanMaps&       m,
+    double                     eps = 1e-5,
+    double                     tol = 1e-4)
+{
+    auto G = euclidean_gradient(mesh, x0, m);
+    const int n = static_cast<int>(G.size());
+
+    std::vector<double> xp = x0, xm = x0;
+    bool ok = true;
+
+    for (int i = 0; i < n; ++i) {
+        std::size_t si = static_cast<std::size_t>(i);
+        xp[si] = x0[si] + eps;
+        xm[si] = x0[si] - eps;
+
+        double Ep = euclidean_energy(mesh, xp, m);
+        double Em = euclidean_energy(mesh, xm, m);
+
+        xp[si] = xm[si] = x0[si];   // restore
+
+        double fd    = (Ep - Em) / (2.0 * eps);
+        double err   = std::abs(G[si] - fd);
+        double scale = std::max(1.0, std::abs(G[si]));
+        if (err / scale > tol) ok = false;
+    }
+    return ok;
+}
+
+} // namespace conformallab

code/include/euclidean_geometry.hpp

@@ -0,0 +1,87 @@
+#pragma once
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
+// euclidean_geometry.hpp
+//
+// Corner-angle formula for Euclidean triangles in the discrete conformal
+// (log-length) parametrisation.
+//
+// Ported from de.varylab.discreteconformal.functional.EuclideanCyclicFunctional.
+//
+// In the discrete conformal parametrisation a Euclidean triangle is described by
+// its three effective log-lengths Λ̃_ij = λ°_ij + u_i + u_j (+ edge DOF).
+// The corresponding side lengths are l_ij = exp(Λ̃_ij / 2).
+//
+// Vertex ordering convention (matches EuclideanCyclicFunctional.java):
+//   v1 is opposite edge l23,  v2 is opposite l31,  v3 is opposite l12.
+//
+// t-value trick (Springborn 2008 §3):
+//   t12 = −l12 + l23 + l31  =  2(s − l12)
+//   t23 = +l12 − l23 + l31  =  2(s − l23)
+//   t31 = +l12 + l23 − l31  =  2(s − l31)
+//   denom = sqrt(t12 · t23 · t31 · l123) = 4 · Area
+//
+//   α_v = 2 · atan2( product of t-values adjacent to v, denom )
+//
+// The centering trick (l_ij ← exp((Λ̃_ij − 2·μ)/2), μ = (Λ̃12+Λ̃23+Λ̃31)/6)
+// rescales all three sides by the same factor, leaving angles unchanged but
+// keeping the arguments of exp in a safe numerical range.
+
+#include <cmath>
+
+namespace conformallab {
+
+/// Interior corner angles of a Euclidean triangle.
+struct EuclideanFaceAngles {
+    double alpha1;   ///< Corner angle at v₁ (opposite l₂₃).
+    double alpha2;   ///< Corner angle at v₂ (opposite l₃₁).
+    double alpha3;   ///< Corner angle at v₃ (opposite l₁₂).
+    bool   valid;    ///< `false` when the triangle is degenerate.
+};
+
+/// Compute the corner angles of a Euclidean triangle from its three
+/// side lengths.  Returns `valid = false` when the triangle inequality
+/// is violated.
+inline EuclideanFaceAngles euclidean_angles_from_lengths(
+    double l12, double l23, double l31)
+{
+    const double t12 = -l12 + l23 + l31;   // 2*(s − l12)
+    const double t23 = +l12 - l23 + l31;   // 2*(s − l23)
+    const double t31 = +l12 + l23 - l31;   // 2*(s − l31)
+
+    if (t12 <= 0.0 || t23 <= 0.0 || t31 <= 0.0)
+        return {0.0, 0.0, 0.0, false};
+
+    const double l123  = l12 + l23 + l31;
+    const double denom2 = t12 * t23 * t31 * l123;  // = (4·Area)²
+    if (denom2 <= 0.0)
+        return {0.0, 0.0, 0.0, false};
+
+    const double denom = std::sqrt(denom2);
+
+    // α at v1 (opposite l23): adjacent t-values are t12 and t31
+    // α at v2 (opposite l31): adjacent t-values are t12 and t23
+    // α at v3 (opposite l12): adjacent t-values are t23 and t31
+    return {
+        2.0 * std::atan2(t12 * t31, denom),
+        2.0 * std::atan2(t12 * t23, denom),
+        2.0 * std::atan2(t23 * t31, denom),
+        true
+    };
+}
+
+/// Compute the corner angles of a Euclidean triangle from its three
+/// effective log-lengths `Λ̃ᵢⱼ`.  Internally centres lengths so that
+/// `l₁₂·l₂₃·l₃₁ = 1` to avoid float overflow for large `|Λ̃|`.
+inline EuclideanFaceAngles euclidean_angles(
+    double lam12, double lam23, double lam31)
+{
+    const double mu   = (lam12 + lam23 + lam31) / 6.0;
+    const double l12  = std::exp((lam12 - 2.0 * mu) * 0.5);
+    const double l23  = std::exp((lam23 - 2.0 * mu) * 0.5);
+    const double l31  = std::exp((lam31 - 2.0 * mu) * 0.5);
+    return euclidean_angles_from_lengths(l12, l23, l31);
+}
+
+} // namespace conformallab

code/include/euclidean_hessian.hpp

@@ -0,0 +1,216 @@
+#pragma once
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
+// euclidean_hessian.hpp
+//
+// Analytical Hessian of the Euclidean discrete conformal energy —
+// the cotangent-Laplace operator.
+//
+// Ported from de.varylab.discreteconformal.functional.EuclideanCyclicFunctional
+// (the hessian() method).
+//
+// ┌──────────────────────────────────────────────────────────────────────────┐
+// │  Hessian formula (vertex DOFs only)                                      │
+// │                                                                          │
+// │  For a face (v1, v2, v3) with effective log-lengths Λ̃ij and             │
+// │  side lengths lij = exp(Λ̃ij/2):                                         │
+// │                                                                          │
+// │    t12 = −l12+l23+l31,  t23 = l12−l23+l31,  t31 = l12+l23−l31         │
+// │    denom2 = 2·sqrt(t12·t23·t31·l123) = 8·Area                           │
+// │                                                                          │
+// │    cot_k = (t_adj1·l123 − t_adj2·t_opp) / denom2                        │
+// │          = cotangent of the angle αk at vertex k                         │
+// │                                                                          │
+// │  Hessian contributions per face:                                         │
+// │    H[vi, vi] += cot_vj + cot_vk         (diagonal, both non-opp angles) │
+// │    H[vi, vj] -= cot_vk                  (off-diagonal, for variable vi,vj│
+// │                                                                          │
+// │  This is exactly the cotangent-Laplace operator from Pinkall–Polthier.   │
+// │                                                                          │
+// │  Pinned vertices (v_idx = −1) contribute to diagonal of neighbours but   │
+// │  do not create a column/row in H themselves.                             │
+// └──────────────────────────────────────────────────────────────────────────┘
+//
+// Requires Eigen (header-only).  The Hessian is returned as an
+// Eigen::SparseMatrix<double> for direct use in the Phase-4 Newton solver
+// (Eigen::SimplicialLDLT).
+//
+// The Hessian is symmetric positive semi-definite for any valid mesh with
+// no degenerate faces.  The null space is spanned by the uniform-shift
+// vector 1 on closed surfaces (Euler characteristic = 0).
+
+#include "euclidean_functional.hpp"
+#include <Eigen/Sparse>
+#include <vector>
+#include <cmath>
+
+namespace conformallab {
+
+// ── Cotangent weight helper ───────────────────────────────────────────────────
+//
+// Given three Euclidean SIDE LENGTHS l12, l23, l31 (already exp(Λ̃/2)),
+// return the three cotangent weights (cot1, cot2, cot3).
+//
+// cot_k = (t_adj·l123 − t_opp·t_other) / (8·Area)
+//
+// Returns {0,0,0} for degenerate faces (triangle inequality violated or Area=0).
+/// Three Euclidean cotangent weights `(cot1, cot2, cot3)` for the
+/// vertices opposite to edges (l₂₃, l₃₁, l₁₂) of a triangle, plus a
+/// `valid` flag that is `false` when the triangle is degenerate.
+struct EuclCotWeights {
+    double cot1; ///< Cotangent at vertex 1 (opposite to l₂₃).
+    double cot2; ///< Cotangent at vertex 2 (opposite to l₃₁).
+    double cot3; ///< Cotangent at vertex 3 (opposite to l₁₂).
+    bool   valid;///< `false` when the triangle is degenerate (triangle inequality violated or area = 0).
+};
+
+/// Compute the three Euclidean cotangent weights from edge lengths.
+/// Returns `{0,0,0,false}` for degenerate triangles.
+inline EuclCotWeights euclidean_cot_weights(double l12, double l23, double l31)
+{
+    const double t12 = -l12 + l23 + l31;
+    const double t23 = +l12 - l23 + l31;
+    const double t31 = +l12 + l23 - l31;
+
+    if (t12 <= 0.0 || t23 <= 0.0 || t31 <= 0.0)
+        return {0.0, 0.0, 0.0, false};
+
+    const double l123   = l12 + l23 + l31;
+    const double denom2_sq = t12 * t23 * t31 * l123;
+    if (denom2_sq <= 0.0) return {0.0, 0.0, 0.0, false};
+
+    // denom2 = 2·sqrt(t12·t23·t31·l123) = 8·Area
+    const double denom2 = 2.0 * std::sqrt(denom2_sq);
+
+    // cot at v1 (opposite l23): adjacent t-values are t12 and t31.
+    // cot at v2 (opposite l31): adjacent t-values are t12 and t23.
+    // cot at v3 (opposite l12): adjacent t-values are t23 and t31.
+    return {
+        (t23 * l123 - t31 * t12) / denom2,  // cot1
+        (t31 * l123 - t12 * t23) / denom2,  // cot2
+        (t12 * l123 - t23 * t31) / denom2,  // cot3
+        true
+    };
+}
+
+/// Analytical Euclidean Hessian (cotangent Laplacian), sparse.
+/// Only vertex DOFs are supported — the function asserts that no edge
+/// DOF is variable.  `x` is used to compute effective log-lengths Λ̃ᵢⱼ.
+inline Eigen::SparseMatrix<double> euclidean_hessian(
+    ConformalMesh&             mesh,
+    const std::vector<double>& x,
+    const EuclideanMaps&       m)
+{
+    const int n = euclidean_dimension(mesh, m);
+
+    // Collect triplets (row, col, value) — setFromTriplets sums duplicates.
+    std::vector<Eigen::Triplet<double>> trips;
+    trips.reserve(static_cast<std::size_t>(n) * 7);   // rough estimate
+
+    for (auto f : mesh.faces()) {
+        Halfedge_index h0 = mesh.halfedge(f);
+        Halfedge_index h1 = mesh.next(h0);
+        Halfedge_index h2 = mesh.next(h1);
+
+        Vertex_index v1 = mesh.source(h0);
+        Vertex_index v2 = mesh.source(h1);
+        Vertex_index v3 = mesh.source(h2);
+
+        Edge_index e12 = mesh.edge(h0);
+        Edge_index e23 = mesh.edge(h1);
+        Edge_index e31 = mesh.edge(h2);
+
+        // Effective log-lengths.
+        double u1 = eucl_dof_val(m.v_idx[v1], x);
+        double u2 = eucl_dof_val(m.v_idx[v2], x);
+        double u3 = eucl_dof_val(m.v_idx[v3], x);
+
+        double lam12 = m.lambda0[e12] + u1 + u2 + eucl_dof_val(m.e_idx[e12], x);
+        double lam23 = m.lambda0[e23] + u2 + u3 + eucl_dof_val(m.e_idx[e23], x);
+        double lam31 = m.lambda0[e31] + u3 + u1 + eucl_dof_val(m.e_idx[e31], x);
+
+        // Side lengths (centered to avoid overflow, same as euclidean_angles).
+        const double mu  = (lam12 + lam23 + lam31) / 6.0;
+        const double l12 = std::exp((lam12 - 2.0 * mu) * 0.5);
+        const double l23 = std::exp((lam23 - 2.0 * mu) * 0.5);
+        const double l31 = std::exp((lam31 - 2.0 * mu) * 0.5);
+
+        auto [cot1, cot2, cot3, valid] = euclidean_cot_weights(l12, l23, l31);
+        if (!valid) continue;
+
+        const int i1 = m.v_idx[v1];
+        const int i2 = m.v_idx[v2];
+        const int i3 = m.v_idx[v3];
+
+        // ── Diagonal contributions ──────────────────────────────────────────
+        // H[v1,v1] += (cot2 + cot3)/2  (Pinkall–Polthier factor of 1/2)
+        // H[v2,v2] += (cot3 + cot1)/2
+        // H[v3,v3] += (cot1 + cot2)/2
+        if (i1 >= 0) trips.emplace_back(i1, i1, (cot2 + cot3) * 0.5);
+        if (i2 >= 0) trips.emplace_back(i2, i2, (cot3 + cot1) * 0.5);
+        if (i3 >= 0) trips.emplace_back(i3, i3, (cot1 + cot2) * 0.5);
+
+        // ── Off-diagonal contributions (only for variable pairs) ────────────
+        // Edge v1-v2 opposite α3: H[v1,v2] -= cot3/2
+        if (i1 >= 0 && i2 >= 0) {
+            trips.emplace_back(i1, i2, -cot3 * 0.5);
+            trips.emplace_back(i2, i1, -cot3 * 0.5);
+        }
+        // Edge v2-v3 opposite α1: H[v2,v3] -= cot1/2
+        if (i2 >= 0 && i3 >= 0) {
+            trips.emplace_back(i2, i3, -cot1 * 0.5);
+            trips.emplace_back(i3, i2, -cot1 * 0.5);
+        }
+        // Edge v3-v1 opposite α2: H[v3,v1] -= cot2/2
+        if (i3 >= 0 && i1 >= 0) {
+            trips.emplace_back(i3, i1, -cot2 * 0.5);
+            trips.emplace_back(i1, i3, -cot2 * 0.5);
+        }
+    }
+
+    Eigen::SparseMatrix<double> H(n, n);
+    H.setFromTriplets(trips.begin(), trips.end());
+    return H;
+}
+
+// ── Finite-difference Hessian check ──────────────────────────────────────────
+/// FD Hessian check for the Euclidean functional.  Compares analytic
+/// `H` column-by-column to `(G(x+εeⱼ) − G(x−εeⱼ)) / (2ε)`; returns
+/// `true` iff max relative error is below `tol`.
+inline bool hessian_check_euclidean(
+    ConformalMesh&             mesh,
+    const std::vector<double>& x0,
+    const EuclideanMaps&       m,
+    double                     eps = 1e-5,
+    double                     tol = 1e-4)
+{
+    const int n = static_cast<int>(x0.size());
+    auto H = euclidean_hessian(mesh, x0, m);
+
+    std::vector<double> xp = x0, xm = x0;
+    bool ok = true;
+
+    for (int j = 0; j < n; ++j) {
+        const std::size_t sj = static_cast<std::size_t>(j);
+        xp[sj] = x0[sj] + eps;
+        xm[sj] = x0[sj] - eps;
+
+        auto Gp = euclidean_gradient(mesh, xp, m);
+        auto Gm = euclidean_gradient(mesh, xm, m);
+
+        xp[sj] = xm[sj] = x0[sj];   // restore
+
+        for (int i = 0; i < n; ++i) {
+            double fd_ij  = (Gp[static_cast<std::size_t>(i)]
+                           - Gm[static_cast<std::size_t>(i)]) / (2.0 * eps);
+            double H_ij   = H.coeff(i, j);
+            double err    = std::abs(H_ij - fd_ij);
+            double scale  = std::max(1.0, std::abs(H_ij));
+            if (err / scale > tol) ok = false;
+        }
+    }
+    return ok;
+}
+
+} // namespace conformallab

code/include/fundamental_domain.hpp

@@ -0,0 +1,219 @@
+#pragma once
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
+// fundamental_domain.hpp
+//
+// Phase 7 — Fundamental domain polygon for closed surfaces.
+//
+// For a closed genus-g surface cut open via a CutGraph + Euclidean layout:
+//
+//   The universal cover is tiled by copies of the cut-open disk.
+//   The fundamental domain is the polygon whose sides are identified in pairs
+//   by the holonomy generators.
+//
+// ─── Genus-1 (flat torus) ────────────────────────────────────────────────────
+//
+//   Parallelogram with vertices  0,  ω_1,  ω_1 + ω_2,  ω_2.
+//   The four edges are identified in pairs:
+//     bottom (0 → ω_1)   ≡  top    (ω_2 → ω_1 + ω_2)   — translation ω_2
+//     left   (0 → ω_2)   ≡  right  (ω_1 → ω_1 + ω_2)   — translation ω_1
+//
+// ─── Genus g > 1 (general) ──────────────────────────────────────────────────
+//
+//   The standard 4g-polygon with sides labelled a_1 b_1 a_1^{-1} b_1^{-1} ...
+//   can be recovered from the layout boundary, but requires walking the
+//   boundary of the cut-open mesh — not yet implemented (see note below).
+//
+//   For now, this file provides the genus-1 parallelogram only.
+//   The polygon vertices for genus-1 are computed from the holonomy generators.
+//
+// ─── API ─────────────────────────────────────────────────────────────────────
+//
+//   FundamentalDomain fd = compute_fundamental_domain_genus1(holonomy);
+//   fd.vertices          — 2D polygon corners  (size = 4 for genus-1)
+//   fd.edge_identifications — pairs (i, j) meaning edge i is identified with j
+//   fd.is_valid()        — true if genus == 1 and data makes sense
+
+#include "layout.hpp"
+#include "period_matrix.hpp"
+#include <vector>
+#include <utility>
+
+namespace conformallab {
+
+// ─────────────────────────────────────────────────────────────────────────────
+// FundamentalDomain
+// ─────────────────────────────────────────────────────────────────────────────
+
+/// Fundamental polygon of a closed surface obtained by cutting along a
+/// `CutGraph`: corner vertices, paired-edge identifications and holonomy
+/// generators.  For genus-1 the polygon is a parallelogram with 4 corners.
+struct FundamentalDomain {
+    /// Polygon corners in order (CCW).  Size = 4 for genus-1.
+    std::vector<Eigen::Vector2d> vertices;
+
+    /// edge_identifications[k] = (i, j) means the edge from vertices[i] to
+    /// vertices[(i+1) % n] is identified with the edge from vertices[j] to
+    /// vertices[(j+1) % n] (with matching orientation).
+    std::vector<std::pair<int, int>> edge_identifications;
+
+    /// Holonomy generators (one per identified edge pair).
+    /// For genus-1: generators[0] = ω_1, generators[1] = ω_2.
+    std::vector<Eigen::Vector2d> generators;
+
+    /// `true` iff the polygon has at least 3 vertices.
+    bool is_valid() const { return vertices.size() >= 3; }
+};
+
+// ─────────────────────────────────────────────────────────────────────────────
+// compute_fundamental_domain_genus1
+//
+// Builds the parallelogram fundamental domain from Euclidean holonomy data
+// with exactly 2 generators ω_1, ω_2.
+//
+// Vertices (CCW):
+//   v0 = (0, 0)
+//   v1 = ω_1
+//   v2 = ω_1 + ω_2
+//   v3 = ω_2
+//
+// Edge identifications:
+//   bottom (v0→v1) ≡ top    (v3→v2)   by ω_2
+//   left   (v3→v0) ≡ right  (v2→v1)   by ω_1   (reversed convention)
+// ─────────────────────────────────────────────────────────────────────────────
+/// Build the parallelogram fundamental domain from genus-1 Euclidean
+/// holonomy data (`hol.translations[0] = ω₁`, `hol.translations[1] = ω₂`).
+inline FundamentalDomain compute_fundamental_domain_genus1(
+    const HolonomyData& hol)
+{
+    FundamentalDomain fd;
+    if (hol.translations.size() < 2) return fd;
+
+    Eigen::Vector2d w1 = hol.translations[0];
+    Eigen::Vector2d w2 = hol.translations[1];
+
+    // Ensure CCW orientation: cross product z-component w1 × w2 > 0
+    double cross = w1.x() * w2.y() - w1.y() * w2.x();
+    if (cross < 0.0) std::swap(w1, w2);
+
+    Eigen::Vector2d origin = Eigen::Vector2d::Zero();
+    fd.vertices = { origin, w1, w1 + w2, w2 };
+
+    // Edge 0: v0→v1  (= bottom),  Edge 2: v3→v2  (= top, reversed)
+    //   Identification: bottom ≡ top translated by w2
+    // Edge 1: v1→v2  (= right),   Edge 3: v0→v3... wait let me use standard labeling:
+    // Edges by index:  0: v0→v1,  1: v1→v2,  2: v2→v3,  3: v3→v0
+    // Identifications: 0 ≡ 2 (reversed: bottom ≡ top by w2)
+    //                  1 ≡ 3 (reversed: right  ≡ left by w1)
+    fd.edge_identifications = { {0, 2}, {1, 3} };
+    fd.generators = { w1, w2 };
+    return fd;
+}
+
+// ─────────────────────────────────────────────────────────────────────────────
+// compute_fundamental_domain
+//
+// Dispatcher: for genus-1 uses compute_fundamental_domain_genus1.
+// For higher genus returns an empty FundamentalDomain (not yet implemented).
+// ─────────────────────────────────────────────────────────────────────────────
+//
+// TODO(Phase 8): Implement the standard 4g-gon fundamental domain for genus g > 1.
+//
+//   Algorithm outline (boundary-walk method):
+//   ─────────────────────────────────────────
+//   1.  Construct the CutGraph on the cut-open mesh (already done upstream).
+//       This yields 2g cut edges; cutting them converts the closed surface into
+//       a topological disk.
+//
+//   2.  Walk the boundary of the cut-open disk in CCW order:
+//       Start from any boundary halfedge and follow `next(h)` along the boundary
+//       (i.e. skip to the next boundary halfedge at each vertex).  Collect the
+//       2·(4g) = 8g boundary halfedges in order.
+//       Each halfedge h_k corresponds to a UV vertex `halfedge_uv[h_k.idx()]`.
+//
+//   3.  Identify paired sides:
+//       The 4g sides of the polygon alternate as  a_1 b_1 a_1^{-1} b_1^{-1} …
+//       For each cut edge e_i (i = 1 … 2g) the two sides that are identified
+//       are those whose source/target vertices match under the holonomy generator
+//       ω_i (Euclidean) or T_i (hyperbolic).
+//       Record the identifications as  edge_identifications[k] = (i, j).
+//
+//   4.  Fill FundamentalDomain:
+//       vertices              = UV corners from the boundary walk.
+//       edge_identifications  = paired-edge list from step 3.
+//       generators            = holonomy.translations (Euclidean) or the
+//                               fixed points of holonomy.mobius_maps (hyperbolic,
+//                               requires computing axis of T_i ∈ SU(1,1)).
+//
+//   References:
+//     Erickson & Whittlesey, "Greedy optimal homotopy and homology generators"
+//       SODA 2005.
+//     Desbrun, Kanso, Tong, "Discrete Differential Forms for Computational
+//       Modeling", in Discrete Differential Geometry (2008).
+//
+//   Note: The Siegel period matrix Ω ∈ H_g (g×g complex symmetric, Im Ω > 0)
+//   for genus g > 1 also requires integration of holomorphic differentials —
+//   this is intentionally deferred and NOT implemented here.
+//   See period_matrix.hpp for the genus-1 case (τ = ω_2/ω_1 ∈ ℍ).
+// ─────────────────────────────────────────────────────────────────────────────
+/// Dispatcher: for genus 1 returns `compute_fundamental_domain_genus1`,
+/// for higher genus returns an empty domain (4g-polygon not yet implemented).
+inline FundamentalDomain compute_fundamental_domain(
+    const HolonomyData& hol)
+{
+    int n = static_cast<int>(hol.translations.size());
+    int g = n / 2;
+    if (g == 1) return compute_fundamental_domain_genus1(hol);
+    // Higher genus: boundary-walk 4g-polygon — not yet implemented (see TODO above).
+    return FundamentalDomain{};
+}
+
+// ─────────────────────────────────────────────────────────────────────────────
+// tiling_copy
+//
+// Given a Layout2D for the cut-open surface and two lattice generators ω_1, ω_2,
+// return a translated copy of the layout shifted by m·ω_1 + n·ω_2.
+// Useful for visualising the tiled universal cover.
+// ─────────────────────────────────────────────────────────────────────────────
+/// Return a translated copy of `layout` shifted by `m·ω₁ + n·ω₂`.
+/// Useful for visualising the tiled universal cover.
+inline Layout2D tiling_copy(const Layout2D& layout,
+                             const Eigen::Vector2d& w1,
+                             const Eigen::Vector2d& w2,
+                             int m, int n)
+{
+    Layout2D copy = layout;
+    Eigen::Vector2d shift = static_cast<double>(m) * w1
+                          + static_cast<double>(n) * w2;
+    for (auto& p : copy.uv) p += shift;
+    return copy;
+}
+
+// ─────────────────────────────────────────────────────────────────────────────
+// tiling_neighbourhood
+//
+// Returns a vector of tiling copies for (m, n) with |m| ≤ m_max, |n| ≤ n_max.
+// The result includes the original (m=0, n=0) at index (m_max)(2*n_max+1)+n_max.
+// ─────────────────────────────────────────────────────────────────────────────
+/// Build a tiling neighbourhood: all `(m, n)` with `|m| ≤ m_max`,
+/// `|n| ≤ n_max`.  The original tile `(0, 0)` is included.
+inline std::vector<Layout2D> tiling_neighbourhood(
+    const Layout2D&    layout,
+    const HolonomyData& hol,
+    int m_max = 2, int n_max = 2)
+{
+    std::vector<Layout2D> tiles;
+    if (hol.translations.size() < 2) {
+        tiles.push_back(layout);
+        return tiles;
+    }
+    const Eigen::Vector2d& w1 = hol.translations[0];
+    const Eigen::Vector2d& w2 = hol.translations[1];
+    for (int m = -m_max; m <= m_max; ++m)
+        for (int n = -n_max; n <= n_max; ++n)
+            tiles.push_back(tiling_copy(layout, w1, w2, m, n));
+    return tiles;
+}
+
+} // namespace conformallab

code/include/gauss_bonnet.hpp

@@ -0,0 +1,159 @@
+#pragma once
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
+// gauss_bonnet.hpp
+//
+// Phase 6 — Gauss–Bonnet consistency check for prescribed target angles.
+//
+// Before calling newton_*() with custom target angles, verify that
+// the angle defect sum matches the topology:
+//
+//   Σ_v (2π − Θ_v)  =  2π · χ(M)         (Euclidean / flat)
+//   Σ_v (2π − Θ_v)  >  0                  (spherical, χ > 0)
+//   Σ_v (2π − Θ_v)  <  0                  (hyperbolic, χ < 0)
+//
+// If this fails, no conformal factor can realise the target angles and
+// Newton will silently fail to converge.
+//
+// API:
+//   int    euler_characteristic(mesh)
+//   int    genus(mesh)
+//   double gauss_bonnet_sum(mesh, maps)      — Σ(2π − Θ_v)
+//   double gauss_bonnet_rhs(mesh)            — 2π · χ(M)
+//   double gauss_bonnet_deficit(mesh, maps)  — lhs − rhs  (0 = satisfied)
+//   void   check_gauss_bonnet(mesh, maps [, tol])   — throws if violated
+//   void   enforce_gauss_bonnet(mesh, maps)  — shifts θ_v by uniform Δ
+
+#include "conformal_mesh.hpp"
+#include "euclidean_functional.hpp"
+#include "spherical_functional.hpp"
+#include "hyper_ideal_functional.hpp"
+#include "constants.hpp"
+
+#include <stdexcept>
+#include <sstream>
+#include <cmath>
+#include <string>
+
+namespace conformallab {
+
+// ── Topology helpers ──────────────────────────────────────────────────────────
+
+/// Euler characteristic χ = V − E + F.
+/// For closed orientable surfaces: χ = 2 − 2g.
+inline int euler_characteristic(const ConformalMesh& mesh)
+{
+    return static_cast<int>(mesh.number_of_vertices())
+         - static_cast<int>(mesh.number_of_edges())
+         + static_cast<int>(mesh.number_of_faces());
+}
+
+/// Genus of a closed orientable surface: g = (2 − χ) / 2.
+/// Returns 0 for open meshes (boundary present) — callers should check.
+inline int genus(const ConformalMesh& mesh)
+{
+    int chi = euler_characteristic(mesh);
+    return (2 - chi) / 2;
+}
+
+// ── Left-hand side  Σ(2π − Θ_v) ─────────────────────────────────────────────
+
+/// Sum `Σ_v (2π − Θ_v)` for a raw vertex → angle property map.
+inline double gauss_bonnet_sum(
+    const ConformalMesh&                                      mesh,
+    const ConformalMesh::Property_map<Vertex_index, double>&  theta)
+{
+    double s = 0.0;
+    for (auto v : mesh.vertices())
+        s += TWO_PI - theta[v];
+    return s;
+}
+
+/// `gauss_bonnet_sum` for the Euclidean-functional property bundle.
+inline double gauss_bonnet_sum(const ConformalMesh& m, const EuclideanMaps&   mp)
+    { return gauss_bonnet_sum(m, mp.theta_v); }
+/// `gauss_bonnet_sum` for the Spherical-functional property bundle.
+inline double gauss_bonnet_sum(const ConformalMesh& m, const SphericalMaps&   mp)
+    { return gauss_bonnet_sum(m, mp.theta_v); }
+/// `gauss_bonnet_sum` for the HyperIdeal-functional property bundle.
+inline double gauss_bonnet_sum(const ConformalMesh& m, const HyperIdealMaps&  mp)
+    { return gauss_bonnet_sum(m, mp.theta_v); }
+
+// ── Right-hand side  2π · χ(M) ───────────────────────────────────────────────
+
+/// Right-hand side of Gauss-Bonnet: `2π · χ(M)`.
+inline double gauss_bonnet_rhs(const ConformalMesh& mesh)
+{
+    return TWO_PI * static_cast<double>(euler_characteristic(mesh));
+}
+
+// ── Deficit: lhs − rhs  (0 = Gauss–Bonnet satisfied) ─────────────────────────
+
+/// Gauss-Bonnet deficit `lhs − rhs`; zero iff the identity is satisfied.
+template <typename Maps>
+inline double gauss_bonnet_deficit(const ConformalMesh& mesh, const Maps& maps)
+{
+    return gauss_bonnet_sum(mesh, maps) - gauss_bonnet_rhs(mesh);
+}
+
+/// Throws `std::runtime_error` if `|lhs − 2π·χ| > tol`.
+/// Overload accepting a precomputed `lhs`.
+inline void check_gauss_bonnet(const ConformalMesh& mesh,
+                                double lhs,
+                                double tol = 1e-8)
+{
+    double rhs = gauss_bonnet_rhs(mesh);
+    double def = lhs - rhs;
+    if (std::abs(def) > tol) {
+        std::ostringstream msg;
+        msg << "Gauss–Bonnet violated:\n"
+            << "  Σ(2π−Θ_v) = " << lhs
+            << "  expected 2π·χ = " << rhs
+            << "  (χ = " << euler_characteristic(mesh)
+            << ", genus = " << genus(mesh) << ")\n"
+            << "  deficit = " << def;
+        throw std::runtime_error(msg.str());
+    }
+}
+
+/// Throws `std::runtime_error` if Gauss-Bonnet is violated by more than `tol`.
+template <typename Maps>
+inline void check_gauss_bonnet(const ConformalMesh& mesh,
+                                const Maps& maps,
+                                double tol = 1e-8)
+{
+    check_gauss_bonnet(mesh, gauss_bonnet_sum(mesh, maps), tol);
+}
+
+// ── enforce_gauss_bonnet — adjust θ_v by uniform Δ ───────────────────────────
+//
+// Adds δ = (rhs − lhs) / V to every θ_v so that Gauss–Bonnet holds exactly.
+// After this call, check_gauss_bonnet() will not throw (up to floating-point).
+// Only modifies free vertices (v_idx[v] >= 0 for EuclideanMaps / SphericalMaps;
+// always all vertices for the raw property-map overload).
+
+/// Distribute the Gauss-Bonnet deficit uniformly across all `Θ_v`:
+/// add `δ = (lhs − rhs) / V` to every entry so that the identity holds
+/// exactly afterwards.  Overload for a raw property map.
+inline void enforce_gauss_bonnet(
+    ConformalMesh&                                           mesh,
+    ConformalMesh::Property_map<Vertex_index, double>&       theta)
+{
+    double lhs   = gauss_bonnet_sum(mesh, theta);
+    double rhs   = gauss_bonnet_rhs(mesh);
+    // Adding δ to every θ_v decreases the sum Σ(2π−θ_v) by V·δ.
+    // We need lhs − V·δ = rhs, so δ = (lhs − rhs) / V.
+    double delta = (lhs - rhs) / static_cast<double>(mesh.number_of_vertices());
+    for (auto v : mesh.vertices())
+        theta[v] += delta;
+}
+
+/// Distribute the Gauss-Bonnet deficit uniformly across `maps.theta_v`.
+template <typename Maps>
+inline void enforce_gauss_bonnet(ConformalMesh& mesh, Maps& maps)
+{
+    enforce_gauss_bonnet(mesh, maps.theta_v);
+}
+
+} // namespace conformallab

code/include/hyper_ideal_functional.hpp

@@ -0,0 +1,476 @@
+#pragma once
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
+// hyper_ideal_functional.hpp
+//
+// Energy and gradient of the hyper-ideal discrete conformal map functional
+// evaluated on a ConformalMesh (CGAL::Surface_mesh).
+//
+// Ported from de.varylab.discreteconformal.functional.HyperIdealFunctional.
+//
+// ┌─────────────────────────────────────────────────────────────────────────┐
+// │  E(x) = Σ_faces U(f)  −  Σ_edges θ_e · a_e  −  Σ_vertices Θ_v · b_v  │
+// │                                                                         │
+// │  ∂E/∂b_v = Σ_{faces adj. v} β_v(face)  −  Θ_v                         │
+// │  ∂E/∂a_e = α_e(face⁺) + α_e(face⁻)    −  θ_e                         │
+// └─────────────────────────────────────────────────────────────────────────┘
+//
+// DOF vector layout (matches getDimension() ordering):
+//   x[v_idx[v]]  = b_v  for each variable vertex   (log scale factor)
+//   x[e_idx[e]]  = a_e  for each variable edge      (intersection angle)
+//   -1 in v_idx / e_idx means "pinned" (ideal / fixed at 0).
+//
+// Usage
+// ─────
+//   auto mesh = make_tetrahedron();
+//   auto maps = setup_hyper_ideal_maps(mesh);
+//   int n = assign_all_dof_indices(mesh, maps);   // all variable
+//   std::vector<double> x(n, 1.0);
+//   auto res = evaluate_hyper_ideal(mesh, x, maps);
+//   // res.energy, res.gradient
+
+#include "conformal_mesh.hpp"
+#include "hyper_ideal_geometry.hpp"
+#include "hyper_ideal_utility.hpp"
+#include <CGAL/boost/graph/iterator.h>
+#include <vector>
+#include <cmath>
+#include <cstdint>
+
+namespace conformallab {
+
+// ── Property-map type aliases ─────────────────────────────────────────────────
+
+/// Property map vertex → `double` (HyperIdeal scalar-per-vertex data).
+using VMapD = ConformalMesh::Property_map<Vertex_index,   double>;
+/// Property map vertex → `int`    (HyperIdeal DOF indices).
+using VMapI = ConformalMesh::Property_map<Vertex_index,   int>;
+/// Property map edge   → `double` (HyperIdeal scalar-per-edge data).
+using EMapD = ConformalMesh::Property_map<Edge_index,     double>;
+/// Property map edge   → `int`    (HyperIdeal DOF indices).
+using EMapI = ConformalMesh::Property_map<Edge_index,     int>;
+
+// ── Persistent map bundle ─────────────────────────────────────────────────────
+
+/// Bundle of the four property maps consumed by the HyperIdeal functional.
+struct HyperIdealMaps {
+    VMapI v_idx;    ///< DOF index per vertex (−1 = pinned / ideal point).
+    EMapI e_idx;    ///< DOF index per edge   (−1 = fixed).
+    VMapD theta_v;  ///< Target cone angle Θᵥ (parameter, not variable).
+    EMapD theta_e;  ///< Target intersection angle θₑ.
+};
+
+/// Attach the four HyperIdeal property maps to `mesh` and return their
+/// handles.
+///
+/// Defaults:
+/// * `v_idx[v] = -1`  (ideal vertex — i.e. the corresponding `b_v` is fixed at 0)
+/// * `e_idx[e] = -1`  (edge DOF fixed at 0)
+/// * `theta_v[v] = 2π`  (regular cone target)
+/// * `theta_e[e] = π`   (orthogonal-circle target)
+///
+/// The map prefix `"v:"` / `"e:"` is intentionally generic for the
+/// HyperIdeal functional — it is the canonical / Phase 3b model.
+/// Other functionals use distinct prefixes (`"ev:"` Euclidean, `"sv:"`
+/// Spherical, `"cf:"`/`"ce:"` CP-Euclidean, `"iv:"`/`"ie:"`
+/// Inversive-Distance) so all models can coexist on the same mesh.
+inline HyperIdealMaps setup_hyper_ideal_maps(ConformalMesh& mesh)
+{
+    HyperIdealMaps m;
+    m.v_idx   = mesh.add_property_map<Vertex_index, int>   ("v:idx",   -1      ).first;
+    m.e_idx   = mesh.add_property_map<Edge_index,   int>   ("e:idx",   -1      ).first;
+    m.theta_v = mesh.add_property_map<Vertex_index, double>("v:theta", 2.0*PI  ).first;
+    m.theta_e = mesh.add_property_map<Edge_index,   double>("e:theta", PI      ).first;
+    return m;
+}
+
+/// Count free DOFs: `#variable_vertices + #variable_edges`.
+inline int hyper_ideal_dimension(const ConformalMesh& mesh, const HyperIdealMaps& m)
+{
+    int dim = 0;
+    for (auto v : mesh.vertices()) if (m.v_idx[v] >= 0) ++dim;
+    for (auto e : mesh.edges())    if (m.e_idx[e] >= 0) ++dim;
+    return dim;
+}
+
+/// Make every vertex hyper-ideal and every edge variable, assigning
+/// sequential DOF indices `0..n-1` (vertices first, edges after).
+///
+/// This is the standard initialisation for the Springborn-2020
+/// hyper-ideal functional — gauge fixing is **not** needed because
+/// the energy is strictly convex on the full DOF space (no
+/// rotational mode for an all-hyper-ideal configuration).
+///
+/// \returns total DOF count = `num_vertices(mesh) + num_edges(mesh)`.
+inline int assign_all_dof_indices(ConformalMesh& mesh, HyperIdealMaps& m)
+{
+    int idx = 0;
+    for (auto v : mesh.vertices()) m.v_idx[v] = idx++;
+    for (auto e : mesh.edges())    m.e_idx[e] = idx++;
+    return idx;
+}
+
+// ── Evaluation result ─────────────────────────────────────────────────────────
+
+/// Output of `evaluate_hyper_ideal()` — the energy value and (optionally)
+/// its gradient evaluated at the current DOF vector.
+struct HyperIdealResult {
+    double              energy   = 0.0; ///< Functional value at the input DOFs.
+    std::vector<double> gradient;       ///< Gradient ∇E; empty when not requested.
+};
+
+// ── Internal helpers ──────────────────────────────────────────────────────────
+
+/// Read the DOF value from `x` for index `idx`; return 0 if pinned (idx < 0).
+static inline double dof_val(int idx, const std::vector<double>& x)
+{
+    return idx >= 0 ? x[static_cast<std::size_t>(idx)] : 0.0;
+}
+
+/// Convert a CGAL half-edge index to a plain `std::size_t` for vector indexing.
+static inline std::size_t hidx(Halfedge_index h)
+{
+    return static_cast<std::size_t>(static_cast<std::uint32_t>(h));
+}
+
+// ── Pure-math face-angle kernel ──────────────────────────────────────────────
+//
+// Computes the six per-face angle outputs (β₁, β₂, β₃, α₁₂, α₂₃, α₃₁) from
+// the six local DOF inputs (b₁, b₂, b₃, a₁₂, a₂₃, a₃₁) and the variability
+// flags (vᵢb).  This is the pure functional core of `compute_face_angles`
+// — no mesh, no property maps, no global x vector.
+//
+// Why exposed as a free function (Phase 9b):
+// ─────────────────────────────────────────
+// The block-FD Hessian (`hyper_ideal_hessian_block_fd`) perturbs only the
+// 6 DOFs adjacent to a single face at a time, recomputes the 6 angle
+// outputs of that face, and uses the local 6×6 Jacobian to scatter into
+// the global Hessian.  Working through a pure 6→6 function (instead of
+// perturbing the full x and re-running the gradient over all faces)
+// reduces the cost of the Hessian from O(F·n) to O(F·36).
+//
+// The clamping logic (negative b → 0.01, negative a → 0) mirrors
+// HyperIdealFunctional.java's defensive behaviour (lines 122-127 of the
+// Java original); this keeps the FD perturbation regime well-defined.
+
+/// Six per-face angle outputs computed from local DOFs (see
+/// `face_angles_from_local_dofs`).  Used by the block-FD Hessian.
+struct FaceAngleOutputs {
+    double beta1;   ///< Interior angle at v₁.
+    double beta2;   ///< Interior angle at v₂.
+    double beta3;   ///< Interior angle at v₃.
+    double alpha12; ///< Dihedral angle at edge e₁₂.
+    double alpha23; ///< Dihedral angle at edge e₂₃.
+    double alpha31; ///< Dihedral angle at edge e₃₁.
+};
+
+/// Pure-math 6→6 kernel: given the six local DOFs (b₁,b₂,b₃,a₁₂,a₂₃,a₃₁)
+/// of one face plus the per-vertex variability flags, return the six
+/// HyperIdeal angle outputs.  No mesh, no property maps — used by the
+/// per-face block-FD Hessian in `hyper_ideal_hessian.hpp`.
+inline FaceAngleOutputs face_angles_from_local_dofs(
+    double b1, double b2, double b3,
+    double a12, double a23, double a31,
+    bool   v1b, bool v2b, bool v3b)
+{
+    // Same defensive clamps as compute_face_angles.
+    if (v1b && v2b && a12 < 0.0) a12 = 0.0;
+    if (v2b && v3b && a23 < 0.0) a23 = 0.0;
+    if (v3b && v1b && a31 < 0.0) a31 = 0.0;
+    if (v1b && b1  < 0.0) b1 = 0.01;
+    if (v2b && b2  < 0.0) b2 = 0.01;
+    if (v3b && b3  < 0.0) b3 = 0.01;
+
+    double l12 = lij(b1, b2, a12, v1b, v2b);
+    double l23 = lij(b2, b3, a23, v2b, v3b);
+    double l31 = lij(b3, b1, a31, v3b, v1b);
+
+    if (l12 < 1E-12 && l23 < 1E-12 && l31 < 1E-12)
+        l12 = l23 = l31 = 1E-12;
+
+    FaceAngleOutputs o;
+
+    if (l12 > l23 + l31) {
+        o.beta1 = 0.0; o.beta2 = 0.0; o.beta3 = PI;
+        o.alpha12 = PI; o.alpha23 = 0.0; o.alpha31 = 0.0;
+    } else if (l23 > l12 + l31) {
+        o.beta1 = PI; o.beta2 = 0.0; o.beta3 = 0.0;
+        o.alpha12 = 0.0; o.alpha23 = PI; o.alpha31 = 0.0;
+    } else if (l31 > l12 + l23) {
+        o.beta1 = 0.0; o.beta2 = PI; o.beta3 = 0.0;
+        o.alpha12 = 0.0; o.alpha23 = 0.0; o.alpha31 = PI;
+    } else {
+        o.beta1 = zeta(l12, l31, l23);
+        o.beta2 = zeta(l23, l12, l31);
+        o.beta3 = zeta(l31, l23, l12);
+        o.alpha12 = alpha_ij(a12, a23, a31, b1, b2, b3,
+                             o.beta1, o.beta2, o.beta3, v1b, v2b, v3b);
+        o.alpha23 = alpha_ij(a23, a31, a12, b2, b3, b1,
+                             o.beta2, o.beta3, o.beta1, v2b, v3b, v1b);
+        o.alpha31 = alpha_ij(a31, a12, a23, b3, b1, b2,
+                             o.beta3, o.beta1, o.beta2, v3b, v1b, v2b);
+    }
+    return o;
+}
+
+// ── Per-face angle kernel ─────────────────────────────────────────────────────
+
+/// Per-face angle bundle returned by `compute_face_angles()`.  Carries
+/// the six output angles plus the six input DOFs (so the energy and
+/// gradient kernels can reuse them without re-reading the mesh).
+struct FaceAngles {
+    double alpha12; ///< Dihedral angle at edge e₁₂.
+    double alpha23; ///< Dihedral angle at edge e₂₃.
+    double alpha31; ///< Dihedral angle at edge e₃₁.
+    double beta1;   ///< Interior angle at vertex v₁.
+    double beta2;   ///< Interior angle at vertex v₂.
+    double beta3;   ///< Interior angle at vertex v₃.
+    double a12;     ///< Edge DOF value at e₁₂.
+    double a23;     ///< Edge DOF value at e₂₃.
+    double a31;     ///< Edge DOF value at e₃₁.
+    double b1;      ///< Vertex DOF value at v₁.
+    double b2;      ///< Vertex DOF value at v₂.
+    double b3;      ///< Vertex DOF value at v₃.
+    bool   v1b;     ///< `true` iff vertex v₁ is variable (not pinned).
+    bool   v2b;     ///< `true` iff vertex v₂ is variable.
+    bool   v3b;     ///< `true` iff vertex v₃ is variable.
+};
+
+/// Compute the six per-face angles (+ remember the input DOFs) for face
+/// `f` of `mesh`, given the current DOF vector `x` and DOF-index maps.
+static FaceAngles compute_face_angles(
+    const ConformalMesh&       mesh,
+    Face_index                 f,
+    const std::vector<double>& x,
+    const HyperIdealMaps&      m)
+{
+    Halfedge_index h0 = mesh.halfedge(f);
+    Halfedge_index h1 = mesh.next(h0);
+    Halfedge_index h2 = mesh.next(h1);
+
+    Vertex_index v1 = mesh.source(h0);
+    Vertex_index v2 = mesh.source(h1);
+    Vertex_index v3 = mesh.source(h2);
+
+    Edge_index e12 = mesh.edge(h0);
+    Edge_index e23 = mesh.edge(h1);
+    Edge_index e31 = mesh.edge(h2);
+
+    FaceAngles fa;
+    fa.v1b = m.v_idx[v1] >= 0;
+    fa.v2b = m.v_idx[v2] >= 0;
+    fa.v3b = m.v_idx[v3] >= 0;
+
+    fa.a12 = dof_val(m.e_idx[e12], x);
+    fa.a23 = dof_val(m.e_idx[e23], x);
+    fa.a31 = dof_val(m.e_idx[e31], x);
+    fa.b1  = dof_val(m.v_idx[v1],  x);
+    fa.b2  = dof_val(m.v_idx[v2],  x);
+    fa.b3  = dof_val(m.v_idx[v3],  x);
+
+    // Clamp invalid inputs (mirrors Java log.warning + clamp)
+    if (fa.v1b && fa.v2b && fa.a12 < 0.0) fa.a12 = 0.0;
+    if (fa.v2b && fa.v3b && fa.a23 < 0.0) fa.a23 = 0.0;
+    if (fa.v3b && fa.v1b && fa.a31 < 0.0) fa.a31 = 0.0;
+    if (fa.v1b && fa.b1  < 0.0)           fa.b1  = 0.01;
+    if (fa.v2b && fa.b2  < 0.0)           fa.b2  = 0.01;
+    if (fa.v3b && fa.b3  < 0.0)           fa.b3  = 0.01;
+
+    double l12 = lij(fa.b1, fa.b2, fa.a12, fa.v1b, fa.v2b);
+    double l23 = lij(fa.b2, fa.b3, fa.a23, fa.v2b, fa.v3b);
+    double l31 = lij(fa.b3, fa.b1, fa.a31, fa.v3b, fa.v1b);
+
+    // Guard degenerate lengths
+    if (l12 < 1E-12 && l23 < 1E-12 && l31 < 1E-12)
+        l12 = l23 = l31 = 1E-12;
+
+    // Check triangle inequalities; degenerate cases get extreme angles
+    if (l12 > l23 + l31) {
+        fa.beta1 = 0.0; fa.beta2 = 0.0; fa.beta3 = PI;
+        fa.alpha12 = PI; fa.alpha23 = 0.0; fa.alpha31 = 0.0;
+    } else if (l23 > l12 + l31) {
+        fa.beta1 = PI; fa.beta2 = 0.0; fa.beta3 = 0.0;
+        fa.alpha12 = 0.0; fa.alpha23 = PI; fa.alpha31 = 0.0;
+    } else if (l31 > l12 + l23) {
+        fa.beta1 = 0.0; fa.beta2 = PI; fa.beta3 = 0.0;
+        fa.alpha12 = 0.0; fa.alpha23 = 0.0; fa.alpha31 = PI;
+    } else {
+        fa.beta1 = zeta(l12, l31, l23);
+        fa.beta2 = zeta(l23, l12, l31);
+        fa.beta3 = zeta(l31, l23, l12);
+
+        fa.alpha12 = alpha_ij(fa.a12, fa.a23, fa.a31,
+                              fa.b1,  fa.b2,  fa.b3,
+                              fa.beta1, fa.beta2, fa.beta3,
+                              fa.v1b, fa.v2b, fa.v3b);
+        fa.alpha23 = alpha_ij(fa.a23, fa.a31, fa.a12,
+                              fa.b2,  fa.b3,  fa.b1,
+                              fa.beta2, fa.beta3, fa.beta1,
+                              fa.v2b, fa.v3b, fa.v1b);
+        fa.alpha31 = alpha_ij(fa.a31, fa.a12, fa.a23,
+                              fa.b3,  fa.b1,  fa.b2,
+                              fa.beta3, fa.beta1, fa.beta2,
+                              fa.v3b, fa.v1b, fa.v2b);
+    }
+    return fa;
+}
+
+/// Per-face energy contribution U(f) before subtracting the θ·a and Θ·b terms.
+static double face_energy(const FaceAngles& fa)
+{
+    double aa = fa.a12*fa.alpha12 + fa.a23*fa.alpha23 + fa.a31*fa.alpha31;
+    double bb = fa.b1 *fa.beta1   + fa.b2 *fa.beta2   + fa.b3 *fa.beta3;
+
+    double V = 0.0;
+    if (fa.v1b && fa.v2b && fa.v3b) {
+        V = calculateTetrahedronVolume(
+                fa.beta1, fa.beta2, fa.beta3,
+                fa.alpha23, fa.alpha31, fa.alpha12);
+    } else if (!fa.v1b) {
+        V = calculateTetrahedronVolumeWithIdealVertexAtGamma(
+                fa.beta1, fa.alpha31, fa.alpha12,
+                fa.alpha23, fa.beta2, fa.beta3);
+    } else if (!fa.v2b) {
+        V = calculateTetrahedronVolumeWithIdealVertexAtGamma(
+                fa.beta2, fa.alpha12, fa.alpha23,
+                fa.alpha31, fa.beta3, fa.beta1);
+    } else {   // !v3b
+        V = calculateTetrahedronVolumeWithIdealVertexAtGamma(
+                fa.beta3, fa.alpha23, fa.alpha31,
+                fa.alpha12, fa.beta1, fa.beta2);
+    }
+    return aa + bb + 2.0 * V;
+}
+
+// ── Full evaluation ───────────────────────────────────────────────────────────
+
+/// Evaluate the HyperIdeal functional at DOF vector `x`.  Returns the
+/// energy value and (optionally) the gradient in a `HyperIdealResult`.
+///
+/// \param mesh           Triangle mesh carrying the DOF-index property maps.
+/// \param x              Current DOF vector (length = `hyper_ideal_dimension(...)`).
+/// \param m              Property-map bundle from `setup_hyper_ideal_maps(...)`.
+/// \param need_energy    If `true`, fill `result.energy` (default: `true`).
+/// \param need_gradient  If `true`, fill `result.gradient` (default: `true`).
+inline HyperIdealResult evaluate_hyper_ideal(
+    ConformalMesh&             mesh,
+    const std::vector<double>& x,
+    const HyperIdealMaps&      m,
+    bool                       need_energy   = true,
+    bool                       need_gradient = true)
+{
+    HyperIdealResult res;
+
+    // Temporary per-halfedge storage for computed angles.
+    // Indexed by the integer value of Halfedge_index.
+    const std::size_t nh = mesh.number_of_halfedges();
+    std::vector<double> h_alpha(nh, 0.0);  // α_ij stored on halfedge
+    std::vector<double> h_beta (nh, 0.0);  // β_i  stored on opposite halfedge
+
+    // ── Pass 1: angles + energy per face ─────────────────────────────────────
+    for (auto f : mesh.faces()) {
+        Halfedge_index h0 = mesh.halfedge(f);
+        Halfedge_index h1 = mesh.next(h0);
+        Halfedge_index h2 = mesh.next(h1);
+
+        FaceAngles fa = compute_face_angles(mesh, f, x, m);
+
+        // Store computed angles into temporary arrays.
+        // h_alpha[h] = α for the edge of h in this face.
+        h_alpha[hidx(h0)] = fa.alpha12;
+        h_alpha[hidx(h1)] = fa.alpha23;
+        h_alpha[hidx(h2)] = fa.alpha31;
+
+        // h_beta[h] = β at the vertex OPPOSITE to h.
+        // β1 (at v1 = source(h0)) is opposite to h1 = e23.
+        h_beta[hidx(h1)] = fa.beta1;   // h1 is across from v1
+        h_beta[hidx(h2)] = fa.beta2;   // h2 is across from v2
+        h_beta[hidx(h0)] = fa.beta3;   // h0 is across from v3
+
+        if (need_energy)
+            res.energy += face_energy(fa);
+    }
+
+    // ── Pass 2: linear energy terms ──────────────────────────────────────────
+    if (need_energy) {
+        for (auto e : mesh.edges()) {
+            int ie = m.e_idx[e];
+            if (ie >= 0) res.energy -= m.theta_e[e] * x[static_cast<std::size_t>(ie)];
+        }
+        for (auto v : mesh.vertices()) {
+            int iv = m.v_idx[v];
+            if (iv >= 0) res.energy -= m.theta_v[v] * x[static_cast<std::size_t>(iv)];
+        }
+    }
+
+    // ── Pass 3: gradient ─────────────────────────────────────────────────────
+    if (need_gradient) {
+        const int n = hyper_ideal_dimension(mesh, m);
+        res.gradient.assign(static_cast<std::size_t>(n), 0.0);
+
+        // ∂E/∂b_v = Σ_{faces adj. v} β_v(face)  −  Θ_v
+        // β_v(face) = h_beta[prev(h)] for the incoming halfedge h to v in that face.
+        for (auto v : mesh.vertices()) {
+            int iv = m.v_idx[v];
+            if (iv < 0) continue;
+            for (auto h : CGAL::halfedges_around_target(v, mesh)) {
+                if (mesh.is_border(h)) continue;
+                res.gradient[static_cast<std::size_t>(iv)] += h_beta[hidx(mesh.prev(h))];
+            }
+            res.gradient[static_cast<std::size_t>(iv)] -= m.theta_v[v];
+        }
+
+        // ∂E/∂a_e = α_e(face⁺) + α_e(face⁻)  −  θ_e
+        for (auto e : mesh.edges()) {
+            int ie = m.e_idx[e];
+            if (ie < 0) continue;
+            auto h  = mesh.halfedge(e);
+            auto ho = mesh.opposite(h);
+            if (!mesh.is_border(h))  res.gradient[static_cast<std::size_t>(ie)] += h_alpha[hidx(h)];
+            if (!mesh.is_border(ho)) res.gradient[static_cast<std::size_t>(ie)] += h_alpha[hidx(ho)];
+            res.gradient[static_cast<std::size_t>(ie)] -= m.theta_e[e];
+        }
+    }
+
+    return res;
+}
+
+/// Finite-difference gradient check (central differences).
+///
+/// Returns `true` iff `|G[i] − fd[i]| / max(1, |G[i]|) < tol` for every
+/// DOF.  Defaults `eps = 1e-5`, `tol = 1e-4` match the Java `FunctionalTest`.
+inline bool gradient_check(
+    ConformalMesh&             mesh,
+    const std::vector<double>& x0,
+    const HyperIdealMaps&      m,
+    double                     eps = 1E-5,
+    double                     tol = 1E-4)
+{
+    // Analytic gradient
+    auto res = evaluate_hyper_ideal(mesh, x0, m, false, true);
+    const auto& G = res.gradient;
+    const int n = static_cast<int>(G.size());
+
+    std::vector<double> xp = x0, xm = x0;
+    bool ok = true;
+
+    for (int i = 0; i < n; ++i) {
+        std::size_t si = static_cast<std::size_t>(i);
+        xp[si] = x0[si] + eps;
+        xm[si] = x0[si] - eps;
+
+        double Ep = evaluate_hyper_ideal(mesh, xp, m, true, false).energy;
+        double Em = evaluate_hyper_ideal(mesh, xm, m, true, false).energy;
+
+        xp[si] = xm[si] = x0[si];
+
+        double fd    = (Ep - Em) / (2.0 * eps);
+        double err   = std::abs(G[si] - fd);
+        double scale = std::max(1.0, std::abs(G[si]));
+        if (err / scale > tol) ok = false;
+    }
+    return ok;
+}
+
+} // namespace conformallab

code/include/hyper_ideal_geometry.hpp

@@ -0,0 +1,139 @@
+#pragma once
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
+// hyper_ideal_geometry.hpp
+//
+// Pure-math building blocks for the hyper-ideal discrete conformal map.
+// Ported from de.varylab.discreteconformal.functional.HyperIdealUtility
+// and the private helpers of HyperIdealFunctional (lij, αij, σi, σij).
+//
+// All functions are independent of the mesh type.
+//
+// Notation follows the original Java / paper:
+//   b_i, b_j  – vertex variables (log scale factors, hyper-ideal vertices)
+//   a_ij      – edge variable (intersection angle between horocycles)
+//   l_ij      – effective hyperbolic edge length in the auxiliary triangle
+//   β_i       – interior angle of the hyperbolic triangle at vertex i
+//   α_ij      – dihedral angle of the tetrahedron at edge ij
+
+#include "constants.hpp"
+#include <cmath>
+#include <algorithm>
+
+namespace conformallab {
+
+// ── Length functions ─────────────────────────────────────────────────────────
+
+/// `ζ(x,y,z)` — interior angle (in radians) in a hyperbolic triangle
+/// with edge lengths `x`, `y`, `z`, opposite to the side of length `z`.
+/// Ports `HyperIdealUtility.ζ(x, y, z)`.
+inline double zeta(double x, double y, double z)
+{
+    double cx = std::cosh(x), cy = std::cosh(y), cz = std::cosh(z);
+    double sx = std::sinh(x), sy = std::sinh(y);
+    double nbd = (cx*cy - cz) / (sx*sy);
+    nbd = std::clamp(nbd, -1.0, 1.0);   // guard floating-point rounding
+    return std::acos(nbd);
+}
+
+/// `ζ₁₃(x,y,z)` — third edge length in a right-angled hyperbolic hexagon.
+/// Ports `HyperIdealUtility.ζ_13(x, y, z)`.
+inline double zeta13(double x, double y, double z)
+{
+    double cx = std::cosh(x), cy = std::cosh(y), cz = std::cosh(z);
+    double sx = std::sinh(x), sy = std::sinh(y);
+    return std::acosh((cx*cy + cz) / (sx*sy));
+}
+
+/// `ζ₁₄(x,y)` — edge length in a hyperbolic pentagon with one ideal vertex.
+/// Ports `HyperIdealUtility.ζ_14(x, y)`.
+inline double zeta14(double x, double y)
+{
+    double cy = std::cosh(y), sy = std::sinh(y);
+    return std::acosh((std::exp(x) + cy) / sy);
+}
+
+/// `ζ₁₅(x)` — length in a hyperbolic quadrilateral with two ideal vertices.
+/// Ports `HyperIdealUtility.ζ_15(x)`.
+inline double zeta15(double x)
+{
+    return 2.0 * std::asinh(std::exp(x / 2.0));
+}
+
+// ── Effective edge length ─────────────────────────────────────────────────────
+
+/// `l_ij`: effective hyperbolic length of edge ij.
+///   * `bi`, `bj` — vertex log scale factors (used only when vertex is hyper-ideal).
+///   * `aij`      — edge intersection-angle variable.
+///   * `vi_var` / `vj_var` — `true` iff the corresponding vertex is hyper-ideal.
+/// Ports `HyperIdealFunctional.lij()`.
+inline double lij(double bi, double bj, double aij, bool vi_var, bool vj_var)
+{
+    if (vi_var && vj_var) return zeta13(bi, bj, aij);
+    if (vi_var)           return zeta14(aij, bi);
+    if (vj_var)           return zeta14(aij, bj);
+    return zeta15(aij);
+}
+
+// ── Auxiliary angle functions ─────────────────────────────────────────────────
+
+/// `σᵢ(aᵢⱼ, aₖᵢ, aⱼₖ, vj_var, vk_var)` — intermediate half-length at vertex i.
+/// Ports `HyperIdealFunctional.σi()`.
+inline double sigma_i(double aij, double aki, double ajk, bool vj_var, bool vk_var)
+{
+    if (vj_var && vk_var) return zeta13(aij, aki, ajk);
+    if (vj_var)           return zeta14(ajk - aki, aij);
+    if (vk_var)           return zeta14(ajk - aij, aki);
+    return zeta15(ajk - aij - aki);
+}
+
+/// `σᵢⱼ(aᵢⱼ, bᵢ, bⱼ, vj_var)` — intermediate half-length for edge ij from vertex i.
+/// Ports `HyperIdealFunctional.σij()`.
+inline double sigma_ij(double aij, double bi, double bj, bool vj_var)
+{
+    if (vj_var) return zeta13(aij, bi, bj);
+    return zeta14(-aij, bi);
+}
+
+/// `α_ij`: computed dihedral angle at edge ij in the face with vertices i, j, k.
+///
+/// Arguments (cyclic role assignment):
+///   * `aij, ajk, aki` — edge variables.
+///   * `bi, bj, bk`    — vertex variables.
+///   * `beta_i, beta_j, beta_k` — interior angles of the auxiliary hyperbolic triangle.
+///   * `vi_var, vj_var, vk_var` — which vertices are hyper-ideal.
+///
+/// Ports `HyperIdealFunctional.αij()` (the private helper).
+/// Note: the `vk_var` case recurses once (never more than one level deep).
+inline double alpha_ij(
+    double aij, double ajk, double aki,
+    double bi,  double bj,  double bk,
+    double beta_i, double beta_j, double beta_k,
+    bool vi_var, bool vj_var, bool vk_var)
+{
+    if (vi_var) {
+        double si  = sigma_i (aij, aki, ajk, vj_var, vk_var);
+        double sij = sigma_ij(aij, bi, bj, vj_var);
+        double sik = sigma_ij(aki, bi, bk, vk_var);
+        return zeta(si, sij, sik);
+    }
+    if (vj_var) {
+        double sj  = sigma_i (ajk, aij, aki, vk_var, vi_var);
+        double sjk = sigma_ij(ajk, bj, bk, vk_var);
+        double sji = sigma_ij(aij, bj, bi, vi_var);
+        return zeta(sj, sji, sjk);
+    }
+    if (vk_var) {
+        // Derive α_ij from α_jk (one level of recursion).
+        double a_jk = alpha_ij(ajk, aki, aij,
+                               bj, bk, bi,
+                               beta_j, beta_k, beta_i,
+                               vj_var, vk_var, vi_var);
+        return PI - a_jk - beta_j;
+    }
+    // All ideal: closed-form formula.
+    return 0.5 * (PI + beta_k - beta_i - beta_j);
+}
+
+} // namespace conformallab

code/include/hyper_ideal_hessian.hpp

@@ -0,0 +1,231 @@
+#pragma once
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
+// hyper_ideal_hessian.hpp
+//
+// Phase 4a — Hessian of the hyper-ideal discrete conformal functional.
+// Phase 9b — Block-finite-difference Hessian (intermediate optimisation).
+//
+// ┌──────────────────────────────────────────────────────────────────────────┐
+// │  Implementation strategy                                                  │
+// │                                                                           │
+// │  The hyper-ideal functional involves angle functions (ζ, σ, α, β)        │
+// │  composed through several nested layers (lij → ζ13/14/15 → β/α).        │
+// │  Deriving closed-form Hessian entries analytically through all these      │
+// │  layers is feasible but lengthy and is deferred to a future PR.           │
+// │                                                                           │
+// │  TWO Hessian implementations are provided here:                           │
+// │                                                                           │
+// │    1.  `hyper_ideal_hessian` — full finite-difference baseline.           │
+// │        Cost  ≈  n × (cost of full gradient evaluation)                    │
+// │             =  O(n · F)         where n = #DOFs and F = #faces.           │
+// │        Used for correctness reference and small meshes.                   │
+// │                                                                           │
+// │    2.  `hyper_ideal_hessian_block_fd` — block-local finite-difference,    │
+// │        Phase 9b.  Exploits the fact that each face contributes to the     │
+// │        gradient through exactly 6 DOFs (3 vertex b_i + 3 edge a_e).       │
+// │        Cost  ≈  F × 6 × (cost of a single face-angle evaluation)          │
+// │             =  O(36 · F).                                                 │
+// │        Speed-up factor ≈ n / 36, i.e. typically 10–50× on V > 200.        │
+// │                                                                           │
+// │  Both produce the same Hessian to O(ε²) and pass identical PSD checks.    │
+// │  The block-FD variant is the production default; the full-FD variant is   │
+// │  kept for cross-validation tests.                                         │
+// │                                                                           │
+// │  An analytic Hessian via Schläfli-type differentiation through the chain  │
+// │     (bᵢ, aₑ) → lᵢⱼ → ζ₁₃/ζ₁₄/ζ₁₅ → αᵢⱼ / βᵢ                            │
+// │  is deferred to a future PR (Phase 9b-analytic).  Speed-up would be       │
+// │  another ~6×, taking the cost to O(F).                                    │
+// │                                                                           │
+// │  The hyper-ideal energy is strictly convex (Springborn 2020), so H is    │
+// │  positive semi-definite everywhere and Eigen::SimplicialLDLT applies      │
+// │  directly to either Hessian variant.                                      │
+// └──────────────────────────────────────────────────────────────────────────┘
+//
+// Note on the Java reference: HyperIdealFunctional.java line 295-298 declares
+//   public boolean hasHessian() { return false; }
+// — i.e. the upstream Java implementation supplies NO Hessian, analytic or
+// numerical.  Both `hyper_ideal_hessian` and `hyper_ideal_hessian_block_fd`
+// are conformallab++ additions beyond Java parity.
+
+#include "hyper_ideal_functional.hpp"
+#include <Eigen/Sparse>
+#include <vector>
+#include <cmath>
+#include <cstdint>
+
+namespace conformallab {
+
+/// Full finite-difference HyperIdeal Hessian (baseline, Phase 4a).
+/// Cost: `n` full-gradient evaluations ≈ `O(n·F)`.  Use for small
+/// meshes or as a correctness reference for the block-FD variant.
+inline Eigen::SparseMatrix<double> hyper_ideal_hessian(
+    ConformalMesh&             mesh,
+    const std::vector<double>& x,
+    const HyperIdealMaps&      m,
+    double                     eps = 1e-5)
+{
+    const int n = hyper_ideal_dimension(mesh, m);
+    std::vector<Eigen::Triplet<double>> trips;
+    trips.reserve(static_cast<std::size_t>(n * n));
+
+    std::vector<double> xp = x, xm = x;
+
+    for (int j = 0; j < n; ++j) {
+        const std::size_t sj = static_cast<std::size_t>(j);
+        xp[sj] = x[sj] + eps;
+        xm[sj] = x[sj] - eps;
+
+        auto Gp = evaluate_hyper_ideal(mesh, xp, m, /*energy=*/false).gradient;
+        auto Gm = evaluate_hyper_ideal(mesh, xm, m, /*energy=*/false).gradient;
+
+        xp[sj] = xm[sj] = x[sj];   // restore
+
+        for (int i = 0; i < n; ++i) {
+            double val = (Gp[static_cast<std::size_t>(i)]
+                        - Gm[static_cast<std::size_t>(i)]) / (2.0 * eps);
+            if (std::abs(val) > 1e-15)
+                trips.emplace_back(i, j, val);
+        }
+    }
+
+    Eigen::SparseMatrix<double> H(n, n);
+    H.setFromTriplets(trips.begin(), trips.end());
+    return H;
+}
+
+/// Symmetrised full-FD HyperIdeal Hessian: returns `(H + Hᵀ) / 2` to
+/// scrub the tiny asymmetries introduced by floating-point rounding.
+inline Eigen::SparseMatrix<double> hyper_ideal_hessian_sym(
+    ConformalMesh&             mesh,
+    const std::vector<double>& x,
+    const HyperIdealMaps&      m,
+    double                     eps = 1e-5)
+{
+    auto H = hyper_ideal_hessian(mesh, x, m, eps);
+    Eigen::SparseMatrix<double> Ht = H.transpose();
+    return (H + Ht) * 0.5;
+}
+
+// ── Block-FD Hessian (Phase 9b) ──────────────────────────────────────────────
+//
+// Computes the Hessian by FD on each face's 6×6 local block.  The 6 local
+// DOFs of a face f are:
+//      (b_{v1}, b_{v2}, b_{v3}, a_{e12}, a_{e23}, a_{e31}).
+// For each face we recompute the 6 output angles (β₁,β₂,β₃,α₁₂,α₂₃,α₃₁)
+// at x ± ε along each local axis and read off the 6×6 Jacobian.  The result
+// scatters into the global Hessian via the DOF-index lookup.
+//
+// Why this is correct:
+// ─────────────────────
+// The global gradient decomposes by face:
+//     G_b_v = Σ_{f ∋ v} β_v(f)  −  Θ_v
+//     G_a_e = Σ_{f ∋ e} α_e(f)  −  θ_e
+// Since β and α at face f depend ONLY on the 6 local DOFs of f, the
+// Hessian also decomposes:
+//     ∂G_x/∂y = Σ_{f: x,y ∈ local(f)}  ∂(β or α)/∂y  at f.
+// So accumulating per-face 6×6 blocks reproduces the full Hessian.
+//
+// Cost: F × 12 face-angle evaluations  (6 DOFs × 2 directions).
+// On a tetrahedron (F=4, n≈10):       48 face evaluations
+//                                     vs full-FD ≈ 80         → ~1.7× speed-up.
+// On cathead.obj (F=248, n≈400):     2976 face evaluations
+//                                     vs full-FD ≈ 99,200    → ~33× speed-up.
+// On brezel.obj (F=13824, n≈14000): 165 888 face evaluations
+//                                     vs full-FD ≈ 193 M     → ~1166× speed-up.
+/// Per-face block-FD HyperIdeal Hessian (Phase 9b).  Uses the locality
+/// lemma `∂G_x/∂y = Σ_{f: x,y ∈ local(f)} ∂(β or α)/∂y` to perturb only
+/// the 6 face-local DOFs at a time, giving an `F·12` face-evaluation
+/// budget vs `n·F` for full-FD (~96× speed-up on brezel.obj).
+inline Eigen::SparseMatrix<double> hyper_ideal_hessian_block_fd(
+    ConformalMesh&             mesh,
+    const std::vector<double>& x,
+    const HyperIdealMaps&      m,
+    double                     eps = 1e-5)
+{
+    const int n = hyper_ideal_dimension(mesh, m);
+    std::vector<Eigen::Triplet<double>> trips;
+    trips.reserve(36 * mesh.number_of_faces());
+
+    for (auto f : mesh.faces()) {
+        Halfedge_index h0 = mesh.halfedge(f);
+        Halfedge_index h1 = mesh.next(h0);
+        Halfedge_index h2 = mesh.next(h1);
+
+        Vertex_index v1 = mesh.source(h0);
+        Vertex_index v2 = mesh.source(h1);
+        Vertex_index v3 = mesh.source(h2);
+        Edge_index   e12 = mesh.edge(h0);
+        Edge_index   e23 = mesh.edge(h1);
+        Edge_index   e31 = mesh.edge(h2);
+
+        // Local DOF indices: (b1, b2, b3, a12, a23, a31).  Pinned slots = -1.
+        const int idx[6] = {
+            m.v_idx[v1], m.v_idx[v2], m.v_idx[v3],
+            m.e_idx[e12], m.e_idx[e23], m.e_idx[e31]
+        };
+        const bool v1b = idx[0] >= 0;
+        const bool v2b = idx[1] >= 0;
+        const bool v3b = idx[2] >= 0;
+
+        // Local DOF values (0 for pinned).
+        const double vals[6] = {
+            dof_val(idx[0], x), dof_val(idx[1], x), dof_val(idx[2], x),
+            dof_val(idx[3], x), dof_val(idx[4], x), dof_val(idx[5], x)
+        };
+
+        // For each free local DOF, evaluate the 6 outputs at ±ε.
+        // We never perturb a pinned DOF (its column would be physically zero
+        // because it is not part of the DOF vector at all).
+        for (int j = 0; j < 6; ++j) {
+            if (idx[j] < 0) continue;
+
+            double vp[6], vm[6];
+            for (int k = 0; k < 6; ++k) { vp[k] = vm[k] = vals[k]; }
+            vp[j] += eps;
+            vm[j] -= eps;
+
+            auto Op = face_angles_from_local_dofs(
+                vp[0], vp[1], vp[2], vp[3], vp[4], vp[5], v1b, v2b, v3b);
+            auto Om = face_angles_from_local_dofs(
+                vm[0], vm[1], vm[2], vm[3], vm[4], vm[5], v1b, v2b, v3b);
+
+            const double Gp[6] = {
+                Op.beta1, Op.beta2, Op.beta3,
+                Op.alpha12, Op.alpha23, Op.alpha31
+            };
+            const double Gm[6] = {
+                Om.beta1, Om.beta2, Om.beta3,
+                Om.alpha12, Om.alpha23, Om.alpha31
+            };
+
+            for (int i = 0; i < 6; ++i) {
+                if (idx[i] < 0) continue;   // pinned: contributes nothing
+                const double val = (Gp[i] - Gm[i]) / (2.0 * eps);
+                if (std::abs(val) > 1e-15)
+                    trips.emplace_back(idx[i], idx[j], val);
+            }
+        }
+    }
+
+    Eigen::SparseMatrix<double> H(n, n);
+    H.setFromTriplets(trips.begin(), trips.end());
+    return H;
+}
+
+/// Symmetrised block-FD HyperIdeal Hessian: returns `(H + Hᵀ) / 2` of
+/// `hyper_ideal_hessian_block_fd(...)` for downstream solvers that
+/// require strict symmetry.
+inline Eigen::SparseMatrix<double> hyper_ideal_hessian_block_fd_sym(
+    ConformalMesh&             mesh,
+    const std::vector<double>& x,
+    const HyperIdealMaps&      m,
+    double                     eps = 1e-5)
+{
+    auto H = hyper_ideal_hessian_block_fd(mesh, x, m, eps);
+    Eigen::SparseMatrix<double> Ht = H.transpose();
+    return (H + Ht) * 0.5;
+}
+
+} // namespace conformallab

code/include/hyper_ideal_utility.hpp

@@ -1,9 +1,13 @@
 #pragma once
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
 
 // Hyperbolic tetrahedron volume formulas.
 // Ported from de.varylab.discreteconformal.functional.HyperIdealUtility (Java).
 
 #include "clausen.hpp"
+#include "constants.hpp"
 
 #include <Eigen/Dense>
 #include <cmath>
@@ -11,15 +15,15 @@
 
 namespace conformallab {
 
-// Volume of a generalized hyperbolic tetrahedron with dihedral angles A..F.
-// Formula: Meyerhoff / Ushijima (Springer 2006).
-// Corresponds to Java HyperIdealUtility.calculateTetrahedronVolume().
+/// Volume of a generalized hyperbolic tetrahedron with dihedral
+/// angles `A,…,F` via the Meyerhoff / Ushijima 2006 formula.
+/// Same as Java `HyperIdealUtility.calculateTetrahedronVolume()`.
 inline double calculateTetrahedronVolume(double A, double B, double C,
                                          double D, double E, double F) {
-    constexpr double pi = 3.14159265358979323846264338328;
+    // PI from constants.hpp (conformallab::PI)
 
     // Degenerate if any angle equals pi.
-    if (A == pi || B == pi || C == pi || D == pi || E == pi || F == pi)
+    if (A == PI || B == PI || C == PI || D == PI || E == PI || F == PI)
         return 0.0;
 
     const double sA = std::sin(A), sB = std::sin(B), sC = std::sin(C);
@@ -72,35 +76,33 @@ inline double calculateTetrahedronVolume(double A, double B, double C,
     return (U(z1) - U(z2)) / 2.0;
 }
 
-// Volume of a hyperideal tetrahedron with one ideal vertex (at gamma).
-// Dihedral angles at the ideal vertex: gamma1, gamma2, gamma3.
-// Dihedral angles at opposite edges: alpha23, alpha31, alpha12.
-// Formula: Kolpakov–Mednykh (arxiv math/0603097).
-// Corresponds to Java HyperIdealUtility.calculateTetrahedronVolumeWithIdealVertexAtGamma().
+/// Volume of a hyperideal tetrahedron with one ideal vertex at γ via
+/// the Kolpakov-Mednykh formula (arxiv math/0603097).  Same as Java
+/// `HyperIdealUtility.calculateTetrahedronVolumeWithIdealVertexAtGamma()`.
 inline double calculateTetrahedronVolumeWithIdealVertexAtGamma(
     double gamma1, double gamma2, double gamma3,
     double alpha23, double alpha31, double alpha12)
 {
-    constexpr double pi = 3.14159265358979323846264338328;
+    // PI from constants.hpp (conformallab::PI)
     auto L = [](double x) { return Lobachevsky(x); };
 
     double result = L(gamma1) + L(gamma2) + L(gamma3);
 
-    result += L((pi + alpha31 - alpha12 - gamma1) / 2.0);
-    result += L((pi + alpha12 - alpha23 - gamma2) / 2.0);
-    result += L((pi + alpha23 - alpha31 - gamma3) / 2.0);
+    result += L((PI + alpha31 - alpha12 - gamma1) / 2.0);
+    result += L((PI + alpha12 - alpha23 - gamma2) / 2.0);
+    result += L((PI + alpha23 - alpha31 - gamma3) / 2.0);
 
-    result += L((pi - alpha31 + alpha12 - gamma1) / 2.0);
-    result += L((pi - alpha12 + alpha23 - gamma2) / 2.0);
-    result += L((pi - alpha23 + alpha31 - gamma3) / 2.0);
+    result += L((PI - alpha31 + alpha12 - gamma1) / 2.0);
+    result += L((PI - alpha12 + alpha23 - gamma2) / 2.0);
+    result += L((PI - alpha23 + alpha31 - gamma3) / 2.0);
 
-    result += L((pi + alpha31 + alpha12 - gamma1) / 2.0);
-    result += L((pi + alpha12 + alpha23 - gamma2) / 2.0);
-    result += L((pi + alpha23 + alpha31 - gamma3) / 2.0);
+    result += L((PI + alpha31 + alpha12 - gamma1) / 2.0);
+    result += L((PI + alpha12 + alpha23 - gamma2) / 2.0);
+    result += L((PI + alpha23 + alpha31 - gamma3) / 2.0);
 
-    result += L((pi - alpha31 - alpha12 - gamma1) / 2.0);
-    result += L((pi - alpha12 - alpha23 - gamma2) / 2.0);
-    result += L((pi - alpha23 - alpha31 - gamma3) / 2.0);
+    result += L((PI - alpha31 - alpha12 - gamma1) / 2.0);
+    result += L((PI - alpha12 - alpha23 - gamma2) / 2.0);
+    result += L((PI - alpha23 - alpha31 - gamma3) / 2.0);
 
     return result / 2.0;
 }

code/include/hyper_ideal_visualization_utility.hpp

@@ -0,0 +1,131 @@
+#pragma once
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
+// Port of the static helper
+//   HyperIdealVisualizationPlugin.getEuclideanCircleFromHyperbolic()
+// from de.varylab.discreteconformal.plugin.
+//
+// Converts a hyperbolic circle (center + radius in the hyperboloid model)
+// to its Euclidean representation (cx, cy, r) in the Poincaré disk model.
+//
+// Mathematical background
+// -----------------------
+// Hyperboloid model: points (x,y,z,w) with w²-x²-y²-z²=1, w>0.
+// Metric signature: g = diag(+1,+1,+1,−1)  (spatial-first, time-last).
+//
+// Hyperbolic translation from the origin e₄=(0,0,0,1) to p=(a,b,c,d):
+//   T = [ I₃ + p'·p'ᵀ/(d+1)   p' ]      p' = (a,b,c)
+//       [        p'ᵀ            d  ]
+// This is the standard Lorentz boost; it is in O(3,1) and maps e₄ → p.
+//
+// Poincaré disk projection (jReality convention):
+//   (x,y,z,w)  →  (x,y) / (w+1)
+//
+// The three reference points on the unit hyperbolic circle (at origin) are
+//   p1 = (sinh r, 0, 0, cosh r)
+//   p2 = (0, sinh r, 0, cosh r)
+//   p3 = (-sinh r, 0, 0, cosh r)
+// After translation and projection to the Poincaré disk their circumcircle
+// equals the image of the original hyperbolic circle.
+
+#include <Eigen/Core>  // downgraded from <Eigen/Dense>: this header only
+                      // uses Matrix/Vector primitives, no decompositions.
+#include <array>
+#include <cmath>
+
+namespace conformallab {
+
+/// Circumcenter of three 2-D points (`a`, `b`, `c`) in the Euclidean plane.
+inline Eigen::Vector2d circumcenter2d(
+    const Eigen::Vector2d& a,
+    const Eigen::Vector2d& b,
+    const Eigen::Vector2d& c)
+{
+    double ax = a.x(), ay = a.y();
+    double bx = b.x(), by = b.y();
+    double cx = c.x(), cy = c.y();
+
+    double D = 2.0 * (ax*(by - cy) + bx*(cy - ay) + cx*(ay - by));
+    double a2 = ax*ax + ay*ay;
+    double b2 = bx*bx + by*by;
+    double c2 = cx*cx + cy*cy;
+
+    double ux = (a2*(by - cy) + b2*(cy - ay) + c2*(ay - by)) / D;
+    double uy = (a2*(cx - bx) + b2*(ax - cx) + c2*(bx - ax)) / D;
+    return {ux, uy};
+}
+
+/// 4×4 Lorentz boost: maps the hyperboloid origin `e₄ = (0,0,0,1)` to
+/// `center`.  Precondition: `center` lies on the hyperboloid.
+inline Eigen::Matrix4d hyperboloidTranslation(const Eigen::Vector4d& center)
+{
+    Eigen::Vector3d p = center.head<3>();
+    double          d = center(3);
+
+    Eigen::Matrix4d T = Eigen::Matrix4d::Identity();
+    // Upper-left 3×3 block: I + p'·p'ᵀ / (d+1)
+    T.block<3,3>(0,0) += p * p.transpose() / (d + 1.0);
+    // Right column and bottom row
+    T.block<3,1>(0,3) = p;
+    T.block<1,3>(3,0) = p.transpose();
+    T(3,3) = d;
+    return T;
+}
+
+/// Project a hyperboloid point `x` onto the Poincaré disk (jReality
+/// convention: add 1 to the w-coordinate, then dehomogenise spatial part).
+inline Eigen::Vector2d toPoincareDisk(const Eigen::Vector4d& x)
+{
+    double w = x(3) + 1.0;
+    return {x(0) / w, x(1) / w};
+}
+
+// ---------------------------------------------------------------------------
+// getEuclideanCircleFromHyperbolic
+//
+// Inputs
+//   center  – point on the hyperboloid, e.g. (0,0,0,1) for the origin
+//   radius  – hyperbolic radius (real number > 0)
+//
+// Output
+//   { euclidean_cx, euclidean_cy, euclidean_radius }
+//   describing the circle in the Poincaré disk that corresponds to the
+//   given hyperbolic circle.
+//
+// Port of HyperIdealVisualizationPlugin.getEuclideanCircleFromHyperbolic()
+// ---------------------------------------------------------------------------
+/// Convert a hyperbolic circle (`center` on the hyperboloid, hyperbolic
+/// `radius`) to the corresponding Euclidean circle in the Poincaré disk;
+/// returns `{cx, cy, r}`.  Port of `HyperIdealVisualizationPlugin
+/// .getEuclideanCircleFromHyperbolic()`.
+inline std::array<double,3> getEuclideanCircleFromHyperbolic(
+    const Eigen::Vector4d& center, double radius)
+{
+    const double s  = std::sinh(radius);
+    const double ch = std::cosh(radius);
+
+    // Three points on the hyperbolic circle centered at the origin
+    Eigen::Vector4d p1( s, 0.0, 0.0, ch);
+    Eigen::Vector4d p2(0.0,  s, 0.0, ch);
+    Eigen::Vector4d p3(-s, 0.0, 0.0, ch);
+
+    // Apply the hyperbolic translation to the target center
+    const Eigen::Matrix4d T = hyperboloidTranslation(center);
+    p1 = T * p1;
+    p2 = T * p2;
+    p3 = T * p3;
+
+    // Project to the Poincaré disk
+    const Eigen::Vector2d q1 = toPoincareDisk(p1);
+    const Eigen::Vector2d q2 = toPoincareDisk(p2);
+    const Eigen::Vector2d q3 = toPoincareDisk(p3);
+
+    // Euclidean circumcircle of the three projected points
+    const Eigen::Vector2d ec = circumcenter2d(q1, q2, q3);
+    const double           r  = (ec - q1).norm();
+
+    return {ec.x(), ec.y(), r};
+}
+
+} // namespace conformallab

code/include/inversive_distance_functional.hpp

@@ -0,0 +1,378 @@
+#pragma once
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
+// inversive_distance_functional.hpp
+//
+// Phase 9a.2 — Inversive-distance circle-packing functional (Luo 2004).
+//
+// VERTEX-based circle packing.  Each vertex carries a circle of radius
+// r_i = exp(u_i).  The inversive distance I_ij between two adjacent
+// circles is a constant of the edge, derived once from the initial
+// geometry via Bowers-Stephenson 2004.
+//
+// This is the FACE-DUAL of CPEuclideanFunctional (Phase 9a.1).  The
+// correspondence is I_ij = cos θ_e (Glickenstein 2011 §5).
+//
+// ┌──────────────────────────────────────────────────────────────────────────┐
+// │  Mathematical model                                                       │
+// │  ──────────────────                                                       │
+// │                                                                           │
+// │  Variables:    u_i = log r_i           (per vertex; r_i is the radius)   │
+// │  Constants:    I_ij                    (per edge; inversive distance)    │
+// │                Θ_v                     (per vertex; target cone angle)   │
+// │                                                                           │
+// │  Bowers-Stephenson (init from initial geometry):                          │
+// │    I_ij = ( ℓ_ij² − r_i² − r_j² ) / ( 2 r_i r_j )                        │
+// │                                                                           │
+// │  Edge length (Luo 2004 §3, Glickenstein 2011 eq. 2.1):                   │
+// │    ℓ_ij(u)²  =  exp(2 u_i) + exp(2 u_j) + 2 I_ij exp(u_i + u_j)         │
+// │             =  r_i² + r_j² + 2 I_ij r_i r_j                              │
+// │                                                                           │
+// │  Triangle angles:    same half-tangent law of cosines as the              │
+// │                      Euclidean functional (numerically stable).           │
+// │                                                                           │
+// │  Gradient  (Luo 2004 Lemma 3.1):                                          │
+// │    ∂E/∂u_v  =  Θ_v  −  Σ_{T ∋ v}  α_v(T)                                 │
+// │                                                                           │
+// │  Energy:     path integral E(u) = ∫₀¹ ⟨G(tu), u⟩ dt                      │
+// │              (Luo's 1-form is closed; we use 10-point Gauss-Legendre      │
+// │               quadrature, identical to euclidean_functional.hpp)          │
+// │                                                                           │
+// │  Hessian:    finite-difference for the MVP port; an analytic form is      │
+// │              given in Glickenstein 2011 eq. (4.6) and may be added        │
+// │              later for performance.                                       │
+// └──────────────────────────────────────────────────────────────────────────┘
+//
+// Relation to euclidean_functional.hpp
+// ────────────────────────────────────
+// The two are structurally identical in:
+//   • DOF layout (per vertex), DOF index sentinel (−1 = pinned)
+//   • Gradient pattern (Θ − Σ α)
+//   • Energy via path integral (same Gauss-Legendre constants)
+//   • Halfedge convention (h0/h1/h2, source pattern, α opposite-edge)
+//
+// They differ ONLY in:
+//   • Per-edge constant: λ°_ij (log²-length) vs I_ij (inversive distance)
+//   • Edge-length formula:
+//       Euclidean:           ℓ_ij  = exp((λ°_ij + u_i + u_j) / 2)
+//       Inversive distance:  ℓ_ij² = exp(2u_i) + exp(2u_j)
+//                                  + 2 I_ij exp(u_i + u_j)
+//
+// In particular at the tangential limit I_ij = 1 the inversive-distance length
+// reduces to (exp(u_i) + exp(u_j))² ⇒ ℓ_ij = r_i + r_j (tangential circles),
+// which is *different* from the Euclidean-conformal length even at the same
+// initial geometry.  The two functionals describe distinct geometric objects.
+//
+// Property-map name prefix: "iv:" (vertex) and "ie:" (edge).
+
+#include "conformal_mesh.hpp"
+#include "constants.hpp"
+#include "euclidean_geometry.hpp"   // euclidean_angles(λ12, λ23, λ31)
+#include <CGAL/boost/graph/iterator.h>
+#include <vector>
+#include <cmath>
+#include <cstdint>
+#include <iostream>
+
+namespace conformallab {
+
+// ── Property-map type aliases ────────────────────────────────────────────────
+
+/// Property map vertex → `int`    for the Inversive-Distance functional.
+using IDVMapI = ConformalMesh::Property_map<Vertex_index, int>;
+/// Property map vertex → `double` for the Inversive-Distance functional.
+using IDVMapD = ConformalMesh::Property_map<Vertex_index, double>;
+/// Property map edge   → `double` for the Inversive-Distance functional.
+using IDEMapD = ConformalMesh::Property_map<Edge_index, double>;
+
+// ── Persistent map bundle ─────────────────────────────────────────────────────
+
+/// Bundle of the four property maps consumed by the Inversive-Distance
+/// circle-packing functional (Luo 2004 / Bowers-Stephenson 2004).
+struct InversiveDistanceMaps {
+    IDVMapI v_idx;       ///< DOF index per vertex (−1 = pinned / u_v = 0)
+    IDVMapD theta_v;     ///< target cone angle Θ_v   (default 2π)
+    IDVMapD r0;          ///< initial radius r_i^(0)  (default 1)
+    IDEMapD I_e;         ///< inversive distance I_ij (per edge, constant)
+};
+
+/// Attach the four inversive-distance property maps to `mesh` and
+/// return their handles.
+///
+/// Defaults are intentionally trivial — every real use of this
+/// functional must call `compute_inversive_distance_init_from_mesh()`
+/// next to populate `r0` and `I_e` from the input geometry.
+/// * `v_idx[v] = -1` (all vertices pinned initially)
+/// * `theta_v[v] = 2π` (regular interior vertex)
+/// * `r0[v] = 1.0` (placeholder)
+/// * `I_e[e] = 1.0` (tangential default — overwritten by init step)
+///
+/// The maps use the `"iv:"` / `"ie:"` prefix so they do not collide
+/// with the Euclidean / Spherical / HyperIdeal / CP-Euclidean maps.
+inline InversiveDistanceMaps setup_inversive_distance_maps(ConformalMesh& mesh)
+{
+    InversiveDistanceMaps m;
+    m.v_idx   = mesh.add_property_map<Vertex_index, int>   ("iv:idx",   -1     ).first;
+    m.theta_v = mesh.add_property_map<Vertex_index, double>("iv:theta", TWO_PI ).first;
+    m.r0      = mesh.add_property_map<Vertex_index, double>("iv:r0",    1.0    ).first;
+    m.I_e     = mesh.add_property_map<Edge_index,   double>("ie:I",     1.0    ).first;
+    return m;
+}
+
+/// Assign sequential DOF indices `0..n-1` to every vertex.
+///
+/// **Note:** this overload does NOT pin a gauge vertex.  The caller
+/// is expected to either:
+/// 1. set one `m.v_idx[v] = -1` *before* calling this function (then
+///    the call is a no-op for that vertex) — OR —
+/// 2. flip one assigned index back to `-1` *after* this function.
+///
+/// For a closed mesh, exactly one pin is required to remove the
+/// global rotational mode.
+inline int assign_inversive_distance_vertex_dof_indices(ConformalMesh&         mesh,
+                                                          InversiveDistanceMaps& m)
+{
+    int idx = 0;
+    for (auto v : mesh.vertices()) m.v_idx[v] = idx++;
+    return idx;
+}
+
+/// Count the free DOFs (vertices with `v_idx >= 0`).
+inline int inversive_distance_dimension(const ConformalMesh&         mesh,
+                                          const InversiveDistanceMaps& m)
+{
+    int dim = 0;
+    for (auto v : mesh.vertices()) if (m.v_idx[v] >= 0) ++dim;
+    return dim;
+}
+
+/// Two-phase initialisation from initial mesh geometry.  Mirrors the
+/// role of `compute_lambda0_from_mesh` in the Euclidean functional, but
+/// adapted to Luo's vertex-based radius parametrisation.
+///
+/// **Phase 1.** Pick a positive radius per vertex:
+/// \code
+///   r_i^(0) = (1/3) · min{ℓ_e : e adjacent to v_i}
+/// \endcode
+/// This is a heuristic — the user may override `m.r0[v]` for any
+/// vertex between `setup_inversive_distance_maps()` and this call.
+///
+/// **Phase 2.** Compute the per-edge inversive distance via the
+/// Bowers-Stephenson 2004 identity:
+/// \code
+///   I_ij = ( ℓ_ij² − r_i² − r_j² ) / ( 2 r_i r_j )
+/// \endcode
+///
+/// \pre  Every edge has positive 3-D length.
+/// \pre  Radii produced in Phase 1 are positive (degenerate isolated
+///       vertices fall back to `r_i = 1`).
+/// \post Every `I_e[e] > -1` for a valid packing.  The chosen
+///       Phase-1 heuristic keeps `I_e > 0` for most real meshes.
+inline void compute_inversive_distance_init_from_mesh(ConformalMesh&         mesh,
+                                                        InversiveDistanceMaps& m)
+{
+    // Phase 1: r_i = (1/3) · min adjacent edge length.
+    for (auto v : mesh.vertices()) {
+        double min_len = std::numeric_limits<double>::infinity();
+        for (auto h : CGAL::halfedges_around_target(v, mesh)) {
+            auto p1 = mesh.point(mesh.source(h));
+            auto p2 = mesh.point(mesh.target(h));
+            double dx = p1.x() - p2.x();
+            double dy = p1.y() - p2.y();
+            double dz = p1.z() - p2.z();
+            double len = std::sqrt(dx*dx + dy*dy + dz*dz);
+            if (len < min_len) min_len = len;
+        }
+        m.r0[v] = (std::isfinite(min_len) && min_len > 1e-15)
+                ? min_len / 3.0
+                : 1.0;
+    }
+
+    // Phase 2: I_ij from initial geometry.
+    for (auto e : mesh.edges()) {
+        auto h  = mesh.halfedge(e);
+        auto vi = mesh.source(h);
+        auto vj = mesh.target(h);
+        auto p1 = mesh.point(vi);
+        auto p2 = mesh.point(vj);
+        double dx = p1.x() - p2.x();
+        double dy = p1.y() - p2.y();
+        double dz = p1.z() - p2.z();
+        double l2 = dx*dx + dy*dy + dz*dz;
+        double ri = m.r0[vi];
+        double rj = m.r0[vj];
+        m.I_e[e] = (l2 - ri*ri - rj*rj) / (2.0 * ri * rj);
+    }
+}
+
+// ── Internal helpers ──────────────────────────────────────────────────────────
+
+namespace id_detail {
+
+inline double dof_val(int idx, const std::vector<double>& x) noexcept
+{
+    return idx >= 0 ? x[static_cast<std::size_t>(idx)] : 0.0;
+}
+
+inline std::size_t hidx(Halfedge_index h) noexcept
+{
+    return static_cast<std::size_t>(static_cast<std::uint32_t>(h));
+}
+
+// Inversive-distance edge length squared: ℓ² = exp(2u_i) + exp(2u_j) + 2 I r_i r_j
+// where r_i = exp(u_i), so:  ℓ² = r_i² + r_j² + 2 I r_i r_j.
+// Returns -1 if the result is non-positive (degenerate; the caller skips the face).
+inline double edge_length_squared(double u_i, double u_j, double I_ij) noexcept
+{
+    double ri = std::exp(u_i);
+    double rj = std::exp(u_j);
+    double l2 = ri*ri + rj*rj + 2.0 * I_ij * ri * rj;
+    return l2 > 0.0 ? l2 : -1.0;
+}
+
+} // namespace id_detail
+
+/// Inversive-Distance gradient `G_v = Θ_v − Σ_faces α_v(face)`.  Same
+/// half-edge corner-angle storage convention as `euclidean_gradient`.
+inline std::vector<double> inversive_distance_gradient(
+    const ConformalMesh&             mesh,
+    const std::vector<double>&       x,
+    const InversiveDistanceMaps&     m)
+{
+    const int n = inversive_distance_dimension(mesh, m);
+    std::vector<double> G(static_cast<std::size_t>(n), 0.0);
+
+    const std::size_t nh = mesh.number_of_halfedges();
+    std::vector<double> h_alpha(nh, 0.0);
+
+    // Pass 1 — per face, compute corner angles via the law of cosines.
+    // We reuse euclidean_angles(λ12, λ23, λ31) which takes 2·log(ℓ) per edge.
+    for (auto f : mesh.faces()) {
+        Halfedge_index h0 = mesh.halfedge(f);
+        Halfedge_index h1 = mesh.next(h0);
+        Halfedge_index h2 = mesh.next(h1);
+
+        Vertex_index v1 = mesh.source(h0);
+        Vertex_index v2 = mesh.source(h1);
+        Vertex_index v3 = mesh.source(h2);
+
+        Edge_index e12 = mesh.edge(h0);
+        Edge_index e23 = mesh.edge(h1);
+        Edge_index e31 = mesh.edge(h2);
+
+        double u1 = id_detail::dof_val(m.v_idx[v1], x);
+        double u2 = id_detail::dof_val(m.v_idx[v2], x);
+        double u3 = id_detail::dof_val(m.v_idx[v3], x);
+
+        double l12sq = id_detail::edge_length_squared(u1, u2, m.I_e[e12]);
+        double l23sq = id_detail::edge_length_squared(u2, u3, m.I_e[e23]);
+        double l31sq = id_detail::edge_length_squared(u3, u1, m.I_e[e31]);
+
+        if (l12sq <= 0 || l23sq <= 0 || l31sq <= 0) continue;
+
+        // euclidean_angles expects 2·log(ℓ) per edge — feed log(ℓ²).
+        auto fa = euclidean_angles(std::log(l12sq), std::log(l23sq), std::log(l31sq));
+        if (!fa.valid) continue;
+
+        h_alpha[id_detail::hidx(h0)] = fa.alpha3;
+        h_alpha[id_detail::hidx(h1)] = fa.alpha1;
+        h_alpha[id_detail::hidx(h2)] = fa.alpha2;
+    }
+
+    // Pass 2 — accumulate vertex gradient.
+    for (auto v : mesh.vertices()) {
+        int iv = m.v_idx[v];
+        if (iv < 0) continue;
+        double sum_alpha = 0.0;
+        for (auto h : CGAL::halfedges_around_target(v, mesh)) {
+            if (mesh.is_border(h)) continue;
+            sum_alpha += h_alpha[id_detail::hidx(mesh.prev(h))];
+        }
+        G[static_cast<std::size_t>(iv)] = m.theta_v[v] - sum_alpha;
+    }
+
+    return G;
+}
+
+/// Inversive-Distance energy `E(u) = ∫₀¹ ⟨G(t·u), u⟩ dt`, evaluated
+/// with 10-point Gauss-Legendre (constants shared with `euclidean_energy`).
+inline double inversive_distance_energy(
+    const ConformalMesh&             mesh,
+    const std::vector<double>&       x,
+    const InversiveDistanceMaps&     m)
+{
+    static const double gl_s[10] = {
+        -0.9739065285171717,  -0.8650633666889845,
+        -0.6794095682990244,  -0.4333953941292472,
+        -0.1488743389816312,   0.1488743389816312,
+         0.4333953941292472,   0.6794095682990244,
+         0.8650633666889845,   0.9739065285171717
+    };
+    static const double gl_w[10] = {
+        0.0666713443086881,  0.1494513491505806,
+        0.2190863625159820,  0.2692667193099963,
+        0.2955242247147529,  0.2955242247147529,
+        0.2692667193099963,  0.2190863625159820,
+        0.1494513491505806,  0.0666713443086881
+    };
+
+    const std::size_t n = x.size();
+    double E = 0.0;
+    std::vector<double> tx(n);
+    for (int k = 0; k < 10; ++k) {
+        double t  = (1.0 + gl_s[k]) * 0.5;
+        double wt = gl_w[k] * 0.5;
+        for (std::size_t i = 0; i < n; ++i) tx[i] = t * x[i];
+        auto G = inversive_distance_gradient(mesh, tx, m);
+        double dot = 0.0;
+        for (std::size_t i = 0; i < n; ++i) dot += G[i] * x[i];
+        E += wt * dot;
+    }
+    return E;
+}
+
+/// FD gradient check for the Inversive-Distance functional (central diff).
+inline bool gradient_check_inversive_distance(
+    const ConformalMesh&             mesh,
+    const std::vector<double>&       x,
+    const InversiveDistanceMaps&     m,
+    double eps = 1e-5,
+    double tol = 1e-6)
+{
+    auto G = inversive_distance_gradient(mesh, x, m);
+    const std::size_t n = G.size();
+
+    for (std::size_t i = 0; i < n; ++i) {
+        std::vector<double> xp = x, xm = x;
+        xp[i] += eps;
+        xm[i] -= eps;
+        double Ep = inversive_distance_energy(mesh, xp, m);
+        double Em = inversive_distance_energy(mesh, xm, m);
+        double fd = (Ep - Em) / (2.0 * eps);
+        if (std::abs(G[i] - fd) > tol) {
+            std::cerr << "[inversive-distance] FD gradient mismatch at DOF " << i
+                      << ": analytic=" << G[i]
+                      << "  FD=" << fd
+                      << "  diff=" << (G[i] - fd) << "\n";
+            return false;
+        }
+    }
+    return true;
+}
+
+/// Newton equilibrium check: returns `true` iff the gradient at `x`
+/// is below `tol` in infinity norm (Σ adj-face angles equal Θ_v).
+inline bool is_inversive_distance_equilibrium(
+    const ConformalMesh&             mesh,
+    const std::vector<double>&       x,
+    const InversiveDistanceMaps&     m,
+    double tol = 1e-8)
+{
+    auto G = inversive_distance_gradient(mesh, x, m);
+    for (double g : G)
+        if (std::abs(g) > tol) return false;
+    return true;
+}
+
+} // namespace conformallab

code/include/layout.hpp

@@ -0,0 +1,909 @@
+#pragma once
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
+// layout.hpp
+//
+// Phase 5/6/7 — Layout / embedding: DOF vector → vertex coordinates in the
+// target geometry via BFS-trilateration.
+//
+// ┌──────────────────────────────────────────────────────────────────────────┐
+// │  Algorithm (all three geometries)                                        │
+// │                                                                          │
+// │  1. For every edge e compute the updated length l_e from the DOF x.     │
+// │  2. Choose root face = largest 3-D area face (quality heuristic).       │
+// │  3. Priority BFS over the dual graph: faces are processed in order of   │
+// │     shortest distance (BFS depth) from the root face, minimising error  │
+// │     accumulation.  Equal depth: arbitrary tie-break.                    │
+// │  4. For each face: trilaterate the one unplaced vertex from the two     │
+// │     already-placed vertices on the shared edge.                          │
+// │  5. If a CutGraph is supplied, cut edges are treated as boundary;       │
+// │     holonomy is recorded for each cut edge.                              │
+// │  6. After the first connected component, unvisited faces start new      │
+// │     BFS sweeps (multi-component support).                               │
+// │                                                                          │
+// │  For open meshes: globally consistent placement.                        │
+// │  For closed meshes without CutGraph: first (shallowest) visit wins,    │
+// │     has_seam = true.                                                    │
+// │  For closed meshes with CutGraph: each vertex placed exactly once;     │
+// │     holonomy = translation (Euclidean) or Möbius map (hyperbolic).     │
+// └──────────────────────────────────────────────────────────────────────────┘
+//
+// Trilateration
+//   Euclidean   — exact (analytic formula in ℝ²)
+//   Spherical   — exact (spherical law of cosines on S²)
+//   Hyperbolic  — exact (hyperbolic law of cosines + Möbius maps,
+//                 Poincaré disk model)
+//
+// Normalisation
+//   normalise_euclidean(layout)   — centroid → origin, major axis → x-axis (PCA)
+//   normalise_hyperbolic(layout)  — face-area-weighted iterative Möbius centering
+//   normalise_spherical(layout)   — rotate centroid to north pole (Rodrigues)
+//
+// Holonomy (genus-g surfaces with CutGraph)
+//   HolonomyData.translations[i]  — translation ω_i   (Euclidean / spherical)
+//   HolonomyData.mobius_maps[i]   — Möbius map T_i    (hyperbolic)
+//   For a flat torus: ω_1, ω_2 are the lattice generators; τ = ω_2/ω_1 ∈ ℍ.
+//
+// Texture atlas
+//   Layout2D.halfedge_uv[h.idx()] — UV of source(h) as seen from face(h).
+//   At seam edges the two opposite halfedges carry different UV values,
+//   enabling proper per-halfedge UV for GPU texture atlasing.
+//
+// Möbius map
+//   MobiusMap::from_three(z1→w1, z2→w2, z3→w3) — fit a Möbius transformation
+//   to three point correspondences (via 3×3 complex linear system).
+
+#include "conformal_mesh.hpp"
+#include "euclidean_functional.hpp"
+#include "spherical_functional.hpp"
+#include "hyper_ideal_functional.hpp"
+#include "cut_graph.hpp"
+#include <Eigen/Dense>
+#include <vector>
+#include <queue>
+#include <complex>
+#include <cmath>
+#include <algorithm>
+#include <limits>
+#include <fstream>
+#include <string>
+
+namespace conformallab {
+
+// ── Möbius map ────────────────────────────────────────────────────────────────
+//
+// T(z) = (a·z + b) / (c·z + d)
+//
+// For hyperbolic holonomy the map is an orientation-preserving isometry of
+// the Poincaré disk (SU(1,1) element).
+// ─────────────────────────────────────────────────────────────────────────────
+/// Möbius transformation `T(z) = (a·z + b) / (c·z + d)` of the Riemann
+/// sphere; restricted to SU(1,1) for hyperbolic holonomy on the
+/// Poincaré disk.
+struct MobiusMap {
+    /// Complex scalar used for all entries.
+    using C = std::complex<double>;
+    C a{1.0, 0.0}; ///< Top-left coefficient.
+    C b{0.0, 0.0}; ///< Top-right coefficient.
+    C c{0.0, 0.0}; ///< Bottom-left coefficient.
+    C d{1.0, 0.0}; ///< Bottom-right coefficient.
+
+    /// Apply the transformation to a complex point.
+    C apply(C z) const { return (a * z + b) / (c * z + d); }
+
+    /// Apply the transformation to a 2-D real point (interpreted as `x + iy`).
+    Eigen::Vector2d apply(const Eigen::Vector2d& p) const {
+        C w = apply(C(p.x(), p.y()));
+        return Eigen::Vector2d(w.real(), w.imag());
+    }
+
+    /// Identity map.
+    static MobiusMap identity() { return {C(1), C(0), C(0), C(1)}; }
+
+    /// Inverse map.
+    MobiusMap inverse()               const { return {d, -b, -c, a}; }
+    /// Composition `(*this) ∘ T`, i.e. apply `T` first then `*this`.
+    MobiusMap compose(const MobiusMap& T) const {
+        return { a*T.a + b*T.c,  a*T.b + b*T.d,
+                 c*T.a + d*T.c,  c*T.b + d*T.d };
+    }
+
+    /// `true` iff the map is the identity up to tolerance `tol`.
+    bool is_identity(double tol = 1e-9) const {
+        if (std::abs(d) < 1e-14) return false;
+        C a_ = a/d, b_ = b/d, c_ = c/d;
+        return std::abs(a_ - C(1)) < tol && std::abs(b_) < tol && std::abs(c_) < tol;
+    }
+
+    /// Fit T to three point correspondences  T(z_i) = w_i.
+    /// Sets d=1 and solves the 3×3 complex linear system.
+    /// Returns identity when the system is (near-)singular.
+    static MobiusMap from_three(C z1, C w1, C z2, C w2, C z3, C w3) {
+        Eigen::Matrix3cd M;
+        M << z1, C(1), -w1*z1,
+             z2, C(1), -w2*z2,
+             z3, C(1), -w3*z3;
+        Eigen::Vector3cd rhs; rhs << w1, w2, w3;
+        Eigen::ColPivHouseholderQR<Eigen::Matrix3cd> qr(M);
+        if (qr.rank() < 3) return identity();
+        Eigen::Vector3cd sol = qr.solve(rhs);
+        return { sol[0], sol[1], sol[2], C(1.0) };
+    }
+};
+
+// ── Result types ──────────────────────────────────────────────────────────────
+
+/// Result of a 2-D layout (`euclidean_layout`, `hyper_ideal_layout`):
+/// per-vertex UV coordinates plus a per-half-edge UV atlas for seamed
+/// textures.
+struct Layout2D {
+    /// uv[v.idx()] — primary 2-D position (first / shallowest-BFS-depth visit).
+    std::vector<Eigen::Vector2d> uv;
+
+    /// halfedge_uv[h.idx()] — UV of source(h) as seen from face(h).
+    ///
+    /// For interior (non-seam) halfedges: equals uv[source(h).idx()].
+    /// For seam halfedges:  carries the UV from the virtual unfolding across
+    ///   the cut (i.e. the trilaterated position that was NOT used as the
+    ///   primary uv).  This gives each face its own copy of a seam vertex,
+    ///   enabling a proper GPU texture atlas without vertex duplication.
+    ///
+    /// Size = mesh.number_of_halfedges().  Border halfedges = (0,0).
+    std::vector<Eigen::Vector2d> halfedge_uv;
+
+    bool success  = false; ///< `true` iff the BFS placed every vertex.
+    bool has_seam = false; ///< `true` when a vertex was reached via two paths.
+};
+
+/// Result of a 3-D layout (`spherical_layout`): per-vertex positions on S².
+struct Layout3D {
+    std::vector<Eigen::Vector3d> pos; ///< Per-vertex spherical positions.
+    bool success  = false; ///< `true` iff the BFS placed every vertex.
+    bool has_seam = false; ///< `true` when a vertex was reached via two paths.
+};
+
+/// Per-cut-edge holonomy.
+///
+/// Euclidean: translations[i] = ω_i  (translation vector for cut_edge_indices[i]).
+///   For a flat torus ω_1, ω_2 are the lattice generators; τ = ω_2/ω_1 ∈ ℍ.
+///
+/// Hyperbolic: mobius_maps[i] = T_i  (Möbius isometry of the Poincaré disk).
+///   T_i(p_actual) = p_virtual  — maps the placed vertex position to the
+///   trilaterated virtual position obtained by continuing the unfolding across
+///   the cut.
+struct HolonomyData {
+    std::vector<Eigen::Vector2d> translations;     ///< Euclidean / spherical translation per cut edge.
+    std::vector<MobiusMap>       mobius_maps;      ///< Hyperbolic Möbius isometry per cut edge (Phase 7).
+    std::vector<std::size_t>     cut_edge_indices; ///< Index (in the cut-graph edge list) of each holonomy entry.
+};
+
+// ── Internal helpers ──────────────────────────────────────────────────────────
+
+namespace detail {
+
+// ─────────────────────────────────────────────────────────────────────────────
+// Priority-BFS queue entry
+// Faces closer to the root (lower depth) are processed first, reducing the
+// accumulation of trilateration errors for later-placed vertices.
+// ─────────────────────────────────────────────────────────────────────────────
+struct BFSEntry {
+    int            depth;  ///< BFS depth = max(depth[v_src], depth[v_tgt]) + 1
+    Halfedge_index h;      ///< halfedge pointing INTO the face to be placed
+    /// min-heap: smaller depth = higher priority
+    bool operator>(const BFSEntry& o) const { return depth > o.depth; }
+};
+
+using BFSQueue = std::priority_queue<BFSEntry,
+                                     std::vector<BFSEntry>,
+                                     std::greater<BFSEntry>>;
+
+// ─────────────────────────────────────────────────────────────────────────────
+// Euclidean trilateration — LEFT (CCW) side of p_a → p_b
+// ─────────────────────────────────────────────────────────────────────────────
+inline Eigen::Vector2d trilaterate_2d(
+    const Eigen::Vector2d& pa, const Eigen::Vector2d& pb,
+    double da, double db)
+{
+    Eigen::Vector2d ab = pb - pa;
+    double d = ab.norm();
+    if (d < 1e-14) return pa;
+    Eigen::Vector2d e1 = ab / d;
+    Eigen::Vector2d e2(-e1.y(), e1.x());
+    double t  = (da*da - db*db + d*d) / (2.0 * d);
+    double s2 = da*da - t*t;
+    double s  = (s2 > 0.0) ? std::sqrt(s2) : 0.0;
+    return pa + t*e1 + s*e2;
+}
+
+// ─────────────────────────────────────────────────────────────────────────────
+// Spherical trilateration (S²) — LEFT side of geodesic arc pa → pb
+// ─────────────────────────────────────────────────────────────────────────────
+inline Eigen::Vector3d trilaterate_sph(
+    const Eigen::Vector3d& pa, const Eigen::Vector3d& pb,
+    double da, double db)
+{
+    double c     = pa.dot(pb);
+    double denom = 1.0 - c*c;
+    if (denom < 1e-14) return pa;
+    double cda = std::cos(da), cdb = std::cos(db);
+    double alpha = (cda - c*cdb) / denom;
+    double beta  = (cdb - c*cda) / denom;
+    Eigen::Vector3d cross  = pa.cross(pb);
+    double          cn2    = cross.squaredNorm();
+    double gamma2 = 1.0 - alpha*alpha - beta*beta - 2.0*alpha*beta*c;
+    double gamma  = (gamma2 > 0.0 && cn2 > 1e-28) ? std::sqrt(gamma2 / cn2) : 0.0;
+    Eigen::Vector3d p = alpha*pa + beta*pb + gamma*cross;
+    double n = p.norm();
+    return (n > 1e-14) ? (p / n) : pa;
+}
+
+// ─────────────────────────────────────────────────────────────────────────────
+// Hyperbolic trilateration — exact via Möbius + hyperbolic law of cosines.
+// LEFT (CCW) side of pa → pb in the Poincaré disk.
+// ─────────────────────────────────────────────────────────────────────────────
+inline Eigen::Vector2d trilaterate_hyp(
+    const Eigen::Vector2d& pa, const Eigen::Vector2d& pb,
+    double D, double da, double db)
+{
+    using C = std::complex<double>;
+    if (D < 1e-12 || da < 1e-12) return pa;
+    if (std::sinh(da) * std::sinh(D) < 1e-14) return pa;
+    C a(pa.x(), pa.y()), b(pb.x(), pb.y());
+    auto fwd = [](C z, C c_) { return (z-c_)/(C(1)-std::conj(c_)*z); };
+    auto inv = [](C w, C c_) { return (w+c_)/(C(1)+std::conj(c_)*w); };
+    double theta_b  = std::arg(fwd(b, a));
+    double cos_alpha = (std::cosh(da)*std::cosh(D) - std::cosh(db))
+                     / (std::sinh(da)*std::sinh(D));
+    cos_alpha = std::max(-1.0, std::min(1.0, cos_alpha));
+    C pc_origin = std::tanh(da*0.5) * std::exp(C(0.0, theta_b + std::acos(cos_alpha)));
+    C pc_c = inv(pc_origin, a);
+    return Eigen::Vector2d(pc_c.real(), pc_c.imag());
+}
+
+// ─────────────────────────────────────────────────────────────────────────────
+// 3-D face area (cross product / 2)
+// ─────────────────────────────────────────────────────────────────────────────
+inline double face_3d_area(const ConformalMesh& mesh, Face_index f)
+{
+    Halfedge_index h = mesh.halfedge(f);
+    auto pA = mesh.point(mesh.source(h));
+    auto pB = mesh.point(mesh.target(h));
+    auto pC = mesh.point(mesh.target(mesh.next(h)));
+    Eigen::Vector3d ab(pB.x()-pA.x(), pB.y()-pA.y(), pB.z()-pA.z());
+    Eigen::Vector3d ac(pC.x()-pA.x(), pC.y()-pA.y(), pC.z()-pA.z());
+    return 0.5 * ab.cross(ac).norm();
+}
+
+// ─────────────────────────────────────────────────────────────────────────────
+// Best root face: largest 3-D area, with 1.5× bonus for interior faces.
+// ─────────────────────────────────────────────────────────────────────────────
+inline Face_index best_root_face(const ConformalMesh& mesh)
+{
+    Face_index best;
+    double best_score = -1.0;
+    for (auto f : mesh.faces()) {
+        double area = face_3d_area(mesh, f);
+        bool interior = true;
+        for (auto h : CGAL::halfedges_around_face(mesh.halfedge(f), mesh))
+            if (mesh.is_border(mesh.opposite(h))) { interior = false; break; }
+        double score = area * (interior ? 1.5 : 1.0);
+        if (score > best_score) { best_score = score; best = f; }
+    }
+    return best;
+}
+
+// ─────────────────────────────────────────────────────────────────────────────
+// Euclidean bounding box of placed vertices (for multi-component offset)
+// ─────────────────────────────────────────────────────────────────────────────
+inline void bbox_2d(const std::vector<Eigen::Vector2d>& uv,
+                    const std::vector<bool>& placed,
+                    double& xmax)
+{
+    xmax = 0.0;
+    for (std::size_t i = 0; i < uv.size(); ++i)
+        if (placed[i]) xmax = std::max(xmax, uv[i].x());
+}
+
+// ─────────────────────────────────────────────────────────────────────────────
+// Möbius centering — unweighted (fallback / Phase 6 compat.)
+// ─────────────────────────────────────────────────────────────────────────────
+inline void center_poincare_disk(std::vector<Eigen::Vector2d>& uv)
+{
+    if (uv.empty()) return;
+    using C = std::complex<double>;
+    C c(0);
+    for (auto& p : uv) c += C(p.x(), p.y());
+    c /= static_cast<double>(uv.size());
+    double r = std::abs(c);
+    if (r > 0.999) c *= 0.999/r;
+    if (r < 1e-12) return;
+    for (auto& p : uv) {
+        C z(p.x(), p.y());
+        C w = (z-c)/(C(1)-std::conj(c)*z);
+        p = {w.real(), w.imag()};
+    }
+}
+
+// ─────────────────────────────────────────────────────────────────────────────
+// Face-area-weighted iterative Möbius centering (Fréchet mean, Phase 7).
+// weights[i] = Voronoi area of vertex i.
+// ─────────────────────────────────────────────────────────────────────────────
+inline void center_poincare_disk_weighted(
+    std::vector<Eigen::Vector2d>& uv, const std::vector<double>& weights)
+{
+    if (uv.empty()) return;
+    using C = std::complex<double>;
+    double total_w = 0.0; for (double w : weights) total_w += w;
+    if (total_w < 1e-14) { center_poincare_disk(uv); return; }
+    C c(0);
+    for (int iter = 0; iter < 30; ++iter) {
+        C mt(0);
+        for (std::size_t i = 0; i < uv.size(); ++i) {
+            C z(uv[i].x(), uv[i].y());
+            mt += weights[i] * (z-c)/(C(1)-std::conj(c)*z);
+        }
+        mt /= total_w;
+        C c_new = (mt+c)/(C(1)+std::conj(c)*mt);
+        if (std::abs(c_new-c) < 1e-12) { c = c_new; break; }
+        c = c_new;
+    }
+    double r = std::abs(c);
+    if (r > 0.999) c *= 0.999/r;
+    if (r < 1e-12) return;
+    for (auto& p : uv) {
+        C z(p.x(), p.y());
+        C w = (z-c)/(C(1)-std::conj(c)*z);
+        p = {w.real(), w.imag()};
+    }
+}
+
+} // namespace detail
+
+/// Compute per-vertex area weights (sum of 1/3 of each adjacent triangle area).
+/// Used by area-weighted layout normalisation routines.
+inline std::vector<double> compute_vertex_area_weights(const ConformalMesh& mesh)
+{
+    std::vector<double> w(mesh.number_of_vertices(), 0.0);
+    for (auto f : mesh.faces()) {
+        double area = detail::face_3d_area(mesh, f);
+        for (auto h : CGAL::halfedges_around_face(mesh.halfedge(f), mesh))
+            w[static_cast<std::size_t>(mesh.target(h).idx())] += area / 3.0;
+    }
+    return w;
+}
+
+// ── Layout normalisation ──────────────────────────────────────────────────────
+
+/// Euclidean canonical normalisation: translate centroid to the origin
+/// and rotate the principal axis of the UV cloud onto the x-axis.
+inline void normalise_euclidean(Layout2D& layout)
+{
+    if (!layout.success || layout.uv.empty()) return;
+    const std::size_t n = layout.uv.size();
+    Eigen::Vector2d mean = Eigen::Vector2d::Zero();
+    for (auto& p : layout.uv) mean += p;
+    mean /= static_cast<double>(n);
+    for (auto& p : layout.uv) p -= mean;
+    for (auto& p : layout.halfedge_uv) p -= mean;
+    Eigen::Matrix2d cov = Eigen::Matrix2d::Zero();
+    for (auto& p : layout.uv) cov += p * p.transpose();
+    cov /= static_cast<double>(n);
+    Eigen::SelfAdjointEigenSolver<Eigen::Matrix2d> eig(cov);
+    Eigen::Vector2d major = eig.eigenvectors().col(1);
+    double angle = -std::atan2(major.y(), major.x());
+    Eigen::Matrix2d R;
+    R << std::cos(angle), -std::sin(angle),
+         std::sin(angle),  std::cos(angle);
+    for (auto& p : layout.uv)         p = R * p;
+    for (auto& p : layout.halfedge_uv) p = R * p;
+}
+
+/// Hyperbolic — face-area-weighted iterative Möbius centering.
+inline void normalise_hyperbolic(Layout2D& layout, const ConformalMesh& mesh)
+{
+    if (!layout.success || layout.uv.empty()) return;
+    auto weights = compute_vertex_area_weights(mesh);
+    detail::center_poincare_disk_weighted(layout.uv, weights);
+    // halfedge_uv follows the same Möbius map
+    detail::center_poincare_disk(layout.halfedge_uv);
+}
+/// Hyperbolic canonical normalisation, mesh-free fallback: uniform
+/// (unweighted) iterative Möbius centring of the Poincaré disk.
+inline void normalise_hyperbolic(Layout2D& layout) // fallback without mesh
+{
+    if (!layout.success || layout.uv.empty()) return;
+    detail::center_poincare_disk(layout.uv);
+    detail::center_poincare_disk(layout.halfedge_uv);
+}
+
+/// Spherical canonical normalisation: rotate the layout so that the
+/// per-vertex centroid (projected back to S²) coincides with the north
+/// pole (Rodrigues rotation).
+inline void normalise_spherical(Layout3D& layout)
+{
+    if (!layout.success || layout.pos.empty()) return;
+    Eigen::Vector3d mean = Eigen::Vector3d::Zero();
+    for (auto& p : layout.pos) mean += p;
+    double len = mean.norm(); if (len < 1e-12) return;
+    mean /= len;
+    Eigen::Vector3d north(0, 0, 1);
+    Eigen::Vector3d axis = mean.cross(north);
+    double sin_a = axis.norm(), cos_a = mean.dot(north);
+    if (sin_a < 1e-12) return;
+    axis /= sin_a;
+    Eigen::Matrix3d K;
+    K <<    0, -axis.z(),  axis.y(),
+       axis.z(),       0, -axis.x(),
+      -axis.y(),  axis.x(),       0;
+    Eigen::Matrix3d Rot = Eigen::Matrix3d::Identity() + sin_a*K + (1-cos_a)*K*K;
+    for (auto& p : layout.pos) p = Rot * p;
+}
+
+// ─────────────────────────────────────────────────────────────────────────────
+// BFS placement helpers shared across all three layout functions.
+//
+// set_face_huv: populate halfedge_uv for the three halfedges of face f,
+//               given that the face was entered via halfedge h_enter.
+//   h_enter: source=v_src, target=v_tgt   → huv = uv[v_src]
+//   next(h_enter): source=v_tgt, target=v_new → huv = uv[v_tgt]
+//   prev(h_enter): source=v_new, target=v_src → huv = p_new  (trilaterated)
+// ─────────────────────────────────────────────────────────────────────────────
+namespace detail {
+
+inline void set_face_huv_2d(
+    std::vector<Eigen::Vector2d>& huv,
+    const ConformalMesh& mesh,
+    Halfedge_index h,
+    const std::vector<Eigen::Vector2d>& uv,
+    const Eigen::Vector2d& p_new)
+{
+    auto s = [](std::size_t x){ return x; };
+    huv[s(static_cast<std::size_t>(h.idx()))]               = uv[static_cast<std::size_t>(mesh.source(h).idx())];
+    huv[s(static_cast<std::size_t>(mesh.next(h).idx()))]    = uv[static_cast<std::size_t>(mesh.target(h).idx())];
+    huv[s(static_cast<std::size_t>(mesh.prev(h).idx()))]    = p_new;
+}
+
+inline void set_root_huv_2d(
+    std::vector<Eigen::Vector2d>& huv,
+    const ConformalMesh& mesh,
+    Face_index f,
+    const std::vector<Eigen::Vector2d>& uv)
+{
+    for (auto hf : CGAL::halfedges_around_face(mesh.halfedge(f), mesh))
+        huv[static_cast<std::size_t>(hf.idx())] = uv[static_cast<std::size_t>(mesh.source(hf).idx())];
+}
+
+} // namespace detail
+
+// ── Euclidean layout ──────────────────────────────────────────────────────────
+
+/// Embed the mesh in ℝ² using the Euclidean metric encoded in x.
+///
+/// Runs priority-BFS trilateration: places vertices in order of BFS depth
+/// from the root face (largest 3D area), so errors accumulate last.
+/// For closed genus-g surfaces a CutGraph must be supplied — otherwise
+/// the layout will have a seam discontinuity (Layout2D::has_seam = true).
+///
+/// \param mesh      Input surface mesh.  Must have lambda0 and v_idx set in maps.
+/// \param x         DOF vector returned by newton_euclidean().
+/// \param maps      EuclideanMaps (lambda0, v_idx, e_idx).
+/// \param cut       Optional cut graph (compute_cut_graph()).  Pass nullptr for
+///                  open meshes or if seams are acceptable.
+/// \param holonomy  If non-null and cut != nullptr, receives the lattice
+///                  translations ω_i ∈ ℂ per cut edge.
+///                  Pass to compute_period_matrix() for the conformal modulus τ.
+/// \param normalise If true, calls normalise_euclidean() on the result:
+///                  centroid → origin, major axis → x-axis (PCA).
+/// \return          Layout2D with .uv[v] (per-vertex UV) and
+///                  .halfedge_uv[h] (per-halfedge UV for texture atlasing).
+///
+/// \note halfedge_uv differs from uv at seam edges: the two sides of a cut
+///       carry different UV coordinates for proper GPU texture atlasing.
+inline Layout2D euclidean_layout(
+    ConformalMesh&             mesh,
+    const std::vector<double>& x,
+    const EuclideanMaps&       maps,
+    const CutGraph*            cut      = nullptr,
+    HolonomyData*              holonomy  = nullptr,
+    bool                       normalise = false)
+{
+    const std::size_t nv = mesh.number_of_vertices();
+    const std::size_t nf = mesh.number_of_faces();
+    const std::size_t nh = mesh.number_of_halfedges();
+    Layout2D result;
+    result.uv.assign(nv, Eigen::Vector2d::Zero());
+    result.halfedge_uv.assign(nh, Eigen::Vector2d::Zero());
+    if (nf == 0) return result;
+
+    auto get_u  = [&](Vertex_index v) {
+        int iv = maps.v_idx[v]; return (iv>=0) ? x[static_cast<std::size_t>(iv)] : 0.0; };
+    auto get_ue = [&](Edge_index e) {
+        int ie = maps.e_idx[e]; return (ie>=0) ? x[static_cast<std::size_t>(ie)] : 0.0; };
+    auto edge_len = [&](Halfedge_index h) {
+        Edge_index e = mesh.edge(h);
+        double lam = maps.lambda0[e] + get_u(mesh.source(h)) + get_u(mesh.target(h)) + get_ue(e);
+        return std::exp(lam * 0.5); };
+
+    std::vector<bool> vertex_placed(nv, false);
+    std::vector<bool> face_placed(nf, false);
+    std::vector<int>  vertex_depth(nv, std::numeric_limits<int>::max());
+
+    auto place_component = [&](Face_index f_root, Eigen::Vector2d offset) {
+        Halfedge_index h0 = mesh.halfedge(f_root);
+        Halfedge_index h1 = mesh.next(h0), h2 = mesh.next(h1);
+        Vertex_index vA = mesh.source(h0), vB = mesh.source(h1), vC = mesh.source(h2);
+        double lAB = edge_len(h0), lBC = edge_len(h1), lCA = edge_len(h2);
+        result.uv[vA.idx()] = offset;
+        result.uv[vB.idx()] = offset + Eigen::Vector2d(lAB, 0.0);
+        result.uv[vC.idx()] = detail::trilaterate_2d(result.uv[vA.idx()], result.uv[vB.idx()], lCA, lBC);
+        vertex_placed[vA.idx()] = vertex_placed[vB.idx()] = vertex_placed[vC.idx()] = true;
+        vertex_depth[vA.idx()]  = vertex_depth[vB.idx()]  = vertex_depth[vC.idx()]  = 0;
+        face_placed[f_root.idx()] = true;
+        detail::set_root_huv_2d(result.halfedge_uv, mesh, f_root, result.uv);
+
+        detail::BFSQueue pq;
+        auto enqueue = [&](Face_index f) {
+            for (auto hf : CGAL::halfedges_around_face(mesh.halfedge(f), mesh)) {
+                Halfedge_index ho = mesh.opposite(hf);
+                if (mesh.is_border(ho)) continue;
+                if (cut && cut->is_cut(mesh.edge(hf))) continue;
+                Face_index fadj = mesh.face(ho);
+                if (!face_placed[fadj.idx()]) {
+                    int d = std::max(vertex_depth[mesh.source(ho).idx()],
+                                     vertex_depth[mesh.target(ho).idx()]) + 1;
+                    pq.push({d, ho});
+                }
+            }
+        };
+        enqueue(f_root);
+
+        while (!pq.empty()) {
+            auto [depth, h] = pq.top(); pq.pop();
+            Face_index f = mesh.face(h);
+            if (face_placed[f.idx()]) continue;
+
+            Vertex_index v_src = mesh.source(h), v_tgt = mesh.target(h);
+            Vertex_index v_new = mesh.target(mesh.next(h));
+            Eigen::Vector2d p = detail::trilaterate_2d(
+                result.uv[v_src.idx()], result.uv[v_tgt.idx()],
+                edge_len(mesh.prev(h)), edge_len(mesh.next(h)));
+
+            if (!vertex_placed[v_new.idx()]) {
+                result.uv[v_new.idx()]     = p;
+                vertex_placed[v_new.idx()] = true;
+                vertex_depth[v_new.idx()]  = depth;
+            } else {
+                result.has_seam = true;
+            }
+            // halfedge_uv: p is the UV of v_new in THIS face (may differ from uv[v_new] at seam)
+            detail::set_face_huv_2d(result.halfedge_uv, mesh, h, result.uv, p);
+            face_placed[f.idx()] = true;
+            enqueue(f);
+        }
+    };
+
+    place_component(detail::best_root_face(mesh), Eigen::Vector2d::Zero());
+    for (auto f : mesh.faces()) {
+        if (face_placed[f.idx()]) continue;
+        double xmax; detail::bbox_2d(result.uv, vertex_placed, xmax);
+        place_component(f, Eigen::Vector2d(xmax * 1.1 + 1.0, 0.0));
+    }
+    result.success = true;
+
+    // ── Holonomy ──────────────────────────────────────────────────────────────
+    if (cut && holonomy) {
+        holonomy->cut_edge_indices = cut->cut_edge_indices;
+        holonomy->translations.clear();
+        holonomy->translations.reserve(cut->cut_edge_indices.size());
+        holonomy->mobius_maps.clear();
+        for (std::size_t ce_idx : cut->cut_edge_indices) {
+            Edge_index e = *std::next(mesh.edges().begin(), static_cast<std::ptrdiff_t>(ce_idx));
+            Halfedge_index h  = mesh.halfedge(e);
+            Halfedge_index ho = mesh.opposite(h);
+            Halfedge_index hx = mesh.is_border(ho) ? h : ho;
+            if (mesh.is_border(hx)) { holonomy->translations.push_back(Eigen::Vector2d::Zero()); continue; }
+            Vertex_index vs = mesh.source(hx), vt = mesh.target(hx), vn = mesh.target(mesh.next(hx));
+            if (!vertex_placed[vn.idx()]) { holonomy->translations.push_back(Eigen::Vector2d::Zero()); continue; }
+            Eigen::Vector2d p_tri = detail::trilaterate_2d(
+                result.uv[vs.idx()], result.uv[vt.idx()],
+                edge_len(mesh.prev(hx)), edge_len(mesh.next(hx)));
+            // Store seam UV for the cut-crossing halfedges
+            detail::set_face_huv_2d(result.halfedge_uv, mesh, hx, result.uv, p_tri);
+            holonomy->translations.push_back(p_tri - result.uv[vn.idx()]);
+        }
+    }
+
+    if (normalise) normalise_euclidean(result);
+    return result;
+}
+
+// ── Spherical layout ──────────────────────────────────────────────────────────
+
+/// Embed the mesh on the unit sphere S² using the spherical metric encoded in x.
+///
+/// Runs priority-BFS trilateration using the spherical law of cosines.
+/// Typical use: genus-0 (sphere-like) surfaces after newton_spherical().
+///
+/// \param mesh      Input genus-0 surface mesh.
+/// \param x         DOF vector returned by newton_spherical().
+/// \param maps      SphericalMaps.
+/// \param cut       Optional cut graph (rarely needed for genus-0).
+/// \param holonomy  If non-null, receives rotational holonomies (spherical).
+/// \param normalise If true, calls normalise_spherical(): rotates the centroid
+///                  to the north pole (Rodrigues rotation formula).
+/// \return          Layout3D with .xyz[v] ∈ S² ⊂ ℝ³ for each vertex.
+inline Layout3D spherical_layout(
+    ConformalMesh&             mesh,
+    const std::vector<double>& x,
+    const SphericalMaps&       maps,
+    const CutGraph*            cut      = nullptr,
+    HolonomyData*              holonomy  = nullptr,
+    bool                       normalise = false)
+{
+    const std::size_t nv = mesh.number_of_vertices();
+    const std::size_t nf = mesh.number_of_faces();
+    Layout3D result;
+    result.pos.assign(nv, Eigen::Vector3d::Zero());
+    if (nf == 0) return result;
+
+    auto get_u  = [&](Vertex_index v) {
+        int iv = maps.v_idx[v]; return (iv>=0) ? x[static_cast<std::size_t>(iv)] : 0.0; };
+    auto get_ue = [&](Edge_index e) {
+        int ie = maps.e_idx[e]; return (ie>=0) ? x[static_cast<std::size_t>(ie)] : 0.0; };
+    auto arc_len = [&](Halfedge_index h) {
+        Edge_index e = mesh.edge(h);
+        double lam = maps.lambda0[e] + get_u(mesh.source(h)) + get_u(mesh.target(h)) + get_ue(e);
+        return 2.0 * std::asin(std::min(std::exp(lam*0.5), 1.0)); };
+
+    std::vector<bool> vertex_placed(nv, false);
+    std::vector<bool> face_placed(nf, false);
+    std::vector<int>  vertex_depth(nv, std::numeric_limits<int>::max());
+
+    auto place_component = [&](Face_index f_root) {
+        Halfedge_index h0 = mesh.halfedge(f_root);
+        Halfedge_index h1 = mesh.next(h0), h2 = mesh.next(h1);
+        Vertex_index vA = mesh.source(h0), vB = mesh.source(h1), vC = mesh.source(h2);
+        double lAB = arc_len(h0), lBC = arc_len(h1), lCA = arc_len(h2);
+        result.pos[vA.idx()] = Eigen::Vector3d(0, 0, 1);
+        result.pos[vB.idx()] = Eigen::Vector3d(std::sin(lAB), 0, std::cos(lAB));
+        result.pos[vC.idx()] = detail::trilaterate_sph(result.pos[vA.idx()], result.pos[vB.idx()], lCA, lBC);
+        vertex_placed[vA.idx()] = vertex_placed[vB.idx()] = vertex_placed[vC.idx()] = true;
+        vertex_depth[vA.idx()]  = vertex_depth[vB.idx()]  = vertex_depth[vC.idx()]  = 0;
+        face_placed[f_root.idx()] = true;
+
+        detail::BFSQueue pq;
+        auto enqueue = [&](Face_index f) {
+            for (auto hf : CGAL::halfedges_around_face(mesh.halfedge(f), mesh)) {
+                Halfedge_index ho = mesh.opposite(hf);
+                if (mesh.is_border(ho)) continue;
+                if (cut && cut->is_cut(mesh.edge(hf))) continue;
+                Face_index fadj = mesh.face(ho);
+                if (!face_placed[fadj.idx()]) {
+                    int d = std::max(vertex_depth[mesh.source(ho).idx()],
+                                     vertex_depth[mesh.target(ho).idx()]) + 1;
+                    pq.push({d, ho});
+                }
+            }
+        };
+        enqueue(f_root);
+
+        while (!pq.empty()) {
+            auto [depth, h] = pq.top(); pq.pop();
+            Face_index f = mesh.face(h);
+            if (face_placed[f.idx()]) continue;
+            Vertex_index vs = mesh.source(h), vt = mesh.target(h), vn = mesh.target(mesh.next(h));
+            Eigen::Vector3d p = detail::trilaterate_sph(
+                result.pos[vs.idx()], result.pos[vt.idx()],
+                arc_len(mesh.prev(h)), arc_len(mesh.next(h)));
+            if (!vertex_placed[vn.idx()]) {
+                result.pos[vn.idx()] = p; vertex_placed[vn.idx()] = true; vertex_depth[vn.idx()] = depth;
+            } else result.has_seam = true;
+            face_placed[f.idx()] = true; enqueue(f);
+        }
+    };
+
+    place_component(detail::best_root_face(mesh));
+    for (auto f : mesh.faces()) if (!face_placed[f.idx()]) place_component(f);
+    result.success = true;
+
+    if (cut && holonomy) {
+        holonomy->cut_edge_indices = cut->cut_edge_indices;
+        holonomy->translations.clear();
+        holonomy->translations.reserve(cut->cut_edge_indices.size());
+        holonomy->mobius_maps.clear();
+        for (std::size_t ce_idx : cut->cut_edge_indices) {
+            Edge_index e = *std::next(mesh.edges().begin(), static_cast<std::ptrdiff_t>(ce_idx));
+            Halfedge_index hx = mesh.is_border(mesh.opposite(mesh.halfedge(e)))
+                               ? mesh.halfedge(e) : mesh.opposite(mesh.halfedge(e));
+            if (mesh.is_border(hx)) { holonomy->translations.push_back(Eigen::Vector2d::Zero()); continue; }
+            Vertex_index vs = mesh.source(hx), vt = mesh.target(hx), vn = mesh.target(mesh.next(hx));
+            if (!vertex_placed[vn.idx()]) { holonomy->translations.push_back(Eigen::Vector2d::Zero()); continue; }
+            Eigen::Vector3d p_tri = detail::trilaterate_sph(
+                result.pos[vs.idx()], result.pos[vt.idx()],
+                arc_len(mesh.prev(hx)), arc_len(mesh.next(hx)));
+            Eigen::Vector3d diff = p_tri - result.pos[vn.idx()];
+            // Note: spherical holonomy is geometrically a 3-D rotation, not a 2-D
+            // translation.  The Vector2d here stores only the (x,y) component of the
+            // S²-position difference across the cut, which is an approximation.
+            // For accurate spherical holonomy (rotation axis + angle) use the full
+            // 3-D positions in result.pos[] directly.  Phase 10+ will replace this
+            // with a proper SO(3) representation.
+            holonomy->translations.push_back(Eigen::Vector2d(diff.x(), diff.y()));
+        }
+    }
+
+    if (normalise) normalise_spherical(result);
+    return result;
+}
+
+// ── HyperIdeal layout (Poincaré disk) — exact trilateration ──────────────────
+
+/// Embed the mesh in the Poincaré disk (H²) using the hyperbolic metric encoded in x.
+///
+/// Runs priority-BFS trilateration using exact Möbius-isometric placement:
+/// each new vertex is located by solving the hyperbolic law of cosines and
+/// applying a Möbius map to position it in the disk.
+/// For closed genus-g surfaces (g ≥ 1) a CutGraph is required.
+///
+/// \param mesh      Input genus-g surface mesh (g ≥ 1).
+/// \param x         DOF vector returned by newton_hyper_ideal()
+///                  (vertex b_v and edge a_e variables).
+/// \param maps      HyperIdealMaps (lambda0, v_idx, e_idx).
+/// \param cut       CutGraph from compute_cut_graph().  Required for closed surfaces.
+/// \param holonomy  If non-null, receives the Möbius maps T_i ∈ SU(1,1) per cut edge.
+///                  Pass to compute_period_matrix() for holonomy analysis.
+/// \param normalise If true, calls normalise_hyperbolic(): iterative face-area-weighted
+///                  Möbius centring (Fréchet mean, 30 iterations) → disk origin.
+/// \return          Layout2D with .uv[v] ∈ Poincaré disk (|uv| < 1) for each vertex,
+///                  and .halfedge_uv[h] for seam-aware texture atlasing.
+///
+/// \note All vertex positions satisfy |uv[v]| < 1 (inside the Poincaré disk)
+///       if the metric is hyperbolic.  Points on or outside the boundary indicate
+///       a non-hyperbolic metric (Gauss–Bonnet violation).
+inline Layout2D hyper_ideal_layout(
+    ConformalMesh&             mesh,
+    const std::vector<double>& x,
+    const HyperIdealMaps&      maps,
+    const CutGraph*            cut      = nullptr,
+    HolonomyData*              holonomy  = nullptr,
+    bool                       normalise = false)
+{
+    const std::size_t nv = mesh.number_of_vertices();
+    const std::size_t nf = mesh.number_of_faces();
+    const std::size_t nh = mesh.number_of_halfedges();
+    Layout2D result;
+    result.uv.assign(nv, Eigen::Vector2d::Zero());
+    result.halfedge_uv.assign(nh, Eigen::Vector2d::Zero());
+    if (nf == 0) return result;
+
+    auto get_b = [&](Vertex_index v) {
+        int iv = maps.v_idx[v]; return (iv>=0) ? x[static_cast<std::size_t>(iv)] : 0.0; };
+    auto get_a = [&](Edge_index e) {
+        int ie = maps.e_idx[e]; return (ie>=0) ? x[static_cast<std::size_t>(ie)] : 0.0; };
+    auto hyp_dist = [&](Halfedge_index h) {
+        double bi = get_b(mesh.source(h)), bj = get_b(mesh.target(h)), a = get_a(mesh.edge(h));
+        return std::acosh(std::max(1.0, std::cosh(bi+a*0.5)*std::cosh(bj+a*0.5)-std::sinh(bi)*std::sinh(bj))); };
+
+    std::vector<bool> vertex_placed(nv, false);
+    std::vector<bool> face_placed(nf, false);
+    std::vector<int>  vertex_depth(nv, std::numeric_limits<int>::max());
+
+    auto place_component = [&](Face_index f_root) {
+        Halfedge_index h0 = mesh.halfedge(f_root);
+        Halfedge_index h1 = mesh.next(h0), h2 = mesh.next(h1);
+        Vertex_index vA = mesh.source(h0), vB = mesh.source(h1), vC = mesh.source(h2);
+        double dAB = hyp_dist(h0), dBC = hyp_dist(h1), dCA = hyp_dist(h2);
+        result.uv[vA.idx()] = Eigen::Vector2d::Zero();
+        result.uv[vB.idx()] = Eigen::Vector2d(std::tanh(dAB*0.5), 0.0);
+        result.uv[vC.idx()] = detail::trilaterate_hyp(result.uv[vA.idx()], result.uv[vB.idx()], dAB, dCA, dBC);
+        vertex_placed[vA.idx()] = vertex_placed[vB.idx()] = vertex_placed[vC.idx()] = true;
+        vertex_depth[vA.idx()]  = vertex_depth[vB.idx()]  = vertex_depth[vC.idx()]  = 0;
+        face_placed[f_root.idx()] = true;
+        detail::set_root_huv_2d(result.halfedge_uv, mesh, f_root, result.uv);
+
+        detail::BFSQueue pq;
+        auto enqueue = [&](Face_index f) {
+            for (auto hf : CGAL::halfedges_around_face(mesh.halfedge(f), mesh)) {
+                Halfedge_index ho = mesh.opposite(hf);
+                if (mesh.is_border(ho)) continue;
+                if (cut && cut->is_cut(mesh.edge(hf))) continue;
+                Face_index fadj = mesh.face(ho);
+                if (!face_placed[fadj.idx()]) {
+                    int d = std::max(vertex_depth[mesh.source(ho).idx()],
+                                     vertex_depth[mesh.target(ho).idx()]) + 1;
+                    pq.push({d, ho});
+                }
+            }
+        };
+        enqueue(f_root);
+
+        while (!pq.empty()) {
+            auto [depth, h] = pq.top(); pq.pop();
+            Face_index f = mesh.face(h);
+            if (face_placed[f.idx()]) continue;
+            Vertex_index vs = mesh.source(h), vt = mesh.target(h), vn = mesh.target(mesh.next(h));
+            double D = hyp_dist(h), da = hyp_dist(mesh.prev(h)), db = hyp_dist(mesh.next(h));
+            Eigen::Vector2d p = detail::trilaterate_hyp(result.uv[vs.idx()], result.uv[vt.idx()], D, da, db);
+            if (!vertex_placed[vn.idx()]) {
+                result.uv[vn.idx()] = p; vertex_placed[vn.idx()] = true; vertex_depth[vn.idx()] = depth;
+            } else result.has_seam = true;
+            detail::set_face_huv_2d(result.halfedge_uv, mesh, h, result.uv, p);
+            face_placed[f.idx()] = true; enqueue(f);
+        }
+    };
+
+    place_component(detail::best_root_face(mesh));
+    for (auto f : mesh.faces()) if (!face_placed[f.idx()]) place_component(f);
+    result.success = true;
+
+    // ── Möbius-map holonomy ───────────────────────────────────────────────────
+    if (cut && holonomy) {
+        holonomy->cut_edge_indices = cut->cut_edge_indices;
+        holonomy->translations.clear();
+        holonomy->mobius_maps.clear();
+        holonomy->mobius_maps.reserve(cut->cut_edge_indices.size());
+        for (std::size_t ce_idx : cut->cut_edge_indices) {
+            Edge_index e = *std::next(mesh.edges().begin(), static_cast<std::ptrdiff_t>(ce_idx));
+            Halfedge_index hcut = mesh.halfedge(e), ho = mesh.opposite(hcut);
+            Halfedge_index hx   = mesh.is_border(ho) ? hcut : ho;
+            if (mesh.is_border(hx)) { holonomy->mobius_maps.push_back(MobiusMap::identity()); continue; }
+            Vertex_index vs = mesh.source(hx), vt = mesh.target(hx), vn = mesh.target(mesh.next(hx));
+            if (!vertex_placed[vn.idx()]) { holonomy->mobius_maps.push_back(MobiusMap::identity()); continue; }
+            double D = hyp_dist(hx), da = hyp_dist(mesh.prev(hx)), db = hyp_dist(mesh.next(hx));
+            Eigen::Vector2d p_tri = detail::trilaterate_hyp(result.uv[vs.idx()], result.uv[vt.idx()], D, da, db);
+            detail::set_face_huv_2d(result.halfedge_uv, mesh, hx, result.uv, p_tri);
+            using C = std::complex<double>;
+            // Möbius deck transformation T across cut edge (vs,vt):
+            // T is the unique Möbius isometry of the Poincaré disk that:
+            //   - fixes vs and vt  (z1=w1, z2=w2: the cut-edge endpoints are
+            //     identified across the seam, so T maps each to itself)
+            //   - maps vn (placed side) → p_tri (virtual side)
+            // This uniquely determines the hyperbolic translation/rotation
+            // along the geodesic through vs and vt.
+            holonomy->mobius_maps.push_back(MobiusMap::from_three(
+                C(result.uv[vs.idx()].x(), result.uv[vs.idx()].y()),
+                C(result.uv[vs.idx()].x(), result.uv[vs.idx()].y()),
+                C(result.uv[vt.idx()].x(), result.uv[vt.idx()].y()),
+                C(result.uv[vt.idx()].x(), result.uv[vt.idx()].y()),
+                C(result.uv[vn.idx()].x(), result.uv[vn.idx()].y()),
+                C(p_tri.x(), p_tri.y())));
+        }
+    }
+
+    if (normalise) normalise_hyperbolic(result, mesh);
+    return result;
+}
+
+// ── Convenience: save layout as OFF ──────────────────────────────────────────
+
+/// Write a 2-D layout to disk in OFF format with z = 0.  Convenience
+/// helper for quickly inspecting the UV result in any OFF viewer.
+inline void save_layout_off(
+    const std::string& path, ConformalMesh& mesh, const Layout2D& layout)
+{
+    std::ofstream ofs(path);
+    ofs << "OFF\n" << mesh.number_of_vertices() << " " << mesh.number_of_faces() << " 0\n";
+    for (auto v : mesh.vertices()) { auto& p = layout.uv[v.idx()]; ofs << p.x() << " " << p.y() << " 0\n"; }
+    for (auto f : mesh.faces()) {
+        ofs << "3";
+        for (auto h : CGAL::halfedges_around_face(mesh.halfedge(f), mesh)) ofs << " " << mesh.target(h).idx();
+        ofs << "\n";
+    }
+}
+
+/// Write a 3-D (spherical) layout to disk in OFF format.
+inline void save_layout_off(
+    const std::string& path, ConformalMesh& mesh, const Layout3D& layout)
+{
+    std::ofstream ofs(path);
+    ofs << "OFF\n" << mesh.number_of_vertices() << " " << mesh.number_of_faces() << " 0\n";
+    for (auto v : mesh.vertices()) { auto& p = layout.pos[v.idx()]; ofs << p.x() << " " << p.y() << " " << p.z() << "\n"; }
+    for (auto f : mesh.faces()) {
+        ofs << "3";
+        for (auto h : CGAL::halfedges_around_face(mesh.halfedge(f), mesh)) ofs << " " << mesh.target(h).idx();
+        ofs << "\n";
+    }
+}
+
+} // namespace conformallab

code/include/matrix_utility.hpp

@@ -1,4 +1,7 @@
 #pragma once
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
 
 // 4x4 mapping matrix from corresponding point pairs.
 // Ported from de.varylab.discreteconformal.math.MatrixUtility (Java).
@@ -7,12 +10,10 @@
 
 namespace conformallab {
 
-// Find the 4×4 matrix R that maps source points to target points.
-// Each row of `from` / `to` is a homogeneous 4-vector (one point per row).
-// Post-condition: R * from.row(i).T == to.row(i).T for all i.
-//
-// Implementation: R = to^T * (from^T)^{-1}
-// Corresponds to Java MatrixUtility.makeMappingMatrix().
+/// Find the 4×4 matrix `R` that maps each row of `from` (homogeneous
+/// 4-vector) to the corresponding row of `to`: `R · fromᵀ = toᵀ`.
+/// Computed as `R = toᵀ · (fromᵀ)⁻¹`.  Same as Java
+/// `MatrixUtility.makeMappingMatrix()`.
 inline Eigen::Matrix4d makeMappingMatrix(const Eigen::Matrix4d& from,
                                          const Eigen::Matrix4d& to) {
     return to.transpose() * from.transpose().inverse();

code/include/mesh_builder.hpp

@@ -0,0 +1,151 @@
+#pragma once
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
+// mesh_builder.hpp
+//
+// Factory functions that build simple reference meshes for testing and examples.
+// All functions return a ConformalMesh (CGAL::Surface_mesh<Point3>).
+//
+// Replaces Java mesh generators:
+//   CoHDS generators (convex hull, hyper-ideal generator) come later (Phase 3c/4).
+//   These builders cover the minimal meshes needed for functional unit tests.
+
+#include "conformal_mesh.hpp"
+#include "constants.hpp"
+#include <cmath>
+#include <vector>
+
+namespace conformallab {
+
+// ── Single triangle ──────────────────────────────────────────────────────────
+//
+//   v2
+//   |  \
+//   |    \
+//   v0 ─ v1
+//
+/// Build a single right-angle triangle in the xy-plane (1 face, 3 vertices, 3 edges).
+// The triangle lies in the xy-plane with a right angle at v0.
+inline ConformalMesh make_triangle(
+    double x0=0, double y0=0,
+    double x1=1, double y1=0,
+    double x2=0, double y2=1)
+{
+    ConformalMesh mesh;
+    auto v0 = mesh.add_vertex(Point3(x0, y0, 0));
+    auto v1 = mesh.add_vertex(Point3(x1, y1, 0));
+    auto v2 = mesh.add_vertex(Point3(x2, y2, 0));
+    mesh.add_face(v0, v1, v2);
+    return mesh;
+}
+
+// ── Regular tetrahedron ──────────────────────────────────────────────────────
+//
+/// Build a regular tetrahedron (4 vertices, 4 faces, 6 edges; sphere topology).
+// Euler characteristic: V - E + F = 4 - 6 + 4 = 2  (sphere topology).
+// Used to test closed-surface traversal.
+inline ConformalMesh make_tetrahedron()
+{
+    ConformalMesh mesh;
+
+    // Vertices of a regular tetrahedron centred at origin, edge length √2·2
+    auto v0 = mesh.add_vertex(Point3( 1,  1,  1));
+    auto v1 = mesh.add_vertex(Point3( 1, -1, -1));
+    auto v2 = mesh.add_vertex(Point3(-1,  1, -1));
+    auto v3 = mesh.add_vertex(Point3(-1, -1,  1));
+
+    // 4 outward-facing triangles (consistent winding)
+    mesh.add_face(v0, v2, v1);  // bottom (z=-1 side)
+    mesh.add_face(v0, v1, v3);  // front  (y=-1 side)
+    mesh.add_face(v0, v3, v2);  // left   (x=-1 side)
+    mesh.add_face(v1, v2, v3);  // back
+
+    return mesh;
+}
+
+// ── Two-triangle strip ───────────────────────────────────────────────────────
+//
+//   v2 ─ v3
+//   |  \ |
+//   v0 ─ v1
+//
+/// Build a two-triangle strip (4 vertices, 2 faces, 5 edges; 1 interior edge).
+/// Useful for testing interior- vs boundary-edge distinction.
+inline ConformalMesh make_quad_strip()
+{
+    ConformalMesh mesh;
+    auto v0 = mesh.add_vertex(Point3(0, 0, 0));
+    auto v1 = mesh.add_vertex(Point3(1, 0, 0));
+    auto v2 = mesh.add_vertex(Point3(0, 1, 0));
+    auto v3 = mesh.add_vertex(Point3(1, 1, 0));
+
+    mesh.add_face(v0, v1, v2);   // lower-left  triangle
+    mesh.add_face(v1, v3, v2);   // upper-right triangle  (shares edge v1–v2)
+
+    return mesh;
+}
+
+// ── Regular flat polygon fan ─────────────────────────────────────────────────
+//
+/// Build a regular flat polygon fan: `n` triangles sharing a central
+/// vertex, with rim vertices on the unit circle (disk topology).
+inline ConformalMesh make_fan(int n)
+{
+    CGAL_precondition(n >= 3);
+    ConformalMesh mesh;
+
+    auto center = mesh.add_vertex(Point3(0, 0, 0));
+
+    const double dtheta = TWO_PI / n;
+    std::vector<Vertex_index> rim(n);
+    for (int i = 0; i < n; ++i) {
+        double a = i * dtheta;
+        rim[i] = mesh.add_vertex(Point3(std::cos(a), std::sin(a), 0));
+    }
+
+    for (int i = 0; i < n; ++i)
+        mesh.add_face(center, rim[i], rim[(i+1) % n]);
+
+    return mesh;
+}
+
+// ── Spherical tetrahedron (vertices on the unit sphere) ───────────────────────
+//
+/// Build a regular tetrahedron with vertices on the unit sphere.
+/// All edge lengths equal `arccos(−1/3) ≈ 1.9106 rad`; used by the
+/// SphericalFunctional tests.
+inline ConformalMesh make_spherical_tetrahedron()
+{
+    ConformalMesh mesh;
+    const double s = 1.0 / std::sqrt(3.0);
+
+    auto v0 = mesh.add_vertex(Point3( s,  s,  s));
+    auto v1 = mesh.add_vertex(Point3( s, -s, -s));
+    auto v2 = mesh.add_vertex(Point3(-s,  s, -s));
+    auto v3 = mesh.add_vertex(Point3(-s, -s,  s));
+
+    mesh.add_face(v0, v2, v1);
+    mesh.add_face(v0, v1, v3);
+    mesh.add_face(v0, v3, v2);
+    mesh.add_face(v1, v2, v3);
+
+    return mesh;
+}
+
+// ── Octahedron face triangle (vertices on the unit sphere) ────────────────────
+//
+/// Build one face of a regular octahedron `(1,0,0)→(0,1,0)→(0,0,1)`:
+/// a right-angled spherical triangle with edge length `π/2` and base
+/// log-length `λ° = −log 2 ≈ −0.6931`.
+inline ConformalMesh make_octahedron_face()
+{
+    ConformalMesh mesh;
+    auto v0 = mesh.add_vertex(Point3(1, 0, 0));
+    auto v1 = mesh.add_vertex(Point3(0, 1, 0));
+    auto v2 = mesh.add_vertex(Point3(0, 0, 1));
+    mesh.add_face(v0, v1, v2);
+    return mesh;
+}
+
+} // namespace conformallab

code/include/mesh_io.hpp

@@ -0,0 +1,66 @@
+#pragma once
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
+// mesh_io.hpp
+//
+// Phase 4b — CGAL::IO wrappers for ConformalMesh.
+//
+// Reads and writes ConformalMesh (CGAL::Surface_mesh) in standard polygon-mesh
+// formats using the CGAL Polygon Mesh I/O utilities (CGAL 5.4+).
+//
+// Supported formats (detected by file extension):
+//   .off   — Object File Format   (read + write)
+//   .obj   — Wavefront OBJ        (read + write)
+//   .ply   — Polygon File Format  (read + write)
+//
+// Usage:
+//   ConformalMesh mesh;
+//   if (!read_mesh("input.off", mesh))  throw std::runtime_error("read failed");
+//   // ... process mesh ...
+//   write_mesh("output.off", mesh);
+//
+// Note: property maps (lambda0, v_idx, etc.) are NOT serialised — they must
+// be re-initialised with setup_*_maps() + compute_lambda0_from_mesh() after
+// reading a file.
+
+#include "conformal_mesh.hpp"
+#include <CGAL/IO/polygon_mesh_io.h>
+#include <string>
+#include <stdexcept>
+
+namespace conformallab {
+
+/// Read a polygon mesh from `filename` into `mesh` (clears existing content).
+/// Returns `true` on success, `false` on failure.
+inline bool read_mesh(const std::string& filename, ConformalMesh& mesh)
+{
+    mesh.clear();
+    return CGAL::IO::read_polygon_mesh(filename, mesh);
+}
+
+/// Write `mesh` to `filename`.  Returns `true` on success.
+inline bool write_mesh(const std::string& filename, const ConformalMesh& mesh)
+{
+    return CGAL::IO::write_polygon_mesh(filename, mesh);
+}
+
+/// Throwing wrapper around `read_mesh`: returns the mesh by value
+/// or throws `std::runtime_error` on read failure.
+inline ConformalMesh load_mesh(const std::string& filename)
+{
+    ConformalMesh mesh;
+    if (!read_mesh(filename, mesh))
+        throw std::runtime_error("conformallab: failed to read mesh from " + filename);
+    return mesh;
+}
+
+/// Throwing wrapper around `write_mesh`; throws `std::runtime_error` on
+/// write failure.
+inline void save_mesh(const std::string& filename, const ConformalMesh& mesh)
+{
+    if (!write_mesh(filename, mesh))
+        throw std::runtime_error("conformallab: failed to write mesh to " + filename);
+}
+
+} // namespace conformallab

code/include/mesh_utils.hpp

@@ -1,14 +1,41 @@
 #pragma once
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
+// mesh_utils.hpp
+//
+// Conversions between CGAL::Surface_mesh and Eigen matrices.  Used
+// primarily by the viewer / example programs to bridge to libigl, which
+// expects (V, F) matrix pairs rather than a halfedge data structure.
+//
+// All functions are templated on the kernel so the same code works
+// with `Simple_cartesian<double>` (production) and with any CGAL
+// `Kernel_d::Point_3` (test scaffolding).
+
 #include <CGAL/Surface_mesh.h>
-#include <Eigen/Dense>
+#include <Eigen/Core>  // downgraded from <Eigen/Dense>: this header only
+                      // uses Matrix/Vector primitives, no decompositions.
 #include <CGAL/Polygon_mesh_processing/triangulate_faces.h>
 
 
 namespace mesh_utils {
+
+/// Copy `mesh` into an Eigen `(V, F)` pair (libigl convention).
+///
+/// **Side effect:** `mesh` is triangulated in place via
+/// `CGAL::Polygon_mesh_processing::triangulate_faces` so the output
+/// `F` is guaranteed to be a 3-column matrix.  If `mesh` is already a
+/// triangle mesh this is a no-op.
+///
+/// \param mesh  Input surface mesh.  **Modified in place** if any face
+///              has more than 3 vertices.
+/// \param V     Output: `(num_vertices, 3)` matrix of vertex positions.
+/// \param F     Output: `(num_faces, 3)` matrix of vertex indices per
+///              face (rows are individual triangles).
 template <typename Kernel>
 void cgal_to_eigen(CGAL::Surface_mesh<typename Kernel::Point_3>& mesh,
                    Eigen::MatrixXd& V, Eigen::MatrixXi& F) {
-  
+
  CGAL::Polygon_mesh_processing::triangulate_faces(mesh);
 
   V.resize(mesh.num_vertices(), 3);
@@ -30,13 +57,38 @@ void cgal_to_eigen(CGAL::Surface_mesh<typename Kernel::Point_3>& mesh,
     face_idx++;
   }
 }
+
+/// Quick interactive visualisation via libigl + GLFW.
+///
+/// **Requires** `WITH_VIEWER=ON` at CMake time (which is implied by
+/// `WITH_CGAL=ON`).  Blocks until the viewer window is closed.
+/// Not suitable for CI / headless contexts.
+///
+/// Typical use:
+/// \code{.cpp}
+///   Eigen::MatrixXd V;  Eigen::MatrixXi F;
+///   mesh_utils::cgal_to_eigen<Kernel>(mesh, V, F);
+///   mesh_utils::simple_visualize_mesh<Kernel>(V, F);
+/// \endcode
 template <typename Kernel>
 void simple_visualize_mesh(Eigen::MatrixXd& V, Eigen::MatrixXi& F) {
   igl::opengl::glfw::Viewer viewer;
   viewer.data().set_mesh(V, F);
   viewer.launch();
 }
-// Zero-Copy Map für V (optional)
+
+/// Zero-copy `Eigen::Map` view of `mesh`'s vertex positions.
+///
+/// Returns a row-major `(N, 3)` `Eigen::Map` that aliases the
+/// `mesh.points()` storage directly — no allocation, O(1).
+///
+/// **Lifetime warning:** the returned `Map` references memory owned by
+/// `mesh`.  Adding or removing vertices may invalidate the underlying
+/// storage; use the `Map` only as long as `mesh` is structurally stable.
+///
+/// This is the read-write counterpart to `cgal_to_eigen` for cases
+/// where the caller wants to *modify* vertex positions through Eigen
+/// (e.g. apply a Möbius transformation) without an intermediate copy.
 template <typename Kernel>
 Eigen::Map<Eigen::Matrix<double, Eigen::Dynamic, 3, Eigen::RowMajor>>
 get_vertex_map(CGAL::Surface_mesh<typename Kernel::Point_3>& mesh) {

code/include/newton_solver.hpp

@@ -0,0 +1,571 @@
+#pragma once
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
+// newton_solver.hpp
+//
+// Phase 4a — Newton solver for all three discrete conformal functionals.
+//
+// Solves G(x) = 0 where G is the gradient of the discrete conformal energy.
+//
+// ┌──────────────────────────────────────────────────────────────────────────┐
+// │  Gradient sign conventions                                               │
+// │    Euclidean / Spherical:  G_v = Θ_v − Σ α_v    (target − actual)      │
+// │    HyperIdeal:             G_v = Σ β_v − Θ_v    (actual − target)      │
+// │                                                                          │
+// │  All solvers use the same Newton step  Δx = −H⁻¹·G                     │
+// │                                                                          │
+// │  Hessian sign at equilibrium                                             │
+// │    Euclidean:   H PSD  → SimplicialLDLT on  H                          │
+// │    Spherical:   H NSD  → SimplicialLDLT on −H  (solve (−H)Δx = G)      │
+// │    HyperIdeal:  H PSD  → SimplicialLDLT on  H  (analytical H: future)  │
+// └──────────────────────────────────────────────────────────────────────────┘
+//
+// SparseQR fallback:
+//   When SimplicialLDLT reports a failure (e.g. singular H on a closed mesh
+//   without a pinned vertex), the solver automatically retries with
+//   Eigen::SparseQR, which finds the minimum-norm Newton step orthogonal to
+//   the null space.  This handles the gauge mode on closed surfaces without
+//   requiring the caller to pin a vertex explicitly.
+//
+// Requires:
+//   Eigen::SimplicialLDLT, Eigen::SparseQR (Eigen sparse module)
+
+#include "euclidean_hessian.hpp"
+#include "spherical_hessian.hpp"
+#include "hyper_ideal_hessian.hpp"
+#include "cp_euclidean_functional.hpp"
+#include "inversive_distance_functional.hpp"
+#include <Eigen/SparseCholesky>
+#include <Eigen/SparseQR>
+#include <Eigen/OrderingMethods>
+#include <Eigen/Dense>
+#include <algorithm>
+#include <cmath>
+
+namespace conformallab {
+
+// ── Result ────────────────────────────────────────────────────────────────────
+
+/// Result of `newton_solve(...)` — converged DOF vector + diagnostics.
+struct NewtonResult {
+    std::vector<double> x;            ///< DOF vector at termination.
+    int                 iterations;   ///< Newton steps taken.
+    double              grad_inf_norm;///< max |Gᵢ| at termination.
+    bool                converged;    ///< `true` iff `grad_inf_norm < tol`.
+};
+
+// ── Internal helpers ──────────────────────────────────────────────────────────
+
+namespace detail {
+
+// Solve A·Δx = rhs with SimplicialLDLT; on failure fall back to SparseQR.
+// Returns Δx.  ok is set to false only if both solvers fail.
+// If fallback_used is non-null, it is set to true iff SparseQR was needed.
+inline Eigen::VectorXd solve_with_fallback(
+    const Eigen::SparseMatrix<double>& A,
+    const Eigen::VectorXd&             rhs,
+    bool&                              ok,
+    bool*                              fallback_used = nullptr)
+{
+    if (fallback_used) *fallback_used = false;
+
+    Eigen::SimplicialLDLT<Eigen::SparseMatrix<double>> ldlt(A);
+    if (ldlt.info() == Eigen::Success) {
+        Eigen::VectorXd dx = ldlt.solve(rhs);
+        if (ldlt.info() == Eigen::Success) { ok = true; return dx; }
+    }
+    // Fallback: SparseQR — handles singular/rank-deficient H (gauge modes).
+    if (fallback_used) *fallback_used = true;
+    Eigen::SparseQR<Eigen::SparseMatrix<double>, Eigen::COLAMDOrdering<int>> qr(A);
+    if (qr.info() == Eigen::Success) {
+        Eigen::VectorXd dx = qr.solve(rhs);
+        if (qr.info() == Eigen::Success) { ok = true; return dx; }
+    }
+    ok = false;
+    return Eigen::VectorXd::Zero(rhs.size());
+}
+
+} // namespace detail
+
+// ── Public linear-system solver (SparseQR fallback) ──────────────────────────
+//
+// Solve A·x = rhs with Eigen::SimplicialLDLT; if that fails (singular or
+// rank-deficient A), retry with Eigen::SparseQR which finds the minimum-norm
+// solution orthogonal to the null space.
+//
+// This is the same primitive used internally by all three Newton solvers.
+// Exposing it publicly lets callers (tests, downstream code) reuse the logic
+// and — via the optional fallback_used pointer — verify which code path ran.
+//
+//   fallback_used  – if non-null, set to true iff SparseQR was invoked
+//   Returns Eigen::VectorXd::Zero(rhs.size()) if both solvers fail.
+/// Solve `A·x = rhs` with the same SimplicialLDLT → SparseQR fallback
+/// strategy used inside all three Newton solvers.  If `fallback_used`
+/// is non-null, it is set to `true` iff the SparseQR fallback ran.
+/// Returns `Eigen::VectorXd::Zero(rhs.size())` if both solvers fail.
+inline Eigen::VectorXd solve_linear_system(
+    const Eigen::SparseMatrix<double>& A,
+    const Eigen::VectorXd&             rhs,
+    bool*                              fallback_used = nullptr)
+{
+    bool ok = false;
+    return detail::solve_with_fallback(A, rhs, ok, fallback_used);
+}
+
+namespace detail {   // re-open for the remaining helpers
+
+// Backtracking line search: find the largest α in {1, 0.5, 0.25, …} such that
+// ||G(x + α·Δx)||₂ < ||G(x)||₂.  Returns the accepted step (α may stay 1).
+template <typename GradFn>
+inline std::vector<double> line_search(
+    const std::vector<double>& x,
+    const Eigen::VectorXd&     dx,
+    double                     norm0,
+    GradFn&&                   grad_fn,
+    int                        max_halvings = 20)
+{
+    const int n = static_cast<int>(x.size());
+    double alpha = 1.0;
+    std::vector<double> xnew(static_cast<std::size_t>(n));
+
+    for (int ls = 0; ls < max_halvings; ++ls) {
+        for (int i = 0; i < n; ++i)
+            xnew[static_cast<std::size_t>(i)] = x[static_cast<std::size_t>(i)]
+                                                + alpha * dx[i];
+        auto Gnew = grad_fn(xnew);
+        double norm_new = 0.0;
+        for (double v : Gnew) norm_new += v * v;
+        norm_new = std::sqrt(norm_new);
+        if (norm_new < norm0) return xnew;
+        alpha *= 0.5;
+    }
+    // No improvement found — return best attempt (full step)
+    for (int i = 0; i < n; ++i)
+        xnew[static_cast<std::size_t>(i)] = x[static_cast<std::size_t>(i)] + dx[i];
+    return xnew;
+}
+
+} // namespace detail
+
+// ── Euclidean Newton solver ────────────────────────────────────────────────────
+
+/// Solve the Euclidean discrete conformal problem: find u ∈ ℝ^V such that
+/// Σ_{faces adj v} α_v(u) = Θ_v for all vertices v.
+///
+/// Starting from x0, Newton's method minimises E(u) (the Euclidean DCE energy,
+/// which is convex) by iterating  u ← u − H⁻¹·G  with backtracking line search.
+/// The Hessian H is the cotangent Laplacian — PSD with one zero eigenvalue on
+/// closed surfaces (gauge mode).  A SparseQR fallback handles this automatically.
+///
+/// \param mesh      Input triangulated surface (edges must carry lambda0 + theta_v).
+/// \param x0        Initial DOF vector (length = number of free vertices).
+///                  Pass all-zeros for a flat start (typical).
+/// \param m         EuclideanMaps: lambda0[e], theta_v[v], v_idx[v] must be set.
+///                  Call setup_euclidean_maps() + compute_euclidean_lambda0_from_mesh()
+///                  + enforce_gauss_bonnet() before passing here.
+/// \param tol       Convergence threshold on max |G_i|.  Default: 1e-8.
+/// \param max_iter  Maximum Newton iterations.  Default: 200.
+/// \return          NewtonResult{x*, iterations, grad_inf_norm, converged}.
+///
+/// \note On closed meshes without a pinned vertex, SimplicialLDLT detects the
+///       gauge singularity and falls back to SparseQR automatically.
+///
+/// \see doc/math/discrete-conformal-theory.md §3 for the mathematical background.
+inline NewtonResult newton_euclidean(
+    ConformalMesh&             mesh,
+    std::vector<double>        x0,
+    const EuclideanMaps&       m,
+    double                     tol      = 1e-8,
+    int                        max_iter = 200)
+{
+    std::vector<double> x = x0;
+    const int n = static_cast<int>(x.size());
+
+    NewtonResult res;
+    res.converged      = false;
+    res.iterations     = 0;
+    res.grad_inf_norm  = 0.0;
+
+    Eigen::SimplicialLDLT<Eigen::SparseMatrix<double>> solver;
+
+    for (int iter = 0; iter < max_iter; ++iter) {
+        // ── Gradient ──────────────────────────────────────────────────────────
+        auto G_std = euclidean_gradient(mesh, x, m);
+        Eigen::Map<const Eigen::VectorXd> G(G_std.data(), n);
+
+        double inf_norm = G.cwiseAbs().maxCoeff();
+        if (inf_norm < tol) {
+            res.converged     = true;
+            res.grad_inf_norm = inf_norm;
+            res.iterations    = iter;
+            res.x             = x;
+            return res;
+        }
+
+        // ── Hessian + solve H·Δx = −G  (SparseQR fallback for singular H) ──
+        auto H = euclidean_hessian(mesh, x, m);
+        bool ok = false;
+        Eigen::VectorXd dx = detail::solve_with_fallback(H, -G, ok);
+        if (!ok) break;
+
+        // ── Backtracking line search ──────────────────────────────────────────
+        double norm0 = G.norm();
+        x = detail::line_search(x, dx, norm0,
+            [&](const std::vector<double>& xnew) {
+                return euclidean_gradient(mesh, xnew, m);
+            });
+
+        res.iterations = iter + 1;
+    }
+
+    // Report final gradient norm
+    auto G_final = euclidean_gradient(mesh, x, m);
+    double inf_final = 0.0;
+    for (double v : G_final) inf_final = std::max(inf_final, std::abs(v));
+    res.grad_inf_norm = inf_final;
+    res.x             = x;
+    return res;
+}
+
+// ── Spherical Newton solver ───────────────────────────────────────────────────
+
+/// Solve the spherical discrete conformal problem: find u ∈ ℝ^V such that
+/// Σ_{faces adj v} α_v(u) = Θ_v for all vertices v (genus-0 / sphere-like surfaces).
+///
+/// The spherical DCE energy is *concave*, so the Hessian H is NSD at the solution.
+/// The solver factorises −H (which is PSD) and solves (−H)·Δx = G.
+/// A gauge vertex must be pinned (set v_idx = -1) to remove the rotational mode.
+///
+/// \param mesh      Input triangulated surface, genus 0.
+/// \param x0        Initial DOF vector (length = free vertices, excluding gauge_vertex).
+///                  All-zeros is a good start.
+/// \param m         SphericalMaps: lambda0[e], theta_v[v], v_idx[v], gauge_vertex set.
+///                  Call setup_spherical_maps() + compute_spherical_lambda0_from_mesh()
+///                  + enforce_gauss_bonnet() (checks Σ(2π-Θ) > 0) before passing here.
+/// \param tol       Convergence threshold on max |G_i|.  Default: 1e-8.
+/// \param max_iter  Maximum Newton iterations.  Default: 200.
+/// \return          NewtonResult{x*, iterations, grad_inf_norm, converged}.
+///
+/// \note Unlike the Euclidean solver, the spherical solver does NOT need a SparseQR
+///       fallback — the gauge vertex pins the null mode directly.
+///
+/// \see doc/math/geometry-modes.md §Spherical for sign-convention details.
+inline NewtonResult newton_spherical(
+    ConformalMesh&             mesh,
+    std::vector<double>        x0,
+    const SphericalMaps&       m,
+    double                     tol      = 1e-8,
+    int                        max_iter = 200)
+{
+    std::vector<double> x = x0;
+    const int n = static_cast<int>(x.size());
+
+    NewtonResult res;
+    res.converged      = false;
+    res.iterations     = 0;
+    res.grad_inf_norm  = 0.0;
+
+    for (int iter = 0; iter < max_iter; ++iter) {
+        // ── Gradient ──────────────────────────────────────────────────────────
+        auto G_std = spherical_gradient(mesh, x, m);
+        Eigen::Map<const Eigen::VectorXd> G(G_std.data(), n);
+
+        double inf_norm = G.cwiseAbs().maxCoeff();
+        if (inf_norm < tol) {
+            res.converged     = true;
+            res.grad_inf_norm = inf_norm;
+            res.iterations    = iter;
+            res.x             = x;
+            return res;
+        }
+
+        // ── Hessian: negate to get PSD; solve (−H)·Δx = G  ──────────────────
+        auto H    = spherical_hessian(mesh, x, m);
+        auto negH = Eigen::SparseMatrix<double>(-H);
+        bool ok   = false;
+        Eigen::VectorXd dx = detail::solve_with_fallback(negH, G, ok);
+        if (!ok) break;
+
+        // ── Backtracking line search ──────────────────────────────────────────
+        double norm0 = G.norm();
+        x = detail::line_search(x, dx, norm0,
+            [&](const std::vector<double>& xnew) {
+                return spherical_gradient(mesh, xnew, m);
+            });
+
+        res.iterations = iter + 1;
+    }
+
+    auto G_final = spherical_gradient(mesh, x, m);
+    double inf_final = 0.0;
+    for (double v : G_final) inf_final = std::max(inf_final, std::abs(v));
+    res.grad_inf_norm = inf_final;
+    res.x             = x;
+    return res;
+}
+
+// ── HyperIdeal Newton solver ──────────────────────────────────────────────────
+
+/// Solve the hyper-ideal discrete conformal problem: find (b, a) ∈ ℝ^{V+E} such that
+/// Σ β_v(b,a) = Θ_v  and  Σ α_e(b,a) = θ_e  for all vertices v and edges e.
+///
+/// Used for genus-g surfaces (g ≥ 1) under hyperbolic cone metrics.
+/// The energy is *strictly convex* (Springborn 2020, Theorem 1.3), so Newton
+/// converges globally from any starting point.
+///
+/// DOF layout: first V_free entries are vertex variables b_v (hyper-ideal radii),
+/// followed by E entries for edge variables a_e (intersection angles).
+/// Use assign_all_dof_indices(mesh, maps) to set v_idx and e_idx automatically —
+/// no vertex needs to be pinned.
+///
+/// \param mesh      Input triangulated surface, genus g ≥ 1.
+/// \param x0        Initial DOF vector (length = V + E).  All-zeros typical.
+/// \param m         HyperIdealMaps: lambda0[e], theta_v[v], v_idx[v], e_idx[e] set.
+///                  Call setup_hyper_ideal_maps() + compute_hyper_ideal_lambda0_from_mesh().
+/// \param tol       Convergence threshold on max |G_i|.  Default: 1e-8.
+/// \param max_iter  Maximum Newton iterations.  Default: 200.
+/// \param hess_eps  Finite-difference step for Hessian approximation.  Default: 1e-5.
+///                  (Phase 9b will replace this with an analytic Hessian.)
+/// \return          NewtonResult{x*, iterations, grad_inf_norm, converged}.
+///
+/// \see Springborn (2020), Theorem 1.3 for the strict convexity proof.
+/// \see doc/math/geometry-modes.md §Hyper-ideal for DOF layout details.
+inline NewtonResult newton_hyper_ideal(
+    ConformalMesh&             mesh,
+    std::vector<double>        x0,
+    const HyperIdealMaps&      m,
+    double                     tol      = 1e-8,
+    int                        max_iter = 200,
+    double                     hess_eps = 1e-5)
+{
+    std::vector<double> x = x0;
+    const int n = static_cast<int>(x.size());
+
+    NewtonResult res;
+    res.converged      = false;
+    res.iterations     = 0;
+    res.grad_inf_norm  = 0.0;
+
+    for (int iter = 0; iter < max_iter; ++iter) {
+        // ── Gradient ──────────────────────────────────────────────────────────
+        auto G_std = evaluate_hyper_ideal(mesh, x, m, /*energy=*/false).gradient;
+        Eigen::Map<const Eigen::VectorXd> G(G_std.data(), n);
+
+        double inf_norm = G.cwiseAbs().maxCoeff();
+        if (inf_norm < tol) {
+            res.converged     = true;
+            res.grad_inf_norm = inf_norm;
+            res.iterations    = iter;
+            res.x             = x;
+            return res;
+        }
+
+        // ── Hessian (numerical FD) + solve H·Δx = −G ─────────────────────────
+        auto H  = hyper_ideal_hessian_sym(mesh, x, m, hess_eps);
+        bool ok = false;
+        Eigen::VectorXd dx = detail::solve_with_fallback(H, -G, ok);
+        if (!ok) break;
+
+        // ── Backtracking line search ──────────────────────────────────────────
+        double norm0 = G.norm();
+        x = detail::line_search(x, dx, norm0,
+            [&](const std::vector<double>& xnew) {
+                return evaluate_hyper_ideal(mesh, xnew, m, false).gradient;
+            });
+
+        res.iterations = iter + 1;
+    }
+
+    auto G_final = evaluate_hyper_ideal(mesh, x, m, false).gradient;
+    double inf_final = 0.0;
+    for (double v : G_final) inf_final = std::max(inf_final, std::abs(v));
+    res.grad_inf_norm = inf_final;
+    res.x             = x;
+    return res;
+}
+
+// ── CP-Euclidean Newton solver (Phase 9a.1) ───────────────────────────────────
+
+/// Solve the CP-Euclidean circle-packing problem: find ρ ∈ ℝ^F such that the
+/// per-face angle sums match φ_f at every free face.
+///
+/// The CP-Euclidean energy (Bobenko-Pinkall-Springborn 2010 §6) is strictly
+/// convex on its open domain of validity, so the Hessian H is PSD and the
+/// solution is unique up to the gauge mode pinned by `f_idx == −1`.
+/// `cp_euclidean_hessian` provides the analytic 2×2-per-edge formula
+/// `h_jk = sin θ / (cosh Δρ − cos θ)`; no FD machinery is required.
+///
+/// \param mesh      Input triangle mesh (closed or with boundary).
+/// \param x0        Initial DOF vector (length = number of free faces).
+///                  All-zeros is a valid start.
+/// \param m         CPEuclideanMaps: f_idx must have one pinned face
+///                  (`f_idx[f0] == −1`); theta_e and phi_f set by the caller.
+/// \param tol       Convergence threshold on `‖G‖∞`.  Default: 1e-8.
+/// \param max_iter  Newton iteration limit.  Default: 200.
+/// \return          NewtonResult{x*, iterations, grad_inf_norm, converged}.
+///
+/// \note  Unlike the Euclidean solver, the CP-Euclidean Hessian is exact
+///        (analytic), so the SparseQR fallback only triggers in genuine
+///        gauge-singular situations (no pinned face).
+/// \see   doc/architecture/phase-9a-validation.md §1 for the BPS-2010 mapping.
+inline NewtonResult newton_cp_euclidean(
+    ConformalMesh&             mesh,
+    std::vector<double>        x0,
+    const CPEuclideanMaps&     m,
+    double                     tol      = 1e-8,
+    int                        max_iter = 200)
+{
+    std::vector<double> x = x0;
+    const int n = static_cast<int>(x.size());
+
+    NewtonResult res;
+    res.converged      = false;
+    res.iterations     = 0;
+    res.grad_inf_norm  = 0.0;
+
+    for (int iter = 0; iter < max_iter; ++iter) {
+        auto G_std = cp_euclidean_gradient(mesh, x, m);
+        Eigen::Map<const Eigen::VectorXd> G(G_std.data(), n);
+
+        double inf_norm = G.cwiseAbs().maxCoeff();
+        if (inf_norm < tol) {
+            res.converged     = true;
+            res.grad_inf_norm = inf_norm;
+            res.iterations    = iter;
+            res.x             = x;
+            return res;
+        }
+
+        auto H = cp_euclidean_hessian(mesh, x, m);
+        bool ok = false;
+        Eigen::VectorXd dx = detail::solve_with_fallback(H, -G, ok);
+        if (!ok) break;
+
+        double norm0 = G.norm();
+        x = detail::line_search(x, dx, norm0,
+            [&](const std::vector<double>& xnew) {
+                return cp_euclidean_gradient(mesh, xnew, m);
+            });
+
+        res.iterations = iter + 1;
+    }
+
+    auto G_final = cp_euclidean_gradient(mesh, x, m);
+    double inf_final = 0.0;
+    for (double v : G_final) inf_final = std::max(inf_final, std::abs(v));
+    res.grad_inf_norm = inf_final;
+    res.x             = x;
+    return res;
+}
+
+// ── Inversive-Distance Newton solver (Phase 9a.2) ─────────────────────────────
+
+/// Solve the inversive-distance circle-packing problem: find u ∈ ℝ^V such that
+/// Σ_{faces adj v} α_v(u) = Θ_v at every free vertex (Luo 2004 Lemma 3.1).
+///
+/// The inversive-distance energy is (locally) strictly convex on the open
+/// domain where every triangle satisfies the inequalities.  Luo's 1-form is
+/// closed there, so the path-integral energy is well-defined.
+///
+/// MVP implementation: the Hessian is computed by **finite differences** of
+/// the analytic gradient (same pattern as the Phase 4a HyperIdeal solver).
+/// An analytic Hessian via Glickenstein 2011 eq. (4.6) is tracked in
+/// `doc/roadmap/research-track.md` as Phase 9a.2-analytic.
+///
+/// \param mesh      Input triangle mesh.
+/// \param x0        Initial DOF vector (length = number of free vertices).
+/// \param m         InversiveDistanceMaps: v_idx has at least one pinned
+///                  vertex; I_e and r0 set by compute_inversive_distance_init.
+/// \param tol       Convergence threshold on `‖G‖∞`.  Default: 1e-8.
+/// \param max_iter  Newton iteration limit.  Default: 200.
+/// \param hess_eps  FD step size for the Hessian.  Default: 1e-5.
+/// \return          NewtonResult{x*, iterations, grad_inf_norm, converged}.
+///
+/// \note  Convergence is sensitive to the initial point: u = 0 is the
+///        natural choice when `compute_inversive_distance_init_from_mesh`
+///        has been called, since the Bowers-Stephenson identity reconstructs
+///        the input edge lengths at u = 0.
+inline NewtonResult newton_inversive_distance(
+    ConformalMesh&                mesh,
+    std::vector<double>           x0,
+    const InversiveDistanceMaps&  m,
+    double                        tol      = 1e-8,
+    int                           max_iter = 200,
+    double                        hess_eps = 1e-5)
+{
+    std::vector<double> x = x0;
+    const int n = static_cast<int>(x.size());
+
+    NewtonResult res;
+    res.converged      = false;
+    res.iterations     = 0;
+    res.grad_inf_norm  = 0.0;
+
+    // Local FD Hessian builder — n × (cost of gradient eval).
+    auto build_hessian = [&](const std::vector<double>& xc) -> Eigen::SparseMatrix<double> {
+        std::vector<Eigen::Triplet<double>> trips;
+        trips.reserve(static_cast<std::size_t>(n) * 16);   // sparse heuristic
+
+        std::vector<double> xp = xc, xm = xc;
+        for (int j = 0; j < n; ++j) {
+            const std::size_t sj = static_cast<std::size_t>(j);
+            xp[sj] = xc[sj] + hess_eps;
+            xm[sj] = xc[sj] - hess_eps;
+
+            auto Gp = inversive_distance_gradient(mesh, xp, m);
+            auto Gm = inversive_distance_gradient(mesh, xm, m);
+
+            xp[sj] = xm[sj] = xc[sj];   // restore
+
+            for (int i = 0; i < n; ++i) {
+                double val = (Gp[static_cast<std::size_t>(i)]
+                            - Gm[static_cast<std::size_t>(i)])
+                             / (2.0 * hess_eps);
+                if (std::abs(val) > 1e-15)
+                    trips.emplace_back(i, j, val);
+            }
+        }
+        Eigen::SparseMatrix<double> H(n, n);
+        H.setFromTriplets(trips.begin(), trips.end());
+        // Symmetrise — FD rounding may introduce tiny asymmetries.
+        Eigen::SparseMatrix<double> Ht = H.transpose();
+        return (H + Ht) * 0.5;
+    };
+
+    for (int iter = 0; iter < max_iter; ++iter) {
+        auto G_std = inversive_distance_gradient(mesh, x, m);
+        Eigen::Map<const Eigen::VectorXd> G(G_std.data(), n);
+
+        double inf_norm = G.cwiseAbs().maxCoeff();
+        if (inf_norm < tol) {
+            res.converged     = true;
+            res.grad_inf_norm = inf_norm;
+            res.iterations    = iter;
+            res.x             = x;
+            return res;
+        }
+
+        auto H = build_hessian(x);
+        bool ok = false;
+        Eigen::VectorXd dx = detail::solve_with_fallback(H, -G, ok);
+        if (!ok) break;
+
+        double norm0 = G.norm();
+        x = detail::line_search(x, dx, norm0,
+            [&](const std::vector<double>& xnew) {
+                return inversive_distance_gradient(mesh, xnew, m);
+            });
+
+        res.iterations = iter + 1;
+    }
+
+    auto G_final = inversive_distance_gradient(mesh, x, m);
+    double inf_final = 0.0;
+    for (double v : G_final) inf_final = std::max(inf_final, std::abs(v));
+    res.grad_inf_norm = inf_final;
+    res.x             = x;
+    return res;
+}
+
+} // namespace conformallab

code/include/p2_utility.hpp

@@ -0,0 +1,98 @@
+#pragma once
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
+
+// 2-D projective geometry utilities for the Euclidean signature.
+// Ported from de.jreality.math.P2 and de.varylab.discreteconformal.math.P2Big.
+//
+// Points and lines are represented as homogeneous 3-vectors (x, y, w).
+// In the Euclidean case a finite point (px, py) is stored as (px, py, 1).
+
+#include <Eigen/Dense>
+#include <cmath>
+
+namespace conformallab {
+
+// ── Point / line duality ──────────────────────────────────────────────────────
+
+/// Cross-product point–line duality in P²: returns the intersection
+/// of two lines (or the line through two points).  Same as Java
+/// `P2.pointFromLines` / `P2.lineFromPoints`.
+inline Eigen::Vector3d pointFromLines(const Eigen::Vector3d& l1,
+                                      const Eigen::Vector3d& l2) {
+    return l1.cross(l2);
+}
+
+// ── Euclidean perpendicular bisector ─────────────────────────────────────────
+
+/// Homogeneous line coordinates `(a, b, c)` of the perpendicular
+/// bisector of `[p, q]` in the Euclidean plane (`ax + by + c = 0`).
+/// Same as Java `P2.perpendicularBisector(p, q, Pn.EUCLIDEAN)`.
+inline Eigen::Vector3d perpendicularBisectorEuclidean(const Eigen::Vector3d& p_h,
+                                                      const Eigen::Vector3d& q_h) {
+    // Dehomogenize
+    Eigen::Vector2d p = p_h.head<2>() / p_h(2);
+    Eigen::Vector2d q = q_h.head<2>() / q_h(2);
+
+    // Direction vector (p → direction, matching jReality sign convention)
+    Eigen::Vector2d d = p - q;
+
+    // Midpoint
+    Eigen::Vector2d m = (p + q) * 0.5;
+
+    // Line: d[0]*(x - m[0]) + d[1]*(y - m[1]) = 0
+    //      = d[0]*x + d[1]*y - (d[0]*m[0] + d[1]*m[1])
+    double c = -(d(0) * m(0) + d(1) * m(1));
+    return {d(0), d(1), c};
+}
+
+/// Euclidean distance between two P² homogeneous points (dehomogenises both).
+inline double euclideanDistanceP2(const Eigen::Vector3d& p_h,
+                                  const Eigen::Vector3d& q_h) {
+    Eigen::Vector2d p = p_h.head<2>() / p_h(2);
+    Eigen::Vector2d q = q_h.head<2>() / q_h(2);
+    return (p - q).norm();
+}
+
+// ── Direct Euclidean isometry from two point-frames ──────────────────────────
+
+/// Build the 3×3 projective frame matrix anchored at `p0` with `p1`
+/// defining the positive x-direction (Euclidean case).  Columns:
+/// `[dehom(p0), unit_dir(p0→p1), perp_dir]`.
+template <typename S>
+Eigen::Matrix<S, 3, 3> makeFrameMatrix(Eigen::Matrix<S, 3, 1> p0_h,
+                                        Eigen::Matrix<S, 3, 1> p1_h) {
+    // Dehomogenize
+    Eigen::Matrix<S, 3, 1> p0 = p0_h / p0_h(2);          // (px, py, 1)
+    Eigen::Matrix<S, 3, 1> p1_d = p1_h / p1_h(2);
+
+    // Unit direction p0 → p1
+    Eigen::Matrix<S, 2, 1> dir2 = (p1_d - p0).template head<2>();
+    dir2.normalize();
+    Eigen::Matrix<S, 3, 1> p1n(dir2(0), dir2(1), S(0));
+
+    // Perpendicular direction
+    Eigen::Matrix<S, 3, 1> p2(-dir2(1), dir2(0), S(0));
+
+    Eigen::Matrix<S, 3, 3> M;
+    M.col(0) = p0;
+    M.col(1) = p1n;
+    M.col(2) = p2;
+    return M;
+}
+
+/// 3×3 Euclidean isometry (as a projective matrix) that maps the
+/// frame `(s1, s2)` to the frame `(t1, t2)`.  Same as Java
+/// `P2.makeDirectIsometryFromFrames(..., Pn.EUCLIDEAN)`.
+template <typename S>
+Eigen::Matrix<S, 3, 3> makeDirectIsometryFromFramesEuclidean(
+    Eigen::Matrix<S, 3, 1> s1, Eigen::Matrix<S, 3, 1> s2,
+    Eigen::Matrix<S, 3, 1> t1, Eigen::Matrix<S, 3, 1> t2)
+{
+    auto toS = makeFrameMatrix<S>(s1, s2);
+    auto toT = makeFrameMatrix<S>(t1, t2);
+    return toT * toS.inverse();
+}
+
+} // namespace conformallab

code/include/period_matrix.hpp

@@ -0,0 +1,166 @@
+#pragma once
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
+// period_matrix.hpp
+//
+// Phase 7 — Period matrix for closed surfaces with Euclidean (flat) metric.
+//
+// For a closed genus-g surface with Euclidean conformal structure the holonomy
+// group is generated by 2g translations ω_1, ..., ω_{2g} ∈ ℂ ≅ ℝ².
+//
+// ─── Genus-1 (flat torus) ────────────────────────────────────────────────────
+//
+//   The lattice Λ = ℤ·ω_1 ⊕ ℤ·ω_2 determines the conformal type.
+//
+//   Period ratio:  τ = ω_2 / ω_1   (as complex numbers)
+//
+//   By convention choose ω_1 such that Im(τ) > 0.
+//   The conformal modulus / Teichmüller parameter is the SL(2,ℤ)-orbit of τ.
+//
+//   Reduction to fundamental domain {|τ| ≥ 1, −½ ≤ Re(τ) < ½, Im(τ) > 0}:
+//     S:  τ ↦ −1/τ        (inversion)
+//     T:  τ ↦  τ + 1      (translation)
+//   Apply S and T repeatedly until τ is in the fundamental domain.
+//
+// ─── Genus g > 1 ─────────────────────────────────────────────────────────────
+//
+//   The full period matrix is a g×g complex symmetric matrix Ω with positive
+//   definite imaginary part (Siegel upper half-space H_g).
+//   Computing Ω from holonomy data requires integration of holomorphic
+//   differentials — not implemented here.  For g > 1, this function returns
+//   only the 2×2 block for the first pair of generators.
+//
+// ─── API ─────────────────────────────────────────────────────────────────────
+//
+//   PeriodData pd = compute_period_matrix(holonomy);
+//   pd.tau          — complex period ratio τ (genus 1)
+//   pd.omega        — holonomy generators as complex numbers (size = 2g)
+//   pd.in_fundamental_domain — whether τ has been reduced
+//
+//   std::complex<double> reduce_to_fundamental_domain(τ)  — apply SL(2,ℤ)
+
+#include "layout.hpp"
+#include <complex>
+#include <cmath>
+#include <vector>
+#include <stdexcept>
+#include <sstream>
+
+namespace conformallab {
+
+// ─────────────────────────────────────────────────────────────────────────────
+// PeriodData
+// ─────────────────────────────────────────────────────────────────────────────
+
+/// Period-matrix data for a genus-g closed surface.  For genus 1 the
+/// conformal type is fully captured by `τ = ω₂ / ω₁ ∈ ℍ`.
+struct PeriodData {
+    /// Lattice generators as complex numbers (one per cut edge).
+    /// omega[i] = translations[i].x() + i·translations[i].y()
+    std::vector<std::complex<double>> omega;
+
+    /// Period ratio τ = omega[1] / omega[0]  (genus-1 only).
+    /// Undefined (NaN) for genus != 1 or if holonomy has fewer than 2 generators.
+    std::complex<double> tau = std::complex<double>(
+        std::numeric_limits<double>::quiet_NaN(), 0.0);
+
+    /// True if τ has been reduced to the standard fundamental domain.
+    bool in_fundamental_domain = false;
+
+    /// Genus of the surface = `|omega| / 2`.
+    int genus() const { return static_cast<int>(omega.size()) / 2; }
+};
+
+// ─────────────────────────────────────────────────────────────────────────────
+// reduce_to_fundamental_domain
+//
+// Applies SL(2,ℤ) generators S: τ↦−1/τ and T: τ↦τ+1 to bring τ into
+//   F = { τ ∈ ℍ : |τ| ≥ 1,  −½ ≤ Re(τ) < ½ }
+//
+// Returns the reduced τ.  Throws if Im(τ) ≤ 0 (not in upper half-plane).
+// ─────────────────────────────────────────────────────────────────────────────
+/// Reduce `τ ∈ ℍ` to the standard SL(2,ℤ) fundamental domain
+/// `F = { τ ∈ ℍ : |τ| ≥ 1, −½ ≤ Re τ < ½ }` via the generators
+/// `S: τ↦−1/τ` and `T: τ↦τ+1`.  Throws if `Im τ ≤ 0`.
+inline std::complex<double> reduce_to_fundamental_domain(std::complex<double> tau)
+{
+    if (tau.imag() <= 0.0) {
+        std::ostringstream msg;
+        msg << "period_matrix: τ = " << tau.real() << " + " << tau.imag()
+            << "i is not in the upper half-plane (Im(τ) must be > 0).";
+        throw std::domain_error(msg.str());
+    }
+
+    // Iterate at most 200 times (convergence is rapid for well-conditioned τ)
+    for (int k = 0; k < 200; ++k) {
+        // T step: shift Re(τ) into [−½, ½)
+        double re = tau.real();
+        long   n  = static_cast<long>(std::floor(re + 0.5));
+        tau -= std::complex<double>(static_cast<double>(n), 0.0);
+
+        // S step: if |τ| < 1, apply τ ← −1/τ
+        if (std::abs(tau) < 1.0 - 1e-12) {
+            tau = -1.0 / tau;
+        } else {
+            break;
+        }
+    }
+    return tau;
+}
+
+// ─────────────────────────────────────────────────────────────────────────────
+// is_in_fundamental_domain — check membership in F with tolerance tol.
+// ─────────────────────────────────────────────────────────────────────────────
+/// `true` iff `τ` lies inside the standard SL(2,ℤ) fundamental domain
+/// with tolerance `tol`.
+inline bool is_in_fundamental_domain(std::complex<double> tau, double tol = 1e-9)
+{
+    if (tau.imag() <= 0.0) return false;
+    if (std::abs(tau.real()) > 0.5 + tol) return false;
+    if (std::abs(tau) < 1.0 - tol) return false;
+    return true;
+}
+
+// ─────────────────────────────────────────────────────────────────────────────
+// compute_period_matrix
+//
+// Computes the period data from the Euclidean holonomy translations.
+// For genus-1 surfaces, also reduces τ to the fundamental domain.
+// ─────────────────────────────────────────────────────────────────────────────
+/// Compute the period data from the Euclidean holonomy translations.
+/// For genus 1, also reduces `τ` to the SL(2,ℤ) fundamental domain
+/// when `reduce` is `true` (default).
+inline PeriodData compute_period_matrix(const HolonomyData& hol, bool reduce = true)
+{
+    PeriodData pd;
+    pd.omega.reserve(hol.translations.size());
+    for (auto& t : hol.translations)
+        pd.omega.push_back(std::complex<double>(t.x(), t.y()));
+
+    if (pd.omega.size() < 2) return pd;   // need at least 2 generators
+
+    // τ = ω_2 / ω_1  — choose ω_1 such that Im(τ) > 0
+    std::complex<double> w1 = pd.omega[0];
+    std::complex<double> w2 = pd.omega[1];
+    if (std::abs(w1) < 1e-14) return pd;
+
+    std::complex<double> tau = w2 / w1;
+    if (tau.imag() < 0.0) {
+        tau = std::conj(tau);              // swap orientation
+        w1  = std::conj(w1);
+        w2  = std::conj(w2);
+        pd.omega[0] = w1;
+        pd.omega[1] = w2;
+    }
+    if (tau.imag() < 0.0) return pd;      // degenerate
+
+    if (reduce) {
+        tau = reduce_to_fundamental_domain(tau);
+        pd.in_fundamental_domain = true;
+    }
+    pd.tau = tau;
+    return pd;
+}
+
+} // namespace conformallab

code/include/projective_math.hpp

@@ -1,10 +1,14 @@
 #pragma once
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
 
 // Projective and hyperbolic geometry utilities.
 // Ported from de.jreality.math.Pn / Rn and
 // de.varylab.discreteconformal.uniformization.SurfaceCurveUtility (Java).
 
-#include <Eigen/Dense>
+#include <Eigen/Core>  // downgraded from <Eigen/Dense>: this header only
+                      // uses Matrix/Vector primitives, no decompositions.
 #include <cmath>
 #include <algorithm>
 #include <array>
@@ -12,17 +16,15 @@
 
 namespace conformallab {
 
-// Divide a homogeneous vector by its last component.
-// Corresponds to Java Pn.dehomogenize().
+/// Dehomogenise: divide a homogeneous vector by its last component.
+/// Same as Java `Pn.dehomogenize()`.
 inline Eigen::VectorXd dehomogenize(const Eigen::VectorXd& p) {
     return p / p(p.size() - 1);
 }
 
-// Hyperbolic distance between two homogeneous vectors of the same dimension.
-// The last component is the "timelike" coordinate (jReality convention).
-// Inner product: <p,q> = -sum_i p_i*q_i + p_last * q_last
-// Distance: arcosh(<p̂, q̂>) where p̂ normalises to the hyperboloid.
-// Corresponds to Java Pn.distanceBetween(p, q, Pn.HYPERBOLIC).
+/// Hyperbolic distance `arcosh(⟨p̂, q̂⟩)` between two homogeneous
+/// vectors (last component = timelike coordinate; jReality convention).
+/// Same as Java `Pn.distanceBetween(p, q, Pn.HYPERBOLIC)`.
 inline double hyperbolicDistance(const Eigen::VectorXd& p,
                                  const Eigen::VectorXd& q) {
     int n = static_cast<int>(p.size());
@@ -34,10 +36,8 @@ inline double hyperbolicDistance(const Eigen::VectorXd& p,
     return std::acosh(std::max(1.0, inner));
 }
 
-// Check whether a homogeneous point p lies on the segment [s[0], s[1]].
-// Works for n-dimensional homogeneous coords; cross product uses the first
-// 3 spatial components after dehomogenization (matching jReality's Rn behaviour).
-// Corresponds to Java SurfaceCurveUtility.isOnSegment().
+/// `true` iff the homogeneous point `p_h` lies on the segment
+/// `[s0_h, s1_h]`.  Same as Java `SurfaceCurveUtility.isOnSegment()`.
 inline bool isOnSegment(const Eigen::VectorXd& p_h,
                         const Eigen::VectorXd& s0_h,
                         const Eigen::VectorXd& s1_h) {
@@ -63,10 +63,10 @@ inline bool isOnSegment(const Eigen::VectorXd& p_h,
     return true;
 }
 
-// Find the point on `target` that corresponds to `p` on `source`.
-// The parameter t is determined by hyperbolic distance ratios on `source`,
-// then applied as a linear interpolation on the dehomogenized `target`.
-// Corresponds to Java SurfaceCurveUtility.getPointOnCorrespondingSegment().
+/// Find the point on the target segment `(tgt0, tgt1)` corresponding
+/// to `p` on the source segment `(src0, src1)`, parametrised by
+/// hyperbolic distance ratios on the source.  Same as Java
+/// `SurfaceCurveUtility.getPointOnCorrespondingSegment()`.
 inline Eigen::VectorXd getPointOnCorrespondingSegment(
     const Eigen::VectorXd& p,
     const Eigen::VectorXd& src0,

code/include/serialization.hpp

@@ -0,0 +1,293 @@
+#pragma once
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
+// serialization.hpp
+//
+// Phase 5 — Save and load conformal map results in JSON and XML formats.
+//
+// JSON (nlohmann/json, bundled in deps/single_includes/json.hpp):
+//   save_result_json / load_result_json
+//
+// XML (hand-written, no external parser dependency):
+//   save_result_xml / load_result_xml
+//
+// Both formats store:
+//   - geometry type string ("euclidean" / "spherical" / "hyper_ideal")
+//   - mesh statistics (vertex/face count)
+//   - solver metadata (converged, iterations, grad_inf_norm)
+//   - DOF vector x
+//   - optional 2D or 3D layout positions
+//
+// XML schema (ConformalResult):
+//
+//   <?xml version="1.0" encoding="UTF-8"?>
+//   <ConformalResult geometry="euclidean" vertices="4" faces="2">
+//     <Solver converged="true" iterations="3" grad_inf_norm="1.43e-13"/>
+//     <DOFVector n="3">0.0 1.23e-13 2.87e-13</DOFVector>
+//     <Layout dim="2" n="4">
+//       0.0 0.0  1.0 0.0  0.5 0.866  1.5 0.866
+//     </Layout>
+//   </ConformalResult>
+
+#include "newton_solver.hpp"
+#include "layout.hpp"
+#include <json.hpp>
+#include <fstream>
+#include <sstream>
+#include <string>
+#include <vector>
+#include <stdexcept>
+#include <iomanip>
+
+namespace conformallab {
+
+// ════════════════════════════════════════════════════════════════════════════
+// JSON
+// ════════════════════════════════════════════════════════════════════════════
+
+/// Save the Newton-solver result (+ optional 2-D layout) to a JSON file.
+inline void save_result_json(
+    const std::string& path,
+    const NewtonResult& res,
+    const std::string&  geometry,
+    int                 n_vertices,
+    int                 n_faces,
+    const Layout2D*     layout2d = nullptr,
+    const Layout3D*     layout3d = nullptr)
+{
+    using json = nlohmann::json;
+    json j;
+    j["geometry"]  = geometry;
+    j["mesh"]      = { {"vertices", n_vertices}, {"faces", n_faces} };
+    j["solver"]    = {
+        {"converged",      res.converged},
+        {"iterations",     res.iterations},
+        {"grad_inf_norm",  res.grad_inf_norm}
+    };
+    j["dof_vector"] = res.x;
+
+    if (layout2d && layout2d->success) {
+        json uv = json::array();
+        for (auto& p : layout2d->uv) uv.push_back({p.x(), p.y()});
+        j["layout"] = { {"dim", 2}, {"uv", uv}, {"has_seam", layout2d->has_seam} };
+    }
+    if (layout3d && layout3d->success) {
+        json pos = json::array();
+        for (auto& p : layout3d->pos) pos.push_back({p.x(), p.y(), p.z()});
+        j["layout"] = { {"dim", 3}, {"pos", pos}, {"has_seam", layout3d->has_seam} };
+    }
+
+    std::ofstream ofs(path);
+    if (!ofs) throw std::runtime_error("Cannot write: " + path);
+    ofs << std::setw(2) << j << "\n";
+}
+
+/// Load a DOF vector from a JSON result file written by
+/// `save_result_json`.  If `res` is non-null its fields are filled too.
+inline std::vector<double> load_result_json(
+    const std::string& path,
+    NewtonResult*       res     = nullptr,
+    std::string*        geom    = nullptr,
+    Layout2D*           layout2d = nullptr)
+{
+    using json = nlohmann::json;
+    std::ifstream ifs(path);
+    if (!ifs) throw std::runtime_error("Cannot open: " + path);
+    json j; ifs >> j;
+
+    if (geom && j.contains("geometry"))
+        *geom = j["geometry"].get<std::string>();
+
+    std::vector<double> x = j.at("dof_vector").get<std::vector<double>>();
+
+    if (res) {
+        res->x             = x;
+        res->converged     = j["solver"]["converged"].get<bool>();
+        res->iterations    = j["solver"]["iterations"].get<int>();
+        res->grad_inf_norm = j["solver"]["grad_inf_norm"].get<double>();
+    }
+
+    if (layout2d && j.contains("layout") && j["layout"]["dim"] == 2) {
+        auto uv = j["layout"]["uv"];
+        layout2d->uv.resize(uv.size());
+        for (std::size_t i = 0; i < uv.size(); ++i)
+            layout2d->uv[i] = { uv[i][0].get<double>(), uv[i][1].get<double>() };
+        layout2d->has_seam = j["layout"].value("has_seam", false);
+        layout2d->success  = true;
+    }
+
+    return x;
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// XML
+// ════════════════════════════════════════════════════════════════════════════
+
+namespace detail_xml {
+
+// Minimal XML attribute escaping
+inline std::string xml_attr(double v)
+{
+    std::ostringstream s;
+    s << std::setprecision(15) << v;
+    return s.str();
+}
+
+// Write a flat vector of doubles as space-separated values
+inline std::string flat_doubles(const std::vector<double>& v)
+{
+    std::ostringstream s;
+    s << std::setprecision(15);
+    for (std::size_t i = 0; i < v.size(); ++i) {
+        if (i) s << ' ';
+        s << v[i];
+    }
+    return s.str();
+}
+
+// Simple attribute parser: find value of key= in a tag string
+inline std::string xml_get_attr(const std::string& tag, const std::string& key)
+{
+    auto pos = tag.find(key + "=\"");
+    if (pos == std::string::npos) return {};
+    pos += key.size() + 2;
+    auto end = tag.find('"', pos);
+    return tag.substr(pos, end - pos);
+}
+
+// Read all text content between the current position and </tag>
+inline std::string xml_read_text(std::istream& is, const std::string& close_tag)
+{
+    std::string buf, line;
+    std::string ctag = "</" + close_tag + ">";
+    while (std::getline(is, line)) {
+        auto pos = line.find(ctag);
+        if (pos != std::string::npos) {
+            buf += line.substr(0, pos);
+            return buf;
+        }
+        buf += line + " ";
+    }
+    return buf;
+}
+
+// Parse space-separated doubles
+inline std::vector<double> parse_doubles(const std::string& s)
+{
+    std::vector<double> v;
+    std::istringstream ss(s);
+    double d;
+    while (ss >> d) v.push_back(d);
+    return v;
+}
+
+} // namespace detail_xml
+
+/// Save the Newton-solver result (+ optional layout) to an XML file.
+inline void save_result_xml(
+    const std::string& path,
+    const NewtonResult& res,
+    const std::string&  geometry,
+    int                 n_vertices,
+    int                 n_faces,
+    const Layout2D*     layout2d = nullptr,
+    const Layout3D*     layout3d = nullptr)
+{
+    std::ofstream ofs(path);
+    if (!ofs) throw std::runtime_error("Cannot write: " + path);
+    ofs << std::setprecision(15);
+
+    ofs << "<?xml version=\"1.0\" encoding=\"UTF-8\"?>\n";
+    ofs << "<ConformalResult"
+        << " geometry=\"" << geometry << "\""
+        << " vertices=\""  << n_vertices << "\""
+        << " faces=\""     << n_faces << "\">\n";
+
+    ofs << "  <Solver"
+        << " converged=\""     << (res.converged ? "true" : "false") << "\""
+        << " iterations=\""    << res.iterations << "\""
+        << " grad_inf_norm=\"" << detail_xml::xml_attr(res.grad_inf_norm) << "\""
+        << "/>\n";
+
+    ofs << "  <DOFVector n=\"" << res.x.size() << "\">"
+        << detail_xml::flat_doubles(res.x)
+        << "</DOFVector>\n";
+
+    if (layout2d && layout2d->success) {
+        ofs << "  <Layout dim=\"2\" n=\"" << layout2d->uv.size()
+            << "\" has_seam=\"" << (layout2d->has_seam ? "true" : "false") << "\">\n";
+        for (auto& p : layout2d->uv)
+            ofs << "    " << p.x() << " " << p.y() << "\n";
+        ofs << "  </Layout>\n";
+    }
+    if (layout3d && layout3d->success) {
+        ofs << "  <Layout dim=\"3\" n=\"" << layout3d->pos.size()
+            << "\" has_seam=\"" << (layout3d->has_seam ? "true" : "false") << "\">\n";
+        for (auto& p : layout3d->pos)
+            ofs << "    " << p.x() << " " << p.y() << " " << p.z() << "\n";
+        ofs << "  </Layout>\n";
+    }
+
+    ofs << "</ConformalResult>\n";
+}
+
+/// Load a DOF vector from an XML result file written by
+/// `save_result_xml`.  If `res`, `geom`, `layout2d` are non-null they
+/// are filled as well.
+inline std::vector<double> load_result_xml(
+    const std::string& path,
+    NewtonResult*       res      = nullptr,
+    std::string*        geom     = nullptr,
+    Layout2D*           layout2d = nullptr)
+{
+    std::ifstream ifs(path);
+    if (!ifs) throw std::runtime_error("Cannot open: " + path);
+
+    std::vector<double> x;
+    std::string line;
+
+    while (std::getline(ifs, line)) {
+        // Root element
+        if (line.find("<ConformalResult") != std::string::npos) {
+            if (geom) *geom = detail_xml::xml_get_attr(line, "geometry");
+        }
+        // Solver metadata
+        else if (line.find("<Solver") != std::string::npos) {
+            if (res) {
+                res->converged     = (detail_xml::xml_get_attr(line, "converged") == "true");
+                res->iterations    = std::stoi(detail_xml::xml_get_attr(line, "iterations"));
+                res->grad_inf_norm = std::stod(detail_xml::xml_get_attr(line, "grad_inf_norm"));
+            }
+        }
+        // DOF vector
+        else if (line.find("<DOFVector") != std::string::npos) {
+            // Text may be on same line: <DOFVector n="...">0 1 2...</DOFVector>
+            auto open_end = line.find('>');
+            auto close    = line.find("</DOFVector>");
+            std::string text;
+            if (close != std::string::npos) {
+                text = line.substr(open_end + 1, close - open_end - 1);
+            } else {
+                text = line.substr(open_end + 1);
+                text += detail_xml::xml_read_text(ifs, "DOFVector");
+            }
+            x = detail_xml::parse_doubles(text);
+            if (res) res->x = x;
+        }
+        // Layout
+        else if (layout2d && line.find("<Layout") != std::string::npos
+                          && detail_xml::xml_get_attr(line, "dim") == "2") {
+            layout2d->has_seam = (detail_xml::xml_get_attr(line, "has_seam") == "true");
+            std::string text   = detail_xml::xml_read_text(ifs, "Layout");
+            auto vals          = detail_xml::parse_doubles(text);
+            layout2d->uv.clear();
+            for (std::size_t i = 0; i + 1 < vals.size(); i += 2)
+                layout2d->uv.push_back({vals[i], vals[i+1]});
+            layout2d->success = true;
+        }
+    }
+    return x;
+}
+
+} // namespace conformallab

code/include/spherical_functional.hpp

@@ -0,0 +1,505 @@
+#pragma once
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
+// spherical_functional.hpp
+//
+// Energy and gradient of the spherical discrete conformal functional
+// evaluated on a ConformalMesh (CGAL::Surface_mesh).
+//
+// Ported from de.varylab.discreteconformal.functional.SphericalFunctional.
+//
+// ┌──────────────────────────────────────────────────────────────────────────┐
+// │  DOFs                                                                    │
+// │    x[v_idx[v]]  = u_v   – conformal factor at vertex v                  │
+// │    x[e_idx[e]]  = λ_e   – edge log-length variable (optional)           │
+// │    -1 means "pinned" (u_v = 0 / λ_e = λ°_e fixed)                      │
+// │                                                                          │
+// │  Effective log-length:  Λ_ij = λ°_ij + u_i + u_j                       │
+// │  Spherical arc length:  l_ij = 2·asin(min(exp(Λ_ij/2), 1))             │
+// │                                                                          │
+// │  Gradient:                                                               │
+// │    ∂E/∂u_v  = Θ_v  –  Σ_{faces adj. v} α_v(face)                       │
+// │    ∂E/∂λ_e  = α_opp(face⁺) + α_opp(face⁻) – π                         │
+// │                                                                          │
+// │  Energy:                                                                 │
+// │    Computed as the Schläfli path integral                                │
+// │      E(x) = ∫₀¹ ⟨G(tx), x⟩ dt                                          │
+// │    using 10-point Gauss-Legendre quadrature. This is mathematically      │
+// │    exact for any conservative (curl-free) gradient G and is numerically  │
+// │    accurate to ~10⁻¹⁰ for smooth angle functions. The gradient check     │
+// │    therefore tests curl-freeness of G, which is the key integrability    │
+// │    condition for the spherical discrete conformal functional.             │
+// └──────────────────────────────────────────────────────────────────────────┘
+
+#include "conformal_mesh.hpp"
+#include "spherical_geometry.hpp"
+#include <CGAL/boost/graph/iterator.h>
+#include <vector>
+#include <cmath>
+#include <cstdint>
+
+namespace conformallab {
+
+// ── Property-map type aliases ─────────────────────────────────────────────────
+
+/// Property map vertex → `double` for the Spherical functional.
+using SpherVMapD = ConformalMesh::Property_map<Vertex_index, double>;
+/// Property map vertex → `int`    for the Spherical functional.
+using SpherVMapI = ConformalMesh::Property_map<Vertex_index, int>;
+/// Property map edge   → `double` for the Spherical functional.
+using SpherEMapD = ConformalMesh::Property_map<Edge_index,   double>;
+/// Property map edge   → `int`    for the Spherical functional.
+using SpherEMapI = ConformalMesh::Property_map<Edge_index,   int>;
+
+// ── Persistent map bundle ─────────────────────────────────────────────────────
+
+/// Bundle of the five property maps consumed by the Spherical functional.
+struct SphericalMaps {
+    SpherVMapI v_idx;    ///< DOF index per vertex (−1 = pinned / u_v = 0).
+    SpherEMapI e_idx;    ///< DOF index per edge   (−1 = no edge DOF).
+    SpherVMapD theta_v;  ///< Target cone angle Θᵥ (default 2π).
+    SpherEMapD theta_e;  ///< Target edge angle θₑ (default π).
+    SpherEMapD lambda0;  ///< Base log-length λ⁰ₑ (default 0).
+};
+
+// Defaults: theta_v = 2π, theta_e = π, lambda0 = 0.
+// lambda0 = 0 means exp(λ°/2)=1, i.e., l=π — degenerate unless u_i<0.
+/// Attach the five spherical property maps to `mesh` and return their
+/// handles.  Mirrors `setup_euclidean_maps` but uses the `"sv:"` /
+/// `"se:"` prefix so the two functionals can coexist on the same mesh
+/// (useful for cross-validation tests).
+///
+/// Defaults match the Euclidean defaults except that `lambda0 = 0` here
+/// gives `l_e = π` which is degenerate on the unit sphere — always call
+/// `compute_lambda0_from_mesh(mesh, m)` next on a real mesh.
+inline SphericalMaps setup_spherical_maps(ConformalMesh& mesh)
+{
+    SphericalMaps m;
+    m.v_idx  = mesh.add_property_map<Vertex_index, int>   ("sv:idx",    -1     ).first;
+    m.e_idx  = mesh.add_property_map<Edge_index,   int>   ("se:idx",    -1     ).first;
+    m.theta_v= mesh.add_property_map<Vertex_index, double>("sv:theta",  2.0*PI_SPHER).first;
+    m.theta_e= mesh.add_property_map<Edge_index,   double>("se:theta",  PI_SPHER    ).first;
+    m.lambda0= mesh.add_property_map<Edge_index,   double>("se:lam0",   0.0    ).first;
+    return m;
+}
+
+/// Assign sequential DOF indices `0..n-1` to all vertices (no edge DOFs).
+/// Caller is expected to pin one gauge vertex with `m.v_idx[v] = -1`.
+inline int assign_vertex_dof_indices(ConformalMesh& mesh, SphericalMaps& m)
+{
+    int idx = 0;
+    for (auto v : mesh.vertices()) m.v_idx[v] = idx++;
+    return idx;
+}
+
+/// Assign DOF indices for all vertices AND all edges (vertex-DOFs first,
+/// then edge-DOFs).  Mirrors `assign_euclidean_all_dof_indices` for the
+/// cyclic spherical formulation.
+inline int assign_all_spherical_dof_indices(ConformalMesh& mesh, SphericalMaps& m)
+{
+    int idx = 0;
+    for (auto v : mesh.vertices()) m.v_idx[v] = idx++;
+    for (auto e : mesh.edges())    m.e_idx[e] = idx++;
+    return idx;
+}
+
+/// Count the free DOFs (vertices + edges with index `≥ 0`).
+inline int spherical_dimension(const ConformalMesh& mesh, const SphericalMaps& m)
+{
+    int dim = 0;
+    for (auto v : mesh.vertices()) if (m.v_idx[v] >= 0) ++dim;
+    for (auto e : mesh.edges())    if (m.e_idx[e] >= 0) ++dim;
+    return dim;
+}
+
+/// Compute `λ°_e` for every edge from the input vertex positions,
+/// assuming `mesh` has vertices on the unit sphere.
+///
+/// Formula: `λ°_e = 2·log(sin(l_e / 2))` where `l_e = arccos(p_i · p_j)`
+/// is the spherical arc length of edge `e`.
+///
+/// \pre  Every vertex `v` of `mesh` lies on the unit sphere (norm = 1).
+/// \pre  No edge is degenerate (`p_i ≠ p_j` and `p_i ≠ -p_j`).
+inline void compute_lambda0_from_mesh(ConformalMesh& mesh, SphericalMaps& m)
+{
+    for (auto e : mesh.edges()) {
+        auto h  = mesh.halfedge(e);
+        auto p1 = mesh.point(mesh.source(h));
+        auto p2 = mesh.point(mesh.target(h));
+        // Dot product (works for unit-sphere vertices).
+        double dot = p1.x()*p2.x() + p1.y()*p2.y() + p1.z()*p2.z();
+        dot = std::max(-1.0, std::min(1.0, dot));
+        double l_e = std::acos(dot);              // spherical arc length
+        double half_sin = std::sin(l_e * 0.5);   // = exp(λ°/2)
+        if (half_sin > 1e-15)
+            m.lambda0[e] = 2.0 * std::log(half_sin);
+        else
+            m.lambda0[e] = -30.0;  // very short edge: essentially 0
+    }
+}
+
+// ── Evaluation result ─────────────────────────────────────────────────────────
+
+/// Output of `evaluate_spherical()` — energy plus optional gradient.
+struct SphericalResult {
+    double              energy   = 0.0; ///< Functional value at input DOFs.
+    std::vector<double> gradient;       ///< Gradient ∇E (empty if not requested).
+};
+
+// ── Internal helpers ──────────────────────────────────────────────────────────
+
+/// Read DOF value from `x` for index `idx`; return 0 if pinned (idx < 0).
+static inline double spher_dof_val(int idx, const std::vector<double>& x)
+{
+    return idx >= 0 ? x[static_cast<std::size_t>(idx)] : 0.0;
+}
+
+/// Convert a CGAL half-edge index to a plain `std::size_t` for vector indexing.
+static inline std::size_t spher_hidx(Halfedge_index h)
+{
+    return static_cast<std::size_t>(static_cast<std::uint32_t>(h));
+}
+
+// ── Gradient only (no energy) ─────────────────────────────────────────────────
+
+/// Compute the Spherical-functional gradient G(x):
+///   * `G_v = Θ_v − Σ_faces α_v(face)`
+///   * `G_e = α_opp(face⁺) + α_opp(face⁻) − θ_e`
+///
+/// The corner angle α_v is stored on half-edges via the convention
+/// `h_alpha[h] = corner angle at the vertex ACROSS FROM the edge of h
+/// in its face`, which makes both gradient accumulators natural.
+inline std::vector<double> spherical_gradient(
+    ConformalMesh&             mesh,
+    const std::vector<double>& x,
+    const SphericalMaps&       m)
+{
+    const int n = spherical_dimension(mesh, m);
+    std::vector<double> G(static_cast<std::size_t>(n), 0.0);
+
+    // Temporary per-halfedge corner-angle storage.
+    // h_alpha[h] = corner angle at the vertex opposite to the edge of h.
+    const std::size_t nh = mesh.number_of_halfedges();
+    std::vector<double> h_alpha(nh, 0.0);
+
+    // ── Pass 1: compute corner angles per face ────────────────────────────────
+    for (auto f : mesh.faces()) {
+        Halfedge_index h0 = mesh.halfedge(f);
+        Halfedge_index h1 = mesh.next(h0);
+        Halfedge_index h2 = mesh.next(h1);
+
+        Vertex_index v1 = mesh.source(h0);
+        Vertex_index v2 = mesh.source(h1);
+        Vertex_index v3 = mesh.source(h2);
+
+        Edge_index e12 = mesh.edge(h0);
+        Edge_index e23 = mesh.edge(h1);
+        Edge_index e31 = mesh.edge(h2);
+
+        // Effective log-length Λ_ij = λ°_ij + u_i + u_j
+        double u1 = spher_dof_val(m.v_idx[v1], x);
+        double u2 = spher_dof_val(m.v_idx[v2], x);
+        double u3 = spher_dof_val(m.v_idx[v3], x);
+
+        double lam12 = m.lambda0[e12] + u1 + u2 + spher_dof_val(m.e_idx[e12], x);
+        double lam23 = m.lambda0[e23] + u2 + u3 + spher_dof_val(m.e_idx[e23], x);
+        double lam31 = m.lambda0[e31] + u3 + u1 + spher_dof_val(m.e_idx[e31], x);
+
+        double l12 = spherical_l(lam12);
+        double l23 = spherical_l(lam23);
+        double l31 = spherical_l(lam31);
+
+        SphericalFaceAngles fa = spherical_angles(l12, l23, l31);
+
+        if (!fa.valid) continue;   // degenerate face: contributes 0
+
+        // Store convention: h_alpha[h] = corner angle at source(prev(h))
+        //   h0 (e12): opposite vertex is v3 → source(prev(h0)) = source(h2) = v3 → α3
+        //   h1 (e23): opposite vertex is v1 → source(prev(h1)) = source(h0) = v1 → α1
+        //   h2 (e31): opposite vertex is v2 → source(prev(h2)) = source(h1) = v2 → α2
+        h_alpha[spher_hidx(h0)] = fa.alpha3;
+        h_alpha[spher_hidx(h1)] = fa.alpha1;
+        h_alpha[spher_hidx(h2)] = fa.alpha2;
+    }
+
+    // ── Pass 2: accumulate gradient ───────────────────────────────────────────
+
+    // Vertex: G_v = Θ_v − Σ h_alpha[prev(h)] for each incoming non-border h to v.
+    for (auto v : mesh.vertices()) {
+        int iv = m.v_idx[v];
+        if (iv < 0) continue;
+        double sum_alpha = 0.0;
+        for (auto h : CGAL::halfedges_around_target(v, mesh)) {
+            if (mesh.is_border(h)) continue;
+            sum_alpha += h_alpha[spher_hidx(mesh.prev(h))];
+        }
+        G[static_cast<std::size_t>(iv)] = m.theta_v[v] - sum_alpha;
+    }
+
+    // Edge: G_e for λ_e additive (Λ_ij = λ°_ij + u_i + u_j + λ_e).
+    //
+    // From the Schläfli identity applied to the spherical face,
+    // the contribution of edge DOF λ_e from face f is:
+    //   a_f = (2·α_opp − S_f) / 2    where S_f = Σ angles in face f.
+    //
+    // Summing over both adjacent faces:
+    //   G_e = a_f+ + a_f−
+    //       = α_opp⁺ + α_opp⁻ − (S_f⁺ + S_f⁻) / 2 − θ_e
+    //
+    // For flat (Euclidean) triangles S_f = π, recovering the familiar
+    // α_opp⁺ + α_opp⁻ − π formula.  For spherical triangles S_f > π.
+    for (auto e : mesh.edges()) {
+        int ie = m.e_idx[e];
+        if (ie < 0) continue;
+        auto h  = mesh.halfedge(e);
+        auto ho = mesh.opposite(h);
+        double sum = 0.0;
+        if (!mesh.is_border(h)) {
+            double alpha_opp = h_alpha[spher_hidx(h)];
+            double S_f = alpha_opp
+                       + h_alpha[spher_hidx(mesh.next(h))]
+                       + h_alpha[spher_hidx(mesh.prev(h))];
+            sum += (2.0 * alpha_opp - S_f) * 0.5;
+        }
+        if (!mesh.is_border(ho)) {
+            double alpha_opp = h_alpha[spher_hidx(ho)];
+            double S_f = alpha_opp
+                       + h_alpha[spher_hidx(mesh.next(ho))]
+                       + h_alpha[spher_hidx(mesh.prev(ho))];
+            sum += (2.0 * alpha_opp - S_f) * 0.5;
+        }
+        G[static_cast<std::size_t>(ie)] = sum - m.theta_e[e];
+    }
+
+    return G;
+}
+
+// ── Energy via Gauss-Legendre path integral ───────────────────────────────────
+//
+// E(x) = ∫₀¹ ⟨G(tx), x⟩ dt
+//
+// This is the correct potential for any conservative G = ∇E.
+// Uses 10-point Gauss-Legendre quadrature; error ≈ O(h²⁰) for smooth G.
+//
+// 10-point GL nodes and weights on [0, 1] (transformed from [-1, 1]):
+//   t_k = (1 + s_k) / 2,  w_k = w_GL_k / 2
+/// Spherical energy `E(x) = ∫₀¹ ⟨G(t·x), x⟩ dt`, evaluated with
+/// 10-point Gauss-Legendre quadrature.  This is the correct potential
+/// for any conservative `G = ∇E`; error ≈ O(h²⁰) for smooth G.
+inline double spherical_energy(
+    ConformalMesh&             mesh,
+    const std::vector<double>& x,
+    const SphericalMaps&       m)
+{
+    // 10-point Gauss-Legendre nodes and weights on [-1, 1].
+    static const double gl_s[10] = {
+        -0.9739065285171717,  -0.8650633666889845,
+        -0.6794095682990244,  -0.4333953941292472,
+        -0.1488743389816312,   0.1488743389816312,
+         0.4333953941292472,   0.6794095682990244,
+         0.8650633666889845,   0.9739065285171717
+    };
+    static const double gl_w[10] = {
+        0.0666713443086881,  0.1494513491505806,
+        0.2190863625159820,  0.2692667193099963,
+        0.2955242247147529,  0.2955242247147529,
+        0.2692667193099963,  0.2190863625159820,
+        0.1494513491505806,  0.0666713443086881
+    };
+
+    const std::size_t n = x.size();
+    double E = 0.0;
+
+    for (int k = 0; k < 10; ++k) {
+        double t   = (1.0 + gl_s[k]) * 0.5;   // node on [0, 1]
+        double wt  = gl_w[k] * 0.5;            // weight on [0, 1]
+
+        // Evaluate G(t·x)
+        std::vector<double> tx(n);
+        for (std::size_t i = 0; i < n; ++i) tx[i] = t * x[i];
+
+        auto G = spherical_gradient(mesh, tx, m);
+
+        // Accumulate ⟨G(tx), x⟩ · wt
+        double dot = 0.0;
+        for (std::size_t i = 0; i < n; ++i) dot += G[i] * x[i];
+        E += wt * dot;
+    }
+    return E;
+}
+
+// ── Full evaluation (energy + gradient) ──────────────────────────────────────
+
+/// Evaluate the Spherical functional at DOFs `x`.  Returns energy and
+/// gradient (toggle via `need_energy` / `need_gradient`).
+inline SphericalResult evaluate_spherical(
+    ConformalMesh&             mesh,
+    const std::vector<double>& x,
+    const SphericalMaps&       m,
+    bool                       need_energy   = true,
+    bool                       need_gradient = true)
+{
+    SphericalResult res;
+    if (need_gradient)
+        res.gradient = spherical_gradient(mesh, x, m);
+    if (need_energy)
+        res.energy = spherical_energy(mesh, x, m);
+    return res;
+}
+
+/// Finite-difference gradient check for the Spherical functional
+/// (central differences).  Same defaults as the Java `FunctionalTest`.
+inline bool gradient_check_spherical(
+    ConformalMesh&             mesh,
+    const std::vector<double>& x0,
+    const SphericalMaps&       m,
+    double                     eps = 1E-5,
+    double                     tol = 1E-4)
+{
+    auto G = spherical_gradient(mesh, x0, m);
+    const int n = static_cast<int>(G.size());
+
+    std::vector<double> xp = x0, xm = x0;
+    bool ok = true;
+
+    for (int i = 0; i < n; ++i) {
+        std::size_t si = static_cast<std::size_t>(i);
+        xp[si] = x0[si] + eps;
+        xm[si] = x0[si] - eps;
+
+        double Ep = spherical_energy(mesh, xp, m);
+        double Em = spherical_energy(mesh, xm, m);
+
+        xp[si] = xm[si] = x0[si];   // restore
+
+        double fd    = (Ep - Em) / (2.0 * eps);
+        double err   = std::abs(G[si] - fd);
+        double scale = std::max(1.0, std::abs(G[si]));
+        if (err / scale > tol) ok = false;
+    }
+    return ok;
+}
+
+// ── Gauge-fix for closed spherical surfaces ───────────────────────────────────
+//
+// On a closed (boundaryless) spherical surface the energy E(u + t·1) has a
+// unique maximum w.r.t. t ∈ ℝ (the "global scale" gauge mode).  Without
+// fixing this gauge the functional is unbounded below and no solver converges.
+//
+// This function returns the scalar shift t* such that
+//   Σ_v G_v(x + t*·1_v)  =  0
+// i.e., the derivative of E(u+t·1) w.r.t. t is zero at t = t*.
+//
+// Apply the shift by adding t* to every vertex DOF in x.
+//
+// Implementation: bisection on f(t) = Σ_v G_v(x + t·1_v).
+//   f is strictly monotone decreasing (second derivative < 0) for a convex
+//   functional, so bisection converges in O(log₂(2·bracket/tol)) iterations.
+//
+// Parameters:
+//   bracket   – initial search interval [−bracket, +bracket]   (default 50)
+//   tol       – absolute tolerance on t*                        (default 1e-8)
+//
+// Returns 0.0 if the zero cannot be bracketed (already at gauge maximum,
+// or open surface — no shift needed).
+/// Find the global-scale gauge shift `t*` for the closed-spherical case
+/// (see comment block above for the maths).  Apply via `apply_spherical_gauge`.
+inline double spherical_gauge_shift(
+    ConformalMesh&             mesh,
+    const std::vector<double>& x,
+    const SphericalMaps&       m,
+    double                     bracket = 50.0,
+    double                     tol     = 1e-8)
+{
+    // Helper: sum of all vertex gradient components at x + t·1_v.
+    auto sum_Gv = [&](double t) -> double {
+        // Build a shifted copy of x (only vertex DOFs shifted).
+        std::vector<double> xt = x;
+        for (auto v : mesh.vertices()) {
+            int iv = m.v_idx[v];
+            if (iv >= 0)
+                xt[static_cast<std::size_t>(iv)] += t;
+        }
+        auto G = spherical_gradient(mesh, xt, m);
+        double sum = 0.0;
+        for (auto v : mesh.vertices()) {
+            int iv = m.v_idx[v];
+            if (iv >= 0)
+                sum += G[static_cast<std::size_t>(iv)];
+        }
+        return sum;
+    };
+
+    // ── Try bisection first (works when there is a sign change in [−bracket, +bracket]) ──
+    double a = -bracket, b = bracket;
+    double fa = sum_Gv(a), fb = sum_Gv(b);
+
+    if (fa * fb <= 0.0) {
+        for (int iter = 0; iter < 120; ++iter) {
+            double c  = 0.5 * (a + b);
+            double fc = sum_Gv(c);
+            if (std::abs(fc) < tol || (b - a) < tol) return c;
+            if (fa * fc < 0.0) { b = c; fb = fc; }
+            else               { a = c; fa = fc; }
+        }
+        return 0.5 * (a + b);
+    }
+
+    // ── No sign change (zero may lie at a domain boundary).  ───────────────────
+    //    Use damped Newton's method with backtracking line search.
+    //    f'(t) estimated by forward finite difference.
+    //    When the Newton step overshoots the valid domain (ΣG_v jumps back up
+    //    because faces become degenerate), backtracking halves the step until
+    //    |f| strictly decreases.
+    const double fd_eps = 1e-5;
+    double t  = 0.0;
+    double ft = sum_Gv(t);
+
+    for (int iter = 0; iter < 120; ++iter) {
+        if (std::abs(ft) < tol) return t;
+
+        double ftp = sum_Gv(t + fd_eps);
+        double dft = (ftp - ft) / fd_eps;
+        if (std::abs(dft) < 1e-14) return t;   // gradient flat — give up
+
+        double dt_raw = -ft / dft;
+
+        // Backtracking line search: halve dt until |f| decreases.
+        double alpha = 1.0;
+        bool improved = false;
+        for (int back = 0; back < 40; ++back) {
+            double t_try  = t + alpha * dt_raw;
+            t_try = std::max(-bracket, std::min(bracket, t_try));
+            double ft_try = sum_Gv(t_try);
+            if (std::abs(ft_try) < std::abs(ft)) {
+                t  = t_try;
+                ft = ft_try;
+                improved = true;
+                break;
+            }
+            alpha *= 0.5;
+        }
+        if (!improved) return t;   // cannot reduce further
+    }
+    return t;
+}
+
+/// Apply the spherical gauge shift in-place: `x_v ← x_v + t*` for every
+/// variable vertex, where `t* = spherical_gauge_shift(mesh, x, m, ...)`.
+inline void apply_spherical_gauge(
+    ConformalMesh&       mesh,
+    std::vector<double>& x,
+    const SphericalMaps& m,
+    double               bracket = 50.0,
+    double               tol     = 1e-8)
+{
+    double t = spherical_gauge_shift(mesh, x, m, bracket, tol);
+    for (auto v : mesh.vertices()) {
+        int iv = m.v_idx[v];
+        if (iv >= 0)
+            x[static_cast<std::size_t>(iv)] += t;
+    }
+}
+
+} // namespace conformallab

code/include/spherical_geometry.hpp

@@ -0,0 +1,76 @@
+#pragma once
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
+// spherical_geometry.hpp
+//
+// Pure-math building blocks for the spherical discrete conformal map.
+// Ported from de.varylab.discreteconformal.functional.SphericalFunctional
+// (the geometry helpers embedded there).
+//
+// Notation:
+//   u_i          – vertex conformal factor (DOF)
+//   λ°_e         – base log-length of edge e (fixed initial value)
+//   λ_ij         – effective log-length = λ°_ij + u_i + u_j
+//   l_ij         – spherical arc length = 2·asin(min(exp(λ_ij/2), 1))
+//   α_k          – interior angle of the spherical triangle at vertex k
+
+#include "constants.hpp"
+#include <cmath>
+#include <algorithm>
+
+namespace conformallab {
+
+/// Backward-compatible alias — prefer conformallab::PI in new code.
+constexpr double PI_SPHER = PI;
+
+// ── Effective spherical arc length ────────────────────────────────────────────
+
+/// Spherical arc length `l(λ) = 2·asin(min(exp(λ/2), 1))`.
+/// Clamps `exp(λ/2)` to `[0, 1]` so `asin` stays in domain.
+inline double spherical_l(double lambda)
+{
+    double half = std::exp(lambda * 0.5);
+    if (half >= 1.0) half = 1.0 - 1e-15;
+    if (half <= 0.0) return 0.0;
+    return 2.0 * std::asin(half);
+}
+
+// ── Interior angles of a spherical triangle ──────────────────────────────────
+
+/// Interior angles of a spherical triangle, plus a `valid` flag.
+struct SphericalFaceAngles {
+    double alpha1; ///< Corner angle at vertex v₁.
+    double alpha2; ///< Corner angle at vertex v₂.
+    double alpha3; ///< Corner angle at vertex v₃.
+    bool   valid;  ///< `false` when the three lengths violate the spherical triangle inequality.
+};
+
+/// Compute the spherical-triangle corner angles `(α₁, α₂, α₃)` from
+/// the three arc lengths `(l₁₂, l₂₃, l₃₁)` using the half-angle form
+/// of the spherical law of cosines.  Returns `valid = false` for
+/// degenerate or out-of-range triangles.
+inline SphericalFaceAngles spherical_angles(double l12, double l23, double l31)
+{
+    double s   = (l12 + l23 + l31) * 0.5;
+    double s12 = s - l12;
+    double s23 = s - l23;
+    double s31 = s - l31;
+
+    // Spherical triangle inequalities: all s-deficiencies > 0 and s < π.
+    if (s12 <= 0.0 || s23 <= 0.0 || s31 <= 0.0 || s >= PI_SPHER)
+        return {0.0, 0.0, 0.0, false};
+
+    const double ss   = std::sin(s);
+    const double ss12 = std::sin(s12);
+    const double ss23 = std::sin(s23);
+    const double ss31 = std::sin(s31);
+
+    double a1 = 2.0 * std::atan2(std::sqrt(ss12 * ss31), std::sqrt(ss * ss23));
+    double a2 = 2.0 * std::atan2(std::sqrt(ss12 * ss23), std::sqrt(ss * ss31));
+    double a3 = 2.0 * std::atan2(std::sqrt(ss23 * ss31), std::sqrt(ss * ss12));
+
+    return {a1, a2, a3, true};
+}
+
+} // namespace conformallab

code/include/spherical_hessian.hpp

@@ -0,0 +1,252 @@
+#pragma once
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
+// spherical_hessian.hpp
+//
+// Analytical Hessian of the spherical discrete conformal energy —
+// the spherical cotangent-Laplace operator.
+//
+// Ported from de.varylab.discreteconformal.functional.SphericalFunctional
+// (the hessian() method, vertex DOFs).
+//
+// ┌──────────────────────────────────────────────────────────────────────────┐
+// │  Hessian formula (vertex DOFs only)                                      │
+// │                                                                          │
+// │  For a spherical face (v1, v2, v3) with vertex angles α1, α2, α3:       │
+// │                                                                          │
+// │    Spherical cotangent weight for edge (vi, vj) with opposite vk:        │
+// │      β_k = (π − αi − αj + αk) / 2                                       │
+// │      w_k  = cot(β_k)  =  1/tan(β_k)                                     │
+// │                                                                          │
+// │    Euclidean limit: α1+α2+α3 → π, β_k → αk, w_k → cot(αk). ✓          │
+// │                                                                          │
+// │  Hessian contributions per face:                                         │
+// │    H[vi, vi] += w_ij + w_ik      (diagonal: weights of incident edges)  │
+// │    H[vi, vj] -= w_ij             (off-diagonal: weight of edge ij)      │
+// │                                                                          │
+// │  where w_ij is the weight of the edge between vi and vj (opposite vk):  │
+// │    w_ij = cot(β_k)  with  β_k = (π − αi − αj + αk) / 2.               │
+// └──────────────────────────────────────────────────────────────────────────┘
+//
+// Requires Eigen (header-only).  Returns Eigen::SparseMatrix<double>.
+
+#include "spherical_functional.hpp"
+#include <Eigen/Sparse>
+#include <vector>
+#include <cmath>
+
+namespace conformallab {
+
+// ── Spherical cotangent weight helper ────────────────────────────────────────
+//
+// Given the three face angles α1, α2, α3 of a spherical triangle, return the
+// three edge cotangent weights w1 (edge opp v1), w2 (edge opp v2), w3 (edge opp v3).
+//
+// w_k = cot(β_k)  where β_k = (π − α_adj1 − α_adj2 + α_opp) / 2
+//                             = (π − αi − αj + αk) / 2  for edge (vi,vj), opposite vk
+//
+// Mapping in our CGAL halfedge convention:
+//   h0 = halfedge(f): edge v1-v2 → opposite v3 → w = cot(β3), β3=(π-α1-α2+α3)/2
+//   h1: edge v2-v3 → opposite v1 → w = cot(β1), β1=(π-α2-α3+α1)/2
+//   h2: edge v3-v1 → opposite v2 → w = cot(β2), β2=(π-α3-α1+α2)/2
+//
+// Returns valid=false if any β_k is out of range (degenerate face).
+/// Three spherical "cotangent" weights for the three edges of a face,
+/// derived from the per-vertex interior angles `α₁, α₂, α₃` via
+/// `w_ij = cot(β_k)` with `β_k = (π − α_i − α_j + α_k) / 2`.
+struct SpherCotWeights {
+    double w12; ///< Weight for edge v₁-v₂ (opposite vertex v₃).
+    double w23; ///< Weight for edge v₂-v₃ (opposite vertex v₁).
+    double w31; ///< Weight for edge v₃-v₁ (opposite vertex v₂).
+    bool   valid; ///< `false` when any β_k is out of `(0, π/2]` (degenerate face).
+};
+
+/// Compute the three spherical cot weights from the three interior
+/// angles `(α₁, α₂, α₃)` of a spherical triangle.  See `SpherCotWeights`.
+inline SpherCotWeights spherical_cot_weights(double alpha1, double alpha2, double alpha3)
+{
+    // β for each edge:
+    //   β3 = (π - α1 - α2 + α3)/2  — weight for edge v1-v2 (opposite v3)
+    //   β1 = (π - α2 - α3 + α1)/2  — weight for edge v2-v3 (opposite v1)
+    //   β2 = (π - α3 - α1 + α2)/2  — weight for edge v3-v1 (opposite v2)
+    const double beta3 = (PI - alpha1 - alpha2 + alpha3) * 0.5;
+    const double beta1 = (PI - alpha2 - alpha3 + alpha1) * 0.5;
+    const double beta2 = (PI - alpha3 - alpha1 + alpha2) * 0.5;
+
+    // Each β_k must be in (0, π/2] for the weight to be positive and well-defined.
+    // For degenerate or very flat triangles some β may be ≤ 0 or ≥ π/2.
+    if (beta1 <= 0.0 || beta2 <= 0.0 || beta3 <= 0.0) return {0.0, 0.0, 0.0, false};
+
+    const double tb1 = std::tan(beta1);
+    const double tb2 = std::tan(beta2);
+    const double tb3 = std::tan(beta3);
+
+    if (std::abs(tb1) < 1e-15 || std::abs(tb2) < 1e-15 || std::abs(tb3) < 1e-15)
+        return {0.0, 0.0, 0.0, false};
+
+    // w_ij = cot(β_k) where β_k is for the edge opposite vk.
+    // w12 is for edge v1-v2 (opposite v3): cot(β3)
+    // w23 is for edge v2-v3 (opposite v1): cot(β1)
+    // w31 is for edge v3-v1 (opposite v2): cot(β2)
+    return {1.0 / tb3, 1.0 / tb1, 1.0 / tb2, true};
+}
+
+/// Analytical Spherical Hessian via `∂α/∂u` from the spherical law of
+/// cosines + chain rule `∂l/∂u = tan(l/2)`; returns an n×n sparse
+/// matrix with `n = spherical_dimension(mesh, m)`.  See block comment
+/// inside the body for the per-face derivation.
+inline Eigen::SparseMatrix<double> spherical_hessian(
+    ConformalMesh&             mesh,
+    const std::vector<double>& x,
+    const SphericalMaps&       m)
+{
+    const int n = spherical_dimension(mesh, m);
+
+    std::vector<Eigen::Triplet<double>> trips;
+    trips.reserve(static_cast<std::size_t>(n) * 9);
+
+    for (auto f : mesh.faces()) {
+        Halfedge_index h0 = mesh.halfedge(f);
+        Halfedge_index h1 = mesh.next(h0);
+        Halfedge_index h2 = mesh.next(h1);
+
+        Vertex_index v1 = mesh.source(h0);
+        Vertex_index v2 = mesh.source(h1);
+        Vertex_index v3 = mesh.source(h2);
+
+        Edge_index e12 = mesh.edge(h0);
+        Edge_index e23 = mesh.edge(h1);
+        Edge_index e31 = mesh.edge(h2);
+
+        // Effective log-lengths.
+        double u1 = spher_dof_val(m.v_idx[v1], x);
+        double u2 = spher_dof_val(m.v_idx[v2], x);
+        double u3 = spher_dof_val(m.v_idx[v3], x);
+
+        double lam12 = m.lambda0[e12] + u1 + u2 + spher_dof_val(m.e_idx[e12], x);
+        double lam23 = m.lambda0[e23] + u2 + u3 + spher_dof_val(m.e_idx[e23], x);
+        double lam31 = m.lambda0[e31] + u3 + u1 + spher_dof_val(m.e_idx[e31], x);
+
+        const double l12 = spherical_l(lam12);
+        const double l23 = spherical_l(lam23);
+        const double l31 = spherical_l(lam31);
+
+        SphericalFaceAngles fa = spherical_angles(l12, l23, l31);
+        if (!fa.valid) continue;
+
+        const double sinl12 = std::sin(l12), cosl12 = std::cos(l12);
+        const double sinl23 = std::sin(l23), cosl23 = std::cos(l23);
+        const double sinl31 = std::sin(l31), cosl31 = std::cos(l31);
+
+        if (sinl12 < 1e-15 || sinl23 < 1e-15 || sinl31 < 1e-15) continue;
+
+        const double cot12 = cosl12 / sinl12;
+        const double cot23 = cosl23 / sinl23;
+        const double cot31 = cosl31 / sinl31;
+
+        // ∂l_ij/∂λ_ij = tan(l_ij/2)
+        const double t12 = std::tan(l12 * 0.5);
+        const double t23 = std::tan(l23 * 0.5);
+        const double t31 = std::tan(l31 * 0.5);
+
+        const double sinA1 = std::sin(fa.alpha1), cosA1 = std::cos(fa.alpha1);
+        const double sinA2 = std::sin(fa.alpha2), cosA2 = std::cos(fa.alpha2);
+        const double sinA3 = std::sin(fa.alpha3), cosA3 = std::cos(fa.alpha3);
+
+        if (sinA1 < 1e-15 || sinA2 < 1e-15 || sinA3 < 1e-15) continue;
+
+        // ∂α1/∂l_jk  (α1 at v1; opposite l23, adjacent l12,l31)
+        const double dA1_dl12 = (cot12 * cosA1 - cot31) / sinA1;
+        const double dA1_dl31 = (cot31 * cosA1 - cot12) / sinA1;
+        const double dA1_dl23 = sinl23 / (sinl12 * sinl31 * sinA1);
+
+        // ∂α2/∂l_jk  (α2 at v2; opposite l31, adjacent l12,l23)
+        const double dA2_dl12 = (cot12 * cosA2 - cot23) / sinA2;
+        const double dA2_dl23 = (cot23 * cosA2 - cot12) / sinA2;
+        const double dA2_dl31 = sinl31 / (sinl12 * sinl23 * sinA2);
+
+        // ∂α3/∂l_jk  (α3 at v3; opposite l12, adjacent l23,l31)
+        const double dA3_dl23 = (cot23 * cosA3 - cot31) / sinA3;
+        const double dA3_dl31 = (cot31 * cosA3 - cot23) / sinA3;
+        const double dA3_dl12 = sinl12 / (sinl23 * sinl31 * sinA3);
+
+        // Chain rule: ∂α_i/∂u_j   (u1 affects l12,l31; u2 affects l12,l23; u3 affects l23,l31)
+        const double dA1_du1 = dA1_dl12 * t12 + dA1_dl31 * t31;
+        const double dA1_du2 = dA1_dl12 * t12 + dA1_dl23 * t23;
+        const double dA1_du3 = dA1_dl23 * t23 + dA1_dl31 * t31;
+
+        const double dA2_du1 = dA2_dl12 * t12 + dA2_dl31 * t31;
+        const double dA2_du2 = dA2_dl12 * t12 + dA2_dl23 * t23;
+        const double dA2_du3 = dA2_dl23 * t23 + dA2_dl31 * t31;
+
+        const double dA3_du1 = dA3_dl12 * t12 + dA3_dl31 * t31;
+        const double dA3_du2 = dA3_dl12 * t12 + dA3_dl23 * t23;
+        const double dA3_du3 = dA3_dl23 * t23 + dA3_dl31 * t31;
+
+        const int i1 = m.v_idx[v1];
+        const int i2 = m.v_idx[v2];
+        const int i3 = m.v_idx[v3];
+
+        // H[vi, vj] -= ∂α_i/∂u_j   (G_v = θ_v − Σ α_v, so ∂G_i/∂u_j = −∂α_i/∂u_j)
+        if (i1 >= 0) trips.emplace_back(i1, i1, -dA1_du1);
+        if (i2 >= 0) trips.emplace_back(i2, i2, -dA2_du2);
+        if (i3 >= 0) trips.emplace_back(i3, i3, -dA3_du3);
+
+        if (i1 >= 0 && i2 >= 0) {
+            trips.emplace_back(i1, i2, -dA1_du2);
+            trips.emplace_back(i2, i1, -dA2_du1);
+        }
+        if (i2 >= 0 && i3 >= 0) {
+            trips.emplace_back(i2, i3, -dA2_du3);
+            trips.emplace_back(i3, i2, -dA3_du2);
+        }
+        if (i3 >= 0 && i1 >= 0) {
+            trips.emplace_back(i3, i1, -dA3_du1);
+            trips.emplace_back(i1, i3, -dA1_du3);
+        }
+    }
+
+    Eigen::SparseMatrix<double> H(n, n);
+    H.setFromTriplets(trips.begin(), trips.end());
+    return H;
+}
+
+/// FD Hessian check for the Spherical functional.  Compares analytic
+/// `H` column-by-column to `(G(x+εeⱼ) − G(x−εeⱼ)) / (2ε)`.
+inline bool hessian_check_spherical(
+    ConformalMesh&             mesh,
+    const std::vector<double>& x0,
+    const SphericalMaps&       m,
+    double                     eps = 1e-5,
+    double                     tol = 1e-4)
+{
+    const int n = static_cast<int>(x0.size());
+    auto H = spherical_hessian(mesh, x0, m);
+
+    std::vector<double> xp = x0, xm = x0;
+    bool ok = true;
+
+    for (int j = 0; j < n; ++j) {
+        const std::size_t sj = static_cast<std::size_t>(j);
+        xp[sj] = x0[sj] + eps;
+        xm[sj] = x0[sj] - eps;
+
+        auto Gp = spherical_gradient(mesh, xp, m);
+        auto Gm = spherical_gradient(mesh, xm, m);
+
+        xp[sj] = xm[sj] = x0[sj];
+
+        for (int i = 0; i < n; ++i) {
+            double fd_ij = (Gp[static_cast<std::size_t>(i)]
+                          - Gm[static_cast<std::size_t>(i)]) / (2.0 * eps);
+            double H_ij  = H.coeff(i, j);
+            double err   = std::abs(H_ij - fd_ij);
+            double scale = std::max(1.0, std::abs(H_ij));
+            if (err / scale > tol) ok = false;
+        }
+    }
+    return ok;
+}
+
+} // namespace conformallab

code/include/viewer_utils.h

@@ -1,11 +1,16 @@
 #pragma once
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
 
 #include <Eigen/Dense>
 #include <igl/opengl/glfw/Viewer.h>
 
 namespace viewer_utils {
 
-// Deklaration (Implementation in viewer.cpp)
+/// Open an interactive libigl OpenGL viewer window showing the mesh
+/// `(V, F)`.  Built only when `WITH_VIEWER=ON`; declaration here, body
+/// in `viewer.cpp`.
 void simple_visualize(Eigen::MatrixXd& V, Eigen::MatrixXi& F);
 
 } 

code/src/apps/v0/conformallab_cli.cpp

@@ -1,107 +1,317 @@
-#include <CGAL/Simple_cartesian.h>
-#include <CGAL/Surface_mesh.h>
-#include <CGAL/IO/polygon_mesh_io.h>
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
 
+// conformallab_cli.cpp
+//
+// ConformalLab++ command-line interface.
+//
+// Usage:
+//   conformallab_core -i input.off [-o layout.off] [-g euclidean|spherical|hyper_ideal]
+//                    [-j result.json] [-x result.xml] [-s] [-v]
+//
+// The tool:
+//   1. Loads an OFF/OBJ/PLY mesh.
+//   2. Sets up DOF maps + computes λ° from the input geometry.
+//   3. Pins one vertex (Euclidean/Spherical) or uses all-free DOFs (HyperIdeal).
+//   4. Runs Newton until convergence.
+//   5. Computes a 2-D (Euclidean / HyperIdeal) or 3-D (Spherical) layout.
+//   6. Saves the layout as an OFF file and optionally serialises the result
+//      to JSON and/or XML.
+//   7. Optionally shows the input mesh in a viewer (-s flag, requires WITH_VIEWER).
 
-#include "viewer_utils.h"
-#include "mesh_utils.hpp"
-
+#include "conformal_mesh.hpp"
+#include "mesh_io.hpp"
+#include "euclidean_functional.hpp"
+#include "spherical_functional.hpp"
+#include "hyper_ideal_functional.hpp"
+#include "newton_solver.hpp"
+#include "layout.hpp"
+#include "serialization.hpp"
 
 #include <CLI11.hpp>
 #include <iostream>
+#include <iomanip>
 #include <string>
+#include <vector>
+#include <cmath>
+#include <algorithm>
 
-
+// Viewer is always available when WITH_CGAL=ON (implies WITH_VIEWER=ON)
+#include "viewer_utils.h"
 #include <Eigen/Core>
 
+namespace cl = conformallab;
+using cl::ConformalMesh;
+using cl::Vertex_index;
+using cl::Edge_index;
 
-using Kernel = CGAL::Simple_cartesian<double>;
-using Point  = Kernel::Point_3;
-using Mesh   = CGAL::Surface_mesh<Point>;
+// ─────────────────────────────────────────────────────────────────────────────
+// Shared helpers (mirroring test_pipeline.cpp patterns)
+// ─────────────────────────────────────────────────────────────────────────────
 
-
-
-
-bool parse_arguments(int argc, char* argv[], std::string& input_file, std::string& output_file, bool& show, bool& verbose) {
-    CLI::App app{"demo of conformallab"};
-    app.add_option("-i,--input",  input_file,  "Input OFF file")->required();
-    app.add_option("-o,--output", output_file, "Output OFF file (optional)");
-    app.add_flag("-s,--show", show, "Show the input mesh in a viewer ");
-    app.add_flag("-v,--verbose",verbose, "Enable verbose output");
-    app.set_help_flag("-h,--help", "Show this help message and exit");
-
-    try {
-        CLI11_PARSE(app, argc, argv);
-    } catch (const CLI::ParseError &e) {
-        return false;
-    }
-
-    if (input_file.empty()) {
-        std::cerr << "Input file is required.\n";
-        return EXIT_FAILURE;
-    }
-
-    if (input_file.substr(input_file.find_last_of('.') + 1) != "off") {
-        std::cerr << "Unsupported file format. Please provide an OFF file.\n";
-        return EXIT_FAILURE;
-    }
-
-    std::cout << "Input file: " << input_file << "\n";
-    std::cout << "Output file: " << output_file << "\n";
-    std::cout << "Show mesh: " << (show ? "Yes" : "No") << "\n";
-    std::cout << "Verbose: " << (verbose ? "Yes" : "No") << "\n";
-
-
-
-    return true;
+// Pin vertex 0, assign 0..n-1 to the rest
+static int pin_first_vertex(ConformalMesh& mesh, cl::EuclideanMaps& maps)
+{
+    auto vit = mesh.vertices().begin();
+    Vertex_index v0 = *vit++;
+    maps.v_idx[v0] = -1;
+    int idx = 0;
+    for (; vit != mesh.vertices().end(); ++vit)
+        maps.v_idx[*vit] = idx++;
+    return idx;
 }
 
+// Natural theta for Euclidean: make x=0 the equilibrium
+static void set_natural_euclidean_theta(ConformalMesh& mesh, cl::EuclideanMaps& maps, int n)
+{
+    std::vector<double> x0(static_cast<std::size_t>(n), 0.0);
+    auto G = cl::euclidean_gradient(mesh, x0, maps);
+    for (auto v : mesh.vertices()) {
+        int iv = maps.v_idx[v];
+        if (iv < 0) continue;
+        maps.theta_v[v] -= G[static_cast<std::size_t>(iv)];
+    }
+}
 
+// Natural theta for HyperIdeal at base point (b=1, a=0.5) to avoid x=0 singularity
+static std::vector<double> set_natural_hyper_ideal_targets(
+    ConformalMesh& mesh, cl::HyperIdealMaps& maps, int n)
+{
+    const auto sz = static_cast<std::size_t>(n);
+    std::vector<double> xbase(sz, 0.0);
+    for (auto v : mesh.vertices()) {
+        int iv = maps.v_idx[v];
+        if (iv >= 0) xbase[static_cast<std::size_t>(iv)] = 1.0;
+    }
+    for (auto e : mesh.edges()) {
+        int ie = maps.e_idx[e];
+        if (ie >= 0) xbase[static_cast<std::size_t>(ie)] = 0.5;
+    }
+    auto G = cl::evaluate_hyper_ideal(mesh, xbase, maps, false).gradient;
+    for (auto v : mesh.vertices()) {
+        int iv = maps.v_idx[v];
+        if (iv < 0) continue;
+        maps.theta_v[v] += G[static_cast<std::size_t>(iv)];
+    }
+    for (auto e : mesh.edges()) {
+        int ie = maps.e_idx[e];
+        if (ie < 0) continue;
+        maps.theta_e[e] += G[static_cast<std::size_t>(ie)];
+    }
+    return xbase;
+}
+
+// ─────────────────────────────────────────────────────────────────────────────
+// Euclidean pipeline
+// ─────────────────────────────────────────────────────────────────────────────
+static int run_euclidean(ConformalMesh& mesh,
+                         const std::string& out_layout,
+                         const std::string& out_json,
+                         const std::string& out_xml,
+                         bool verbose)
+{
+    // Setup
+    auto maps = cl::setup_euclidean_maps(mesh);
+    cl::compute_euclidean_lambda0_from_mesh(mesh, maps);
+
+    // DOF assignment: pin vertex 0
+    int n = pin_first_vertex(mesh, maps);
+    if (n <= 0) { std::cerr << "Error: mesh has only one vertex.\n"; return 1; }
+
+    // Natural target angles
+    set_natural_euclidean_theta(mesh, maps, n);
+
+    // Newton
+    std::vector<double> x0(static_cast<std::size_t>(n), 0.0);
+    auto res = cl::newton_euclidean(mesh, x0, maps);
+
+    if (!res.converged && verbose)
+        std::cerr << "[warn] Newton did not converge (|grad|=" << res.grad_inf_norm << ")\n";
+
+    // Layout
+    cl::Layout2D layout = cl::euclidean_layout(mesh, res.x, maps);
+
+    // Output
+    if (!out_layout.empty()) cl::save_layout_off(out_layout, mesh, layout);
+    if (!out_json.empty())
+        cl::save_result_json(out_json, res, "euclidean",
+                             static_cast<int>(mesh.number_of_vertices()),
+                             static_cast<int>(mesh.number_of_faces()),
+                             &layout);
+    if (!out_xml.empty())
+        cl::save_result_xml(out_xml, res, "euclidean",
+                            static_cast<int>(mesh.number_of_vertices()),
+                            static_cast<int>(mesh.number_of_faces()),
+                            &layout);
+
+    std::cout << std::fixed << std::setprecision(6);
+    std::cout << "Euclidean: converged=" << (res.converged ? "yes" : "no")
+              << "  iter=" << res.iterations
+              << "  |grad|_inf=" << std::scientific << std::setprecision(3)
+              << res.grad_inf_norm << "\n";
+    if (!out_layout.empty()) std::cout << "  layout → " << out_layout << "\n";
+    if (!out_json.empty())   std::cout << "  json   → " << out_json   << "\n";
+    if (!out_xml.empty())    std::cout << "  xml    → " << out_xml    << "\n";
+    return 0;
+}
+
+// ─────────────────────────────────────────────────────────────────────────────
+// Spherical pipeline
+// ─────────────────────────────────────────────────────────────────────────────
+static int run_spherical(ConformalMesh& mesh,
+                          const std::string& out_layout,
+                          const std::string& out_json,
+                          const std::string& out_xml,
+                          bool verbose)
+{
+    auto maps = cl::setup_spherical_maps(mesh);
+    cl::compute_lambda0_from_mesh(mesh, maps);
+    int n = cl::assign_vertex_dof_indices(mesh, maps);
+
+    std::vector<double> x0(static_cast<std::size_t>(n), 0.0);
+    auto res = cl::newton_spherical(mesh, x0, maps);
+
+    if (!res.converged && verbose)
+        std::cerr << "[warn] Newton did not converge (|grad|=" << res.grad_inf_norm << ")\n";
+
+    cl::Layout3D layout = cl::spherical_layout(mesh, res.x, maps);
+
+    if (!out_layout.empty()) cl::save_layout_off(out_layout, mesh, layout);
+    if (!out_json.empty())
+        cl::save_result_json(out_json, res, "spherical",
+                             static_cast<int>(mesh.number_of_vertices()),
+                             static_cast<int>(mesh.number_of_faces()),
+                             nullptr, &layout);
+    if (!out_xml.empty())
+        cl::save_result_xml(out_xml, res, "spherical",
+                            static_cast<int>(mesh.number_of_vertices()),
+                            static_cast<int>(mesh.number_of_faces()),
+                            nullptr, &layout);
+
+    std::cout << "Spherical: converged=" << (res.converged ? "yes" : "no")
+              << "  iter=" << res.iterations
+              << "  |grad|_inf=" << std::scientific << std::setprecision(3)
+              << res.grad_inf_norm << "\n";
+    if (!out_layout.empty()) std::cout << "  layout → " << out_layout << "\n";
+    if (!out_json.empty())   std::cout << "  json   → " << out_json   << "\n";
+    if (!out_xml.empty())    std::cout << "  xml    → " << out_xml    << "\n";
+    return 0;
+}
+
+// ─────────────────────────────────────────────────────────────────────────────
+// HyperIdeal pipeline
+// ─────────────────────────────────────────────────────────────────────────────
+static int run_hyper_ideal(ConformalMesh& mesh,
+                            const std::string& out_layout,
+                            const std::string& out_json,
+                            const std::string& out_xml,
+                            bool verbose)
+{
+    auto maps = cl::setup_hyper_ideal_maps(mesh);
+    int n = cl::assign_all_dof_indices(mesh, maps);
+
+    // Natural targets at base point (b=1, a=0.5)
+    auto xbase = set_natural_hyper_ideal_targets(mesh, maps, n);
+
+    // Perturb slightly from the base and solve back
+    std::vector<double> x0 = xbase;
+    for (auto& v : x0) v += 0.3;
+
+    auto res = cl::newton_hyper_ideal(mesh, x0, maps);
+
+    if (!res.converged && verbose)
+        std::cerr << "[warn] Newton did not converge (|grad|=" << res.grad_inf_norm << ")\n";
+
+    cl::Layout2D layout = cl::hyper_ideal_layout(mesh, res.x, maps);
+
+    if (!out_layout.empty()) cl::save_layout_off(out_layout, mesh, layout);
+    if (!out_json.empty())
+        cl::save_result_json(out_json, res, "hyper_ideal",
+                             static_cast<int>(mesh.number_of_vertices()),
+                             static_cast<int>(mesh.number_of_faces()),
+                             &layout);
+    if (!out_xml.empty())
+        cl::save_result_xml(out_xml, res, "hyper_ideal",
+                            static_cast<int>(mesh.number_of_vertices()),
+                            static_cast<int>(mesh.number_of_faces()),
+                            &layout);
+
+    std::cout << "HyperIdeal: converged=" << (res.converged ? "yes" : "no")
+              << "  iter=" << res.iterations
+              << "  |grad|_inf=" << std::scientific << std::setprecision(3)
+              << res.grad_inf_norm << "\n";
+    if (!out_layout.empty()) std::cout << "  layout → " << out_layout << "\n";
+    if (!out_json.empty())   std::cout << "  json   → " << out_json   << "\n";
+    if (!out_xml.empty())    std::cout << "  xml    → " << out_xml    << "\n";
+    return 0;
+}
+
+// ─────────────────────────────────────────────────────────────────────────────
+// main
+// ─────────────────────────────────────────────────────────────────────────────
 int main(int argc, char* argv[])
 {
+    CLI::App app{"ConformalLab++ — discrete conformal mapping"};
+
     std::string input_file;
-    std::string output_file; 
-    bool show = false;
+    std::string out_layout;
+    std::string out_json;
+    std::string out_xml;
+    std::string geometry = "euclidean";
+    bool show    = false;
     bool verbose = false;
 
-    if(!parse_arguments(argc, argv, input_file, output_file, show, verbose)) {
-        std::cerr << "Failed to parse arguments.\n";
+    app.add_option("-i,--input",    input_file, "Input mesh (OFF/OBJ/PLY)")->required();
+    app.add_option("-o,--output",   out_layout, "Output layout OFF file");
+    app.add_option("-j,--json",     out_json,   "Save result as JSON");
+    app.add_option("-x,--xml",      out_xml,    "Save result as XML");
+    app.add_option("-g,--geometry", geometry,   "Target geometry: euclidean|spherical|hyper_ideal")
+        ->check(CLI::IsMember({"euclidean", "spherical", "hyper_ideal"}));
+    app.add_flag("-s,--show",    show,    "Visualise input mesh (requires WITH_VIEWER)");
+    app.add_flag("-v,--verbose", verbose, "Verbose output");
+
+    CLI11_PARSE(app, argc, argv);
+
+    // ── Load mesh ─────────────────────────────────────────────────────────────
+    ConformalMesh mesh;
+    if (!cl::read_mesh(input_file, mesh) || mesh.is_empty()) {
+        std::cerr << "Error: cannot load mesh from '" << input_file << "'\n";
         return EXIT_FAILURE;
     }
+    std::cout << "Loaded: " << input_file
+              << "  (" << mesh.number_of_vertices() << "v, "
+              << mesh.number_of_faces() << "f)\n";
 
-    Mesh surface_mesh;
-    if (!CGAL::IO::read_polygon_mesh(input_file, surface_mesh) || surface_mesh.is_empty()) {
-        std::cerr << "Invalid input file: " << input_file << "\n";
-        return EXIT_FAILURE;
-    }
-  
+    // ── Optional viewer ───────────────────────────────────────────────────────
+    if (show) {
+        // Convert ConformalMesh to Eigen matrices for the libigl viewer
+        const std::size_t nv = mesh.number_of_vertices();
+        const std::size_t nf = mesh.number_of_faces();
+        Eigen::MatrixXd V(static_cast<Eigen::Index>(nv), 3);
+        Eigen::MatrixXi F(static_cast<Eigen::Index>(nf), 3);
 
-
-    
-    // #TODO: later: here we would call the unwrapping code, e.g.:
-    // UnwrapSettings settings;
-    // settings.target_geometry = TargetGeometry::Euclidean;
-    // UnwrapJob job(surface_mesh, settings);
-    // auto result = job.run();
-    // Mesh unwrapped = result.surface_unwrapped;
-
-
-    // CGAL -> libigl
-    if(show){
-    std::cout << "Visualizing input mesh...\n";
-    Eigen::MatrixXd V;
-    Eigen::MatrixXi F;
-   
-    mesh_utils::cgal_to_eigen<Kernel>(surface_mesh, V, F);
-    viewer_utils::simple_visualize(V, F);
-
-    }
-    
-    // for now, we just write the input mesh to the output file as a placeholder
-    if (!output_file.empty() && !CGAL::IO::write_polygon_mesh(output_file, surface_mesh)) {
-        std::cerr << "Failed to write output file: " << output_file << "\n";
-        return EXIT_FAILURE;
+        for (auto v : mesh.vertices()) {
+            auto p = mesh.point(v);
+            V.row(v.idx()) << p.x(), p.y(), p.z();
+        }
+        Eigen::Index fi = 0;
+        for (auto f : mesh.faces()) {
+            int ci = 0;
+            for (auto h : CGAL::halfedges_around_face(mesh.halfedge(f), mesh))
+                F(fi, ci++) = static_cast<int>(mesh.target(h).idx());
+            ++fi;
+        }
+        viewer_utils::simple_visualize(V, F);
     }
 
-    return EXIT_SUCCESS;
+    // ── Dispatch ──────────────────────────────────────────────────────────────
+    if (geometry == "euclidean")
+        return run_euclidean(mesh, out_layout, out_json, out_xml, verbose);
+    if (geometry == "spherical")
+        return run_spherical(mesh, out_layout, out_json, out_xml, verbose);
+    if (geometry == "hyper_ideal")
+        return run_hyper_ideal(mesh, out_layout, out_json, out_xml, verbose);
+
+    std::cerr << "Unknown geometry: " << geometry << "\n";
+    return EXIT_FAILURE;
 }

code/src/viewer/simple_viewer.cpp

@@ -1,3 +1,6 @@
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
 #include "viewer_utils.h"
 
 namespace viewer_utils {

code/tests/CMakeLists.txt

@@ -1,8 +1,20 @@
 add_executable(conformallab_tests
+    # ── Pure-math test suite (no CGAL, no mesh — runs on every branch) ─────
     test_clausen.cpp
     test_hyper_ideal_utility.cpp
     test_matrix_utility.cpp
     test_surface_curve_utility.cpp
+    test_discrete_elliptic_utility.cpp
+    test_p2_utility.cpp
+    test_hyper_ideal_visualization_utility.cpp
+    #
+    # Stale stub files were removed in v0.9.0:
+    #   test_hyper_ideal_functional.cpp
+    #   test_hyper_ideal_hyperelliptic_utility.cpp
+    #   test_spherical_functional.cpp
+    # They referenced a "HDS port (Phase 4)" that never happened —
+    # CoHDS was intentionally replaced by CGAL::Surface_mesh, and the
+    # functionals + tests live in code/tests/cgal/test_*_functional.cpp.
 )
 
 target_include_directories(conformallab_tests SYSTEM PRIVATE
@@ -15,5 +27,21 @@ target_include_directories(conformallab_tests PRIVATE
 
 target_link_libraries(conformallab_tests PRIVATE GTest::gtest_main)
 
+# Fast test-build mode (lever #10): -O0 -g overrides the inherited
+# Release-mode -O3 + -DNDEBUG.  Applies only to this test target;
+# library/installable code is never affected.
+if(CONFORMALLAB_FAST_TEST_BUILD)
+    target_compile_options(conformallab_tests PRIVATE
+        $<$<CXX_COMPILER_ID:GNU,Clang,AppleClang>:-O0 -g -UNDEBUG>
+    )
+endif()
+
 include(GoogleTest)
 gtest_discover_tests(conformallab_tests DISCOVERY_TIMEOUT 60)
+
+# ── CGAL test suite ──────────────────────────────────────────────────────────
+# Built with -DWITH_CGAL_TESTS=ON (headless CI, no viewer) or
+#            -DWITH_CGAL=ON       (full build with viewer + CLI).
+if(WITH_CGAL OR WITH_CGAL_TESTS)
+    add_subdirectory(cgal)
+endif()

code/tests/cgal/CMakeLists.txt

@@ -0,0 +1,191 @@
+# tests/cgal/CMakeLists.txt
+#
+# CGAL-dependent test target.  Only built when -DWITH_CGAL=ON.
+# Requires Boost (find_package(Boost REQUIRED) is called in the root CMakeLists).
+#
+# Run with:
+#   cmake -S code -B build -DWITH_CGAL=ON
+#   cmake --build build --target conformallab_cgal_tests
+#   ctest --test-dir build -R cgal
+
+add_executable(conformallab_cgal_tests
+    # ── Phase 3a: mesh infrastructure ──────────────────────────────────────
+    test_conformal_mesh.cpp
+
+    # ── Phase 3b: HyperIdealFunctional ─────────────────────────────────────
+    test_hyper_ideal_functional.cpp
+
+    # ── Phase 3c + 3e: SphericalFunctional + gauge-fix ────────────────────
+    test_spherical_functional.cpp
+
+    # ── Phase 3d: EuclideanCyclicFunctional ───────────────────────────────
+    test_euclidean_functional.cpp
+
+    # ── Phase 3f: Hessians (cotangent Laplacian, spherical + Euclidean) ───
+    test_euclidean_hessian.cpp
+    test_spherical_hessian.cpp
+
+    # ── Phase 9b: Hyper-ideal Hessian — block-FD vs full-FD validation ───
+    #    Verifies the O(F·36) block-local Hessian agrees with the
+    #    O(F·n) full-FD baseline.  Java upstream has no Hessian at all
+    #    (HyperIdealFunctional.hasHessian() returns false) — both
+    #    variants are conformallab++ extensions beyond the port.
+    test_hyper_ideal_hessian.cpp
+
+    # ── Phase 4a: Newton solver ────────────────────────────────────────────
+    test_newton_solver.cpp
+
+    # ── Phase 4b: Mesh I/O (CGAL::IO) ─────────────────────────────────────
+    test_mesh_io.cpp
+
+    # ── Phase 4c: End-to-end pipeline + user examples ─────────────────────
+    test_pipeline.cpp
+
+    # ── Phase 5: Layout / embedding + serialization ────────────────────────
+    test_layout.cpp
+
+    # ── Phase 6: Gauss–Bonnet, cut graph, exact trilateration, normalisation
+    test_phase6.cpp
+
+    # ── Phase 7: Java-parity layout — MobiusMap, priority BFS, halfedge_uv,
+    #             period matrix, fundamental domain, tiling
+    test_phase7.cpp
+
+    # ── Java parity: geometry utility tests ──────────────────────────────────
+    #    Ported from CuttinUtilityTest, UnwrapUtilityTest,
+    #    ConvergenceUtilityTests, HomologyTest.  All tests active —
+    #    the v0.7.0 genus-2 homology stub was implemented in Phase 7
+    #    (HomologyGenerators.Genus2_FourCutEdges, brezel2.obj).
+    test_geometry_utils.cpp
+
+    # ── Scalability smoke tests ────────────────────────────────────────────────
+    #    Newton convergence on large real-world meshes (cathead, brezel, brezel2).
+    #    Assert correctness only (< 30 iterations, ||G|| < 1e-8).
+    #    Wall-clock time is printed for documentation but NOT asserted,
+    #    so the tests remain stable on slow CI hardware (Raspberry Pi ARM64).
+    test_scalability_smoke.cpp
+
+    # ── Phase 8 MVP: new CGAL-style public API ────────────────────────────────
+    #    First client of Conformal_map_traits.h + Discrete_conformal_map.h.
+    #    Acceptance probe before Phase 9a (Inversive-Distance) lands.
+    test_cgal_traits_mvp.cpp
+
+    # ── Phase 9a.1: CPEuclideanFunctional (BPS 2010 circle packing) ──────────
+    #    Face-based circle-packing functional ported from
+    #    CPEuclideanFunctional.java.  Reference: Bobenko-Pinkall-Springborn 2010.
+    test_cp_euclidean_functional.cpp
+
+    # ── Phase 9a.2: InversiveDistance (Luo 2004 + Glickenstein 2011) ─────────
+    #    Vertex-based inversive-distance circle-packing functional.  No Java
+    #    reference; implemented from the literature.  Cross-validated against
+    #    EuclideanCyclicFunctional at the natural initial geometry (u = 0).
+    test_inversive_distance_functional.cpp
+
+    # ── Phase 9a: Newton solvers for the two new circle-packing functionals ──
+    #    Convergence tests for newton_cp_euclidean (analytic Hessian) and
+    #    newton_inversive_distance (FD Hessian).
+    test_newton_phase9a.cpp
+
+    # ── Phase 8b-Lite: CGAL entry wrappers for the 4 non-Euclidean modes ─────
+    #    Spherical, HyperIdeal, CircleP-Euclidean, Inversive-Distance via
+    #    <CGAL/Discrete_*.h> public API + Conformal_layout.h wrapper.
+    test_cgal_phase8b_lite.cpp
+)
+
+target_include_directories(conformallab_cgal_tests SYSTEM PRIVATE
+    ${CMAKE_SOURCE_DIR}/deps/eigen-3.4.0
+    ${CMAKE_SOURCE_DIR}/deps/CGAL-6.1.1/include
+    ${CMAKE_SOURCE_DIR}/deps/single_includes
+    ${Boost_INCLUDE_DIRS}
+)
+
+target_include_directories(conformallab_cgal_tests PRIVATE
+    ${CMAKE_SOURCE_DIR}/include
+)
+
+target_compile_definitions(conformallab_cgal_tests PRIVATE
+    CGAL_DISABLE_GMP
+    CGAL_DISABLE_MPFR
+    # Data directory — absolute path to code/data/ at build time.
+    # Used by tests that load real mesh files (cathead.obj, brezel2.obj, …).
+    CONFORMALLAB_DATA_DIR="${CMAKE_SOURCE_DIR}/data"
+)
+
+# Suppress warnings from CGAL/Boost headers
+target_compile_options(conformallab_cgal_tests PRIVATE
+    $<$<CXX_COMPILER_ID:GNU,Clang,AppleClang>:-Wno-unused-parameter>
+)
+
+# Fast test-build mode (lever #10): -O0 -g overrides the inherited
+# Release-mode -O3 + -DNDEBUG.  Applies only to this test target.
+if(CONFORMALLAB_FAST_TEST_BUILD)
+    target_compile_options(conformallab_cgal_tests PRIVATE
+        $<$<CXX_COMPILER_ID:GNU,Clang,AppleClang>:-O0 -g -UNDEBUG>
+    )
+endif()
+
+target_link_libraries(conformallab_cgal_tests PRIVATE GTest::gtest_main)
+
+# ── Compile-time speed-up: precompiled headers ───────────────────────────────
+#
+# The CGAL+Eigen template soup dominates every TU in this target:
+# measured at 5.9 s per minimal "include <CGAL/Discrete_conformal_map.h>"
+# TU on Apple M1.  A shared PCH absorbs that cost once, slashing the
+# total wall-clock from ~78 s (j8) to ~25 s (3×).
+#
+# Opt-out with -DCONFORMALLAB_USE_PCH=OFF if the PCH itself misbehaves
+# (e.g. older toolchains that don't share PCH across translation units
+# reliably) — falls back to the historical "every TU re-parses CGAL"
+# build mode.
+option(CONFORMALLAB_USE_PCH
+    "Enable precompiled headers for the CGAL test target." ON)
+
+if(CONFORMALLAB_USE_PCH)
+    # Per-target Unity Build property takes precedence over the global
+    # CMAKE_UNITY_BUILD; honour CONFORMALLAB_DEV_BUILD's preference here
+    # so `-DCONFORMALLAB_DEV_BUILD=ON` truly turns Unity Build off for
+    # incremental-rebuild workflows.
+    if(NOT CONFORMALLAB_DEV_BUILD)
+        set_target_properties(conformallab_cgal_tests PROPERTIES
+            # Unity-builds amortise the per-TU CGAL+Eigen header cost
+            # across several tests in the same compile.  Batch size 4
+            # keeps gtest's TEST(...) macros + per-file `using
+            # namespace …` from colliding while still cutting parser
+            # cost ~4×.
+            UNITY_BUILD ON
+            UNITY_BUILD_MODE BATCH
+            UNITY_BUILD_BATCH_SIZE 4)
+    endif()
+
+    target_precompile_headers(conformallab_cgal_tests PRIVATE
+        # CGAL headers that every test transitively includes.
+        <CGAL/Surface_mesh.h>
+        <CGAL/Simple_cartesian.h>
+        <CGAL/Kernel_traits.h>
+        <CGAL/boost/graph/iterator.h>
+        <CGAL/Polygon_mesh_processing/triangulate_faces.h>
+
+        # Eigen blocks that drive the slowest template instantiations
+        # (SelfAdjointEigenSolver<Matrix<2,2>>, ColPivHouseholderQR<
+        # Matrix<complex,3,3>>, sparse Cholesky + QR fallback).
+        <Eigen/Dense>
+        <Eigen/Sparse>
+        <Eigen/SparseCholesky>
+        <Eigen/SparseQR>
+
+        # GoogleTest itself; every test includes it.
+        <gtest/gtest.h>
+
+        # std headers that appear in every test.
+        <vector>
+        <string>
+        <cmath>
+        <complex>
+    )
+endif()
+
+include(GoogleTest)
+gtest_discover_tests(conformallab_cgal_tests
+    TEST_PREFIX "cgal."
+    DISCOVERY_TIMEOUT 60
+)

code/tests/cgal/test_cgal_phase8b_lite.cpp

@@ -0,0 +1,429 @@
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
+// test_cgal_phase8b_lite.cpp
+//
+// Phase 8b-Lite — Smoke tests for the four new CGAL-style entry functions
+// added on top of the Phase 8a MVP (`discrete_conformal_map_euclidean`).
+//
+// All entries are thin wrappers around the legacy Newton solvers; the
+// purpose of these tests is to verify:
+//   • the wrapper compiles + dispatches correctly
+//   • named parameters pass through (gradient_tolerance, max_iterations)
+//   • the returned Result struct contains the expected DOF vector
+//   • Newton convergence happens end-to-end via the public API
+
+#include <CGAL/Discrete_conformal_map.h>
+#include <CGAL/Discrete_circle_packing.h>
+#include <CGAL/Discrete_inversive_distance.h>
+#include <CGAL/Conformal_layout.h>
+
+#include "mesh_builder.hpp"
+#include "conformal_mesh.hpp"
+
+#include <gtest/gtest.h>
+#include <cmath>
+
+using namespace conformallab;
+
+namespace {
+
+// Mesh helper — closed regular tetrahedron, used for spherical / hyper-ideal /
+// circle-packing tests.
+inline ConformalMesh make_closed_tet() { return make_tetrahedron(); }
+
+// Open 3-face tetrahedron-minus-face, for layout testing.
+inline ConformalMesh make_open_3face()
+{
+    ConformalMesh mesh;
+    auto v0 = mesh.add_vertex(Point3( 1,  1,  1));
+    auto v1 = mesh.add_vertex(Point3( 1, -1, -1));
+    auto v2 = mesh.add_vertex(Point3(-1,  1, -1));
+    auto v3 = mesh.add_vertex(Point3(-1, -1,  1));
+    mesh.add_face(v0, v2, v1);
+    mesh.add_face(v0, v1, v3);
+    mesh.add_face(v0, v3, v2);
+    return mesh;
+}
+
+} // anonymous
+
+// ════════════════════════════════════════════════════════════════════════════
+// 1.  Spherical entry — closed genus-0 tetrahedron, natural-theta default
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(CGALPhase8bLite, Spherical_ClosedTetrahedron_NaturalThetaConverges)
+{
+    auto mesh = make_closed_tet();
+    auto res  = CGAL::discrete_conformal_map_spherical(mesh);
+
+    EXPECT_TRUE(res.converged);
+    EXPECT_LT(res.gradient_norm, 1e-8);
+    EXPECT_EQ(res.u_per_vertex.size(), num_vertices(mesh));
+    // Natural-theta ⇒ u = 0 is the equilibrium ⇒ all values ≈ 0.
+    for (double u : res.u_per_vertex) EXPECT_NEAR(u, 0.0, 1e-8);
+}
+
+TEST(CGALPhase8bLite, Spherical_NamedParametersTakeEffect)
+{
+    auto mesh = make_closed_tet();
+    auto res  = CGAL::discrete_conformal_map_spherical(
+        mesh,
+        CGAL::parameters::max_iterations(0));
+    EXPECT_EQ(res.iterations, 0);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// 2.  Hyper-ideal entry — wrapper compiles + runs, returns both b_v and a_e
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(CGALPhase8bLite, HyperIdeal_Tetrahedron_ReturnsBothVertexAndEdgeDOFs)
+{
+    auto mesh = make_closed_tet();
+    auto res  = CGAL::discrete_conformal_map_hyper_ideal(
+        mesh,
+        CGAL::parameters::max_iterations(20));
+
+    // Newton on default targets (Θ=2π, θ=π) from the "natural" b=1, a=0.5
+    // start may or may not converge in 20 iterations — but the wrapper must
+    // populate the result struct in any case.
+    EXPECT_EQ(res.b_per_vertex.size(), num_vertices(mesh));
+    EXPECT_EQ(res.a_per_edge.size(),    num_edges   (mesh));
+    EXPECT_GE(res.iterations, 0);
+    EXPECT_TRUE(std::isfinite(res.gradient_norm));
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// 3.  Circle-packing (face-based) entry — natural-phi convergence
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(CGALPhase8bLite, CirclePacking_ClosedTetrahedron_NaturalPhiConverges)
+{
+    auto mesh = make_closed_tet();
+    auto res  = CGAL::discrete_circle_packing_euclidean(mesh);
+
+    EXPECT_TRUE(res.converged);
+    EXPECT_LT(res.gradient_norm, 1e-8);
+    EXPECT_EQ(res.rho_per_face.size(), num_faces(mesh));
+    // Pinned face is at index 0 (first iterated face); its ρ is 0 by gauge.
+    // After natural-phi the equilibrium is ρ_f = 0 for every face.
+    for (double r : res.rho_per_face) EXPECT_NEAR(r, 0.0, 1e-8);
+}
+
+TEST(CGALPhase8bLite, CirclePacking_GradientToleranceTakesEffect)
+{
+    auto mesh = make_closed_tet();
+    auto res_loose = CGAL::discrete_circle_packing_euclidean(
+        mesh,
+        CGAL::parameters::gradient_tolerance(1e-4));
+    EXPECT_TRUE(res_loose.converged);
+
+    auto mesh2 = make_closed_tet();
+    auto res_strict = CGAL::discrete_circle_packing_euclidean(
+        mesh2,
+        CGAL::parameters::gradient_tolerance(1e-12));
+    EXPECT_TRUE(res_strict.converged);
+    EXPECT_LT(res_strict.gradient_norm, 1e-10);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// 4.  Inversive-distance (vertex-based) entry — natural-theta convergence
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(CGALPhase8bLite, InversiveDistance_Triangle_NaturalThetaConverges)
+{
+    auto mesh = make_triangle();
+    auto res  = CGAL::discrete_inversive_distance_map(mesh);
+
+    EXPECT_TRUE(res.converged);
+    EXPECT_LT(res.gradient_norm, 1e-8);
+    EXPECT_EQ(res.u_per_vertex.size(), num_vertices(mesh));
+    for (double u : res.u_per_vertex) EXPECT_NEAR(u, 0.0, 1e-8);
+}
+
+TEST(CGALPhase8bLite, InversiveDistance_QuadStrip_NamedParametersWork)
+{
+    auto mesh = make_quad_strip();
+    // Named-parameter chaining (`a.b().c()`) is not currently supported on
+    // the package-local tags; pass one parameter per call instead.
+    auto res  = CGAL::discrete_inversive_distance_map(
+        mesh,
+        CGAL::parameters::max_iterations(50));
+    EXPECT_TRUE(res.converged);
+    EXPECT_LE(res.iterations, 50);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// 5.  Layout wrapper — end-to-end through CGAL API on an open mesh
+//
+// Uses the legacy maps explicitly because the wrappers return the
+// Newton-converged x vector but not the maps.  This exercises that the
+// `CGAL::euclidean_layout` shim works as expected.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(CGALPhase8bLite, Layout_EuclideanWrapper_RoundTrip)
+{
+    auto mesh = make_open_3face();
+
+    // Set up the maps + run Newton via the CGAL Euclidean entry.
+    auto res  = CGAL::discrete_conformal_map_euclidean(mesh);
+    ASSERT_TRUE(res.converged);
+
+    // The wrapper does its own DOF assignment internally; we re-fetch
+    // the (now-populated) EuclideanMaps from the mesh's property maps
+    // to feed the layout wrapper.
+    auto maps = setup_euclidean_maps(mesh);
+    compute_euclidean_lambda0_from_mesh(mesh, maps);
+    // Pin first vertex (mirrors the wrapper's gauge choice).
+    auto vit = mesh.vertices().begin();
+    maps.v_idx[*vit++] = -1;
+    int idx = 0;
+    for (; vit != mesh.vertices().end(); ++vit) maps.v_idx[*vit] = idx++;
+    std::vector<double> x(idx, 0.0);   // wrapper's natural-theta equilibrium
+
+    auto layout = CGAL::euclidean_layout(mesh, x, maps);
+    EXPECT_EQ(layout.uv.size(), num_vertices(mesh));
+    // All UVs finite — basic sanity that the layout ran.
+    for (auto& uv : layout.uv) {
+        EXPECT_TRUE(std::isfinite(uv.x()));
+        EXPECT_TRUE(std::isfinite(uv.y()));
+    }
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// 6.  output_uv_map named parameter — integrated layout step
+//
+// Phase 8b-Lite extension (2026-05-22): if the caller supplies a property
+// map via `CGAL::parameters::output_uv_map(pmap)`, the entry function runs
+// the appropriate `*_layout()` after Newton and writes the per-vertex
+// coordinates into `pmap`.  This closes the prior UX gap where users had
+// to call the wrapper, then re-set up maps, then call the legacy layout
+// API separately.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(CGALPhase8bLite, OutputUvMap_Euclidean_PopulatesPmap)
+{
+    using K = CGAL::Simple_cartesian<double>;
+    auto mesh = make_quad_strip();
+
+    auto uv_map = mesh.add_property_map<Vertex_index, K::Point_2>(
+        "v:test_uv", K::Point_2(0, 0)).first;
+
+    auto res = CGAL::discrete_conformal_map_euclidean(
+        mesh,
+        CGAL::parameters::output_uv_map(uv_map));
+
+    ASSERT_TRUE(res.converged);
+
+    // The map must be populated with finite values.
+    for (auto v : mesh.vertices()) {
+        const auto& p = uv_map[v];
+        EXPECT_TRUE(std::isfinite(p.x())) << "non-finite UV.x at vertex " << v.idx();
+        EXPECT_TRUE(std::isfinite(p.y())) << "non-finite UV.y at vertex " << v.idx();
+    }
+
+    // At least one vertex must have moved off the origin (the layout
+    // did NOT just return defaults).
+    bool any_nonzero = false;
+    for (auto v : mesh.vertices()) {
+        const auto& p = uv_map[v];
+        if (std::abs(p.x()) + std::abs(p.y()) > 1e-10) { any_nonzero = true; break; }
+    }
+    EXPECT_TRUE(any_nonzero) << "every UV is exactly (0,0) — layout did not run";
+}
+
+TEST(CGALPhase8bLite, OutputUvMap_Spherical_PopulatesXyz)
+{
+    using K = CGAL::Simple_cartesian<double>;
+    auto mesh = make_tetrahedron();
+
+    auto xyz_map = mesh.add_property_map<Vertex_index, K::Point_3>(
+        "v:test_xyz", K::Point_3(0, 0, 0)).first;
+
+    auto res = CGAL::discrete_conformal_map_spherical(
+        mesh,
+        CGAL::parameters::output_uv_map(xyz_map));
+
+    ASSERT_TRUE(res.converged);
+
+    // Every output point must lie on (or very near) the unit sphere.
+    for (auto v : mesh.vertices()) {
+        const auto& p = xyz_map[v];
+        const double r = std::sqrt(p.x()*p.x() + p.y()*p.y() + p.z()*p.z());
+        EXPECT_NEAR(r, 1.0, 1e-6) << "vertex " << v.idx() << " not on unit sphere";
+    }
+}
+
+TEST(CGALPhase8bLite, OutputUvMap_HyperIdeal_PointsInPoincareDisk)
+{
+    using K = CGAL::Simple_cartesian<double>;
+    auto mesh = make_tetrahedron();
+
+    auto uv_map = mesh.add_property_map<Vertex_index, K::Point_2>(
+        "v:test_uv_hyp", K::Point_2(0, 0)).first;
+
+    // Named-parameter chaining is not supported yet — pass output_uv_map only.
+    auto res = CGAL::discrete_conformal_map_hyper_ideal(
+        mesh,
+        CGAL::parameters::output_uv_map(uv_map));
+
+    // The wrapper must complete and return a well-formed result struct
+    // regardless of whether Newton fully converges with the default
+    // Θ/θ targets in 200 iterations.  We only verify that *if* the
+    // layout step ran (which happens only on converged Newton), the
+    // output is finite — Poincaré-disk geometric check is conditional.
+    EXPECT_EQ(res.b_per_vertex.size(), num_vertices(mesh));
+    EXPECT_EQ(res.a_per_edge.size(),    num_edges(mesh));
+
+    if (res.converged) {
+        for (auto v : mesh.vertices()) {
+            const auto& p = uv_map[v];
+            const double r2 = p.x()*p.x() + p.y()*p.y();
+            EXPECT_LE(r2, 1.0 + 1e-6)
+                << "vertex " << v.idx() << " outside Poincaré disk (|p|² = " << r2 << ")";
+        }
+    }
+    // (else: Newton did not reach equilibrium; UV pmap is left at its
+    //  default (0,0) per the wrapper's "if (nr.converged)" guard.
+    //  No assertion needed; this is documented behaviour.)
+}
+
+TEST(CGALPhase8bLite, OutputUvMap_InversiveDistance_PopulatesPmap)
+{
+    // Inversive-Distance: per-vertex u_i = log r_i.  With output_uv_map
+    // the entry function reconstructs effective Euclidean edge lengths via
+    // the Bowers-Stephenson identity and reuses the euclidean_layout
+    // priority-BFS to populate per-vertex Point_2 coordinates.
+    using K = CGAL::Simple_cartesian<double>;
+    auto mesh = make_quad_strip();
+
+    auto uv_map = mesh.add_property_map<Vertex_index, K::Point_2>(
+        "v:test_uv_id", K::Point_2(0, 0)).first;
+
+    auto res = CGAL::discrete_inversive_distance_map(
+        mesh, CGAL::parameters::output_uv_map(uv_map));
+
+    ASSERT_TRUE(res.converged) << "ID Newton did not converge on quad_strip";
+    EXPECT_EQ(res.u_per_vertex.size(), num_vertices(mesh));
+    // Every UV must be finite; not all zero.
+    bool any_nonzero = false;
+    for (auto v : mesh.vertices()) {
+        const auto& p = uv_map[v];
+        ASSERT_TRUE(std::isfinite(p.x()));
+        ASSERT_TRUE(std::isfinite(p.y()));
+        if (std::abs(p.x()) > 1e-9 || std::abs(p.y()) > 1e-9) any_nonzero = true;
+    }
+    EXPECT_TRUE(any_nonzero) << "all UVs are zero — layout did not run";
+}
+
+TEST(CGALPhase8bLite, OutputUvMap_CPEuclidean_ThrowsClearly)
+{
+    // CP-Euclidean is face-based; its natural layout is a per-face
+    // circle packing in ℝ², not a per-vertex Point_2 map.  The entry
+    // throws std::runtime_error with a helpful message rather than
+    // silently producing nonsense.  See doc/architecture/locked-vs-flexible.md.
+    using K = CGAL::Simple_cartesian<double>;
+    auto mesh = make_quad_strip();
+
+    auto uv_map = mesh.add_property_map<Vertex_index, K::Point_2>(
+        "v:test_uv_cp", K::Point_2(0, 0)).first;
+
+    EXPECT_THROW(
+        CGAL::discrete_circle_packing_euclidean(
+            mesh, CGAL::parameters::output_uv_map(uv_map)),
+        std::runtime_error)
+        << "expected discrete_circle_packing_euclidean to reject "
+           "`output_uv_map(...)` (face-based DOF, Phase 9c).";
+
+    // Sanity: without output_uv_map the entry function still works fine.
+    auto res = CGAL::discrete_circle_packing_euclidean(mesh);
+    // Convergence depends on the mesh; we only check no-throw + a sane
+    // shape of the result struct.
+    EXPECT_EQ(res.rho_per_face.size(), num_faces(mesh));
+}
+
+TEST(CGALPhase8bLite, OutputUvMap_Absent_DoesNotRunLayout)
+{
+    // Sanity: without the parameter, no layout work happens.  Verified
+    // here only via the fact that the call still succeeds and produces
+    // the same u-vector as before.
+    auto mesh = make_quad_strip();
+    auto res = CGAL::discrete_conformal_map_euclidean(mesh);
+    EXPECT_TRUE(res.converged);
+    EXPECT_EQ(res.u_per_vertex.size(), num_vertices(mesh));
+}
+
+TEST(CGALPhase8bLite, OutputUvMap_NormaliseLayout_TakesEffect)
+{
+    using K = CGAL::Simple_cartesian<double>;
+    auto mesh = make_quad_strip();
+
+    auto uv_raw  = mesh.add_property_map<Vertex_index, K::Point_2>(
+        "v:test_uv_raw",  K::Point_2(0, 0)).first;
+    auto uv_norm = mesh.add_property_map<Vertex_index, K::Point_2>(
+        "v:test_uv_norm", K::Point_2(0, 0)).first;
+
+    auto res1 = CGAL::discrete_conformal_map_euclidean(
+        mesh, CGAL::parameters::output_uv_map(uv_raw));
+    auto res2 = CGAL::discrete_conformal_map_euclidean(
+        mesh, CGAL::parameters::output_uv_map(uv_norm));
+    // (We can only pass one named parameter at a time without chaining;
+    //  test the toggle by running the wrapper twice and verifying the
+    //  raw call works.  The normalise_layout flag is exercised in
+    //  internal unit tests via direct calls to normalise_euclidean.)
+    ASSERT_TRUE(res1.converged);
+    ASSERT_TRUE(res2.converged);
+    // Both maps populated to finite values.
+    for (auto v : mesh.vertices()) {
+        EXPECT_TRUE(std::isfinite(uv_raw[v].x()));
+        EXPECT_TRUE(std::isfinite(uv_norm[v].x()));
+    }
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// 7.  Named-parameter chaining via pipe-operator
+//
+// CGAL's `.a().b().c()` chaining requires modifying CGAL upstream, which
+// we don't do.  conformallab++ provides a `|` operator that achieves the
+// same effect by left-to-right composition.  These tests verify that the
+// chain is read back correctly by the entry functions.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(CGALPhase8bLite, NamedParamPipe_MultipleParamsTakeEffect)
+{
+    using K = CGAL::Simple_cartesian<double>;
+    auto mesh = make_quad_strip();
+
+    auto uv = mesh.add_property_map<Vertex_index, K::Point_2>(
+        "v:pipe_uv", K::Point_2(0, 0)).first;
+
+    // Chain three parameters using `|`.
+    auto params = CGAL::parameters::gradient_tolerance(1e-12)
+                | CGAL::parameters::max_iterations(500)
+                | CGAL::parameters::output_uv_map(uv);
+
+    auto res = CGAL::discrete_conformal_map_euclidean(mesh, params);
+
+    EXPECT_TRUE(res.converged);
+    EXPECT_LT(res.gradient_norm, 1e-10);   // tight tolerance applied
+    EXPECT_LE(res.iterations, 500);
+    // UV pmap was populated.
+    bool any_nonzero = false;
+    for (auto v : mesh.vertices()) {
+        if (std::abs(uv[v].x()) + std::abs(uv[v].y()) > 1e-10) {
+            any_nonzero = true;
+            break;
+        }
+    }
+    EXPECT_TRUE(any_nonzero);
+}
+
+TEST(CGALPhase8bLite, NamedParamPipe_TwoParams)
+{
+    // Pipe two parameters and verify both take effect.
+    auto mesh = make_triangle();
+    auto params = CGAL::parameters::max_iterations(0)
+                | CGAL::parameters::gradient_tolerance(1e-6);
+    auto res = CGAL::discrete_conformal_map_euclidean(mesh, params);
+    EXPECT_EQ(res.iterations, 0);   // max_iterations(0) blocks the loop
+}

code/tests/cgal/test_cgal_traits_mvp.cpp

@@ -0,0 +1,249 @@
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
+// test_cgal_traits_mvp.cpp
+//
+// Phase 8 MVP — first tests for the new CGAL-style public API.
+//
+// Validates:
+//   1. Default_conformal_map_traits<Surface_mesh, K> compiles and
+//      provides all advertised types and property-map accessors.
+//   2. discrete_conformal_map_euclidean() runs end-to-end on a small mesh.
+//   3. Named-parameter overrides (gradient_tolerance, max_iterations)
+//      change the Newton behaviour as expected.
+//   4. The result agrees with the legacy newton_euclidean() at the
+//      same DOF assignment — proving the wrapper is non-destructive.
+//
+// These tests are the Phase 8 MVP acceptance probe.  Phase 9a
+// (Inversive-Distance) will become the next, deeper validation by
+// implementing a new functional against this same trait API.
+
+#include <CGAL/Conformal_map_traits.h>
+#include <CGAL/Discrete_conformal_map.h>
+#include <CGAL/Kernel_traits.h>
+
+#include "mesh_builder.hpp"   // make_triangle, make_quad_strip, make_tetrahedron
+#include "euclidean_functional.hpp"
+#include "newton_solver.hpp"
+
+#include <gtest/gtest.h>
+#include <cmath>
+
+using namespace conformallab;
+
+// ════════════════════════════════════════════════════════════════════════════
+// 1.  Traits class: compile-time type sanity
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(CGALConformalTraits, DefaultTraitsTypes)
+{
+    using K     = CGAL::Simple_cartesian<double>;
+    using Mesh  = CGAL::Surface_mesh<K::Point_3>;
+    using Tr    = CGAL::Default_conformal_map_traits<Mesh, K>;
+
+    // FT comes from the kernel.
+    static_assert(std::is_same_v<typename Tr::FT, double>);
+
+    // Descriptors come from boost::graph_traits, not from Surface_mesh directly.
+    static_assert(std::is_same_v<typename Tr::Triangle_mesh, Mesh>);
+    static_assert(std::is_same_v<
+        typename Tr::Vertex_descriptor,
+        typename boost::graph_traits<Mesh>::vertex_descriptor>);
+
+    // Property-map types should match Surface_mesh::Property_map for the
+    // appropriate key.
+    static_assert(std::is_same_v<
+        typename Tr::Theta_pmap,
+        typename Mesh::template Property_map<typename Tr::Vertex_descriptor, double>>);
+    static_assert(std::is_same_v<
+        typename Tr::Vertex_index_pmap,
+        typename Mesh::template Property_map<typename Tr::Vertex_descriptor, int>>);
+    static_assert(std::is_same_v<
+        typename Tr::Lambda0_pmap,
+        typename Mesh::template Property_map<typename Tr::Edge_descriptor, double>>);
+
+    // Default kernel: Simple_cartesian<double>.
+    using TrDefault = CGAL::Default_conformal_map_traits<Mesh>;
+    static_assert(std::is_same_v<typename TrDefault::Kernel,
+                                 CGAL::Simple_cartesian<double>>);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// 2.  Traits property-map accessors are non-destructive
+//
+// Setting up via the trait helpers and via setup_euclidean_maps() must yield
+// the same property map (Surface_mesh deduplicates by name).
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(CGALConformalTraits, AccessorsReuseExistingMaps)
+{
+    using K   = CGAL::Simple_cartesian<double>;
+    using Mesh = CGAL::Surface_mesh<K::Point_3>;
+    using Tr  = CGAL::Default_conformal_map_traits<Mesh, K>;
+
+    auto mesh = make_triangle();
+    auto maps = setup_euclidean_maps(mesh);
+
+    auto theta_via_traits   = Tr::theta_map(mesh);
+    auto idx_via_traits     = Tr::vertex_index_map(mesh);
+    auto lambda0_via_traits = Tr::lambda0_map(mesh);
+
+    // Surface_mesh property maps with the same key type are equality-comparable
+    // by name lookup — accessing through the traits class must return the
+    // same map that setup_euclidean_maps() created.
+    EXPECT_EQ(theta_via_traits,   maps.theta_v);
+    EXPECT_EQ(idx_via_traits,     maps.v_idx);
+    EXPECT_EQ(lambda0_via_traits, maps.lambda0);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// 3.  End-to-end: discrete_conformal_map_euclidean() on a small open mesh
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(CGALDiscreteConformalMap, SingleTriangleConverges)
+{
+    auto mesh = make_triangle();
+    auto result = CGAL::discrete_conformal_map_euclidean(mesh);
+
+    EXPECT_TRUE(result.converged)
+        << "Newton did not converge on a single triangle";
+    EXPECT_LT(result.gradient_norm, 1e-8);
+    EXPECT_GE(result.iterations, 0);
+    EXPECT_EQ(result.u_per_vertex.size(), num_vertices(mesh));
+
+    // With the default flat-disc target curvature and the first vertex pinned,
+    // the natural-theta equilibrium is at u = 0 — Newton should accept x0=0.
+    for (double u : result.u_per_vertex)
+        EXPECT_NEAR(u, 0.0, 1e-8);
+}
+
+TEST(CGALDiscreteConformalMap, QuadStripConverges)
+{
+    auto mesh = make_quad_strip();
+    auto result = CGAL::discrete_conformal_map_euclidean(mesh);
+
+    EXPECT_TRUE(result.converged);
+    EXPECT_LT(result.gradient_norm, 1e-8);
+    EXPECT_EQ(result.u_per_vertex.size(), num_vertices(mesh));
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// 4.  Named-parameter overrides take effect
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(CGALDiscreteConformalMap, MaxIterationsTakesEffect)
+{
+    auto mesh = make_triangle();
+
+    // max_iterations(0) forces Newton to give up immediately.
+    auto result = CGAL::discrete_conformal_map_euclidean(
+        mesh,
+        CGAL::parameters::max_iterations(0));
+
+    EXPECT_EQ(result.iterations, 0);
+    // Trivial natural-theta case: gradient is already zero at x=0,
+    // so even 0 iterations may report "converged" depending on the
+    // initial gradient check.  The point is just that the parameter
+    // was *read* — verified by EXPECT_EQ on iterations above.
+}
+
+TEST(CGALDiscreteConformalMap, GradientToleranceTakesEffect)
+{
+    auto mesh = make_quad_strip();
+
+    // Loose tolerance — must still converge, but possibly in fewer steps.
+    auto result_loose = CGAL::discrete_conformal_map_euclidean(
+        mesh,
+        CGAL::parameters::gradient_tolerance(1e-4));
+    EXPECT_TRUE(result_loose.converged);
+
+    // Strict tolerance — also must converge, gradient norm must be tighter.
+    auto result_strict = CGAL::discrete_conformal_map_euclidean(
+        mesh,
+        CGAL::parameters::gradient_tolerance(1e-12));
+    EXPECT_TRUE(result_strict.converged);
+    EXPECT_LT(result_strict.gradient_norm, 1e-10);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// 5.  Wrapper agrees with the legacy newton_euclidean() at the same setup
+//
+// This is the cross-API consistency check: same mesh, same default settings
+// (first vertex pinned, x0=0) — the u-vector returned by the wrapper must
+// match what newton_euclidean produces directly.
+// ════════════════════════════════════════════════════════════════════════════
+
+// ════════════════════════════════════════════════════════════════════════════
+// 6.  Kernel deduction: the wrapper must NOT hard-code Simple_cartesian
+//
+// Regression guard: the wrapper deduces its kernel from the mesh point type
+// via `CGAL::Kernel_traits`.  If anyone re-introduces a hard-coded
+// `Simple_cartesian<double>` in the wrapper, the static_asserts here still
+// pass (the legacy ConformalMesh uses that kernel) — but Phase 9a or any
+// user with a different kernel-backed Surface_mesh would fail to compile.
+// This test pins the deduction *contract* explicitly.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(CGALDiscreteConformalMap, KernelIsDeducedFromMeshPointType)
+{
+    using Mesh = ConformalMesh;
+    using P    = typename Mesh::Point;
+
+    using DeducedKernel = typename CGAL::Kernel_traits<P>::Kernel;
+    using DeducedTraits = CGAL::Default_conformal_map_traits<Mesh, DeducedKernel>;
+
+    static_assert(std::is_same_v<DeducedKernel, CGAL::Simple_cartesian<double>>,
+                  "ConformalMesh point type must deduce to Simple_cartesian<double>");
+    static_assert(std::is_same_v<typename DeducedTraits::FT, double>);
+    static_assert(std::is_same_v<typename DeducedTraits::Triangle_mesh, Mesh>);
+
+    // Run-time sanity: the wrapper accepts the deduced-kernel mesh end-to-end.
+    auto mesh = make_quad_strip();
+    auto result = CGAL::discrete_conformal_map_euclidean(mesh);
+    EXPECT_TRUE(result.converged);
+}
+
+TEST(CGALDiscreteConformalMap, WrapperMatchesLegacyAPI)
+{
+    auto mesh = make_quad_strip();
+
+    // ── New API: applies natural-theta automatically ───────────────────────
+    auto result_new = CGAL::discrete_conformal_map_euclidean(mesh);
+
+    // ── Legacy API on a fresh mesh — must replicate the *same* preparation
+    //    that the wrapper performs internally (pin first vertex, assign
+    //    DOFs, apply natural-theta).  Otherwise the comparison is unfair
+    //    (Newton would diverge without natural-theta on these meshes). ────
+    auto mesh_legacy = make_quad_strip();
+    auto maps = setup_euclidean_maps(mesh_legacy);
+    compute_euclidean_lambda0_from_mesh(mesh_legacy, maps);
+
+    // Pin first vertex (gauge), assign sequential DOFs to the rest.
+    auto vit = mesh_legacy.vertices().begin();
+    maps.v_idx[*vit++] = -1;
+    int idx = 0;
+    for (; vit != mesh_legacy.vertices().end(); ++vit)
+        maps.v_idx[*vit] = idx++;
+
+    // Natural-theta: shift Θ so that x = 0 is the natural equilibrium.
+    std::vector<double> x0(idx, 0.0);
+    auto G0 = euclidean_gradient(mesh_legacy, x0, maps);
+    for (auto v : mesh_legacy.vertices()) {
+        int j = maps.v_idx[v];
+        if (j >= 0) maps.theta_v[v] -= G0[static_cast<std::size_t>(j)];
+    }
+
+    auto nr = newton_euclidean(mesh_legacy, x0, maps, 1e-10, 200);
+
+    // ── Compare ────────────────────────────────────────────────────────────
+    EXPECT_EQ(result_new.converged, nr.converged);
+    EXPECT_NEAR(result_new.gradient_norm, nr.grad_inf_norm, 1e-12);
+
+    // Pinned vertex u is 0 in both; for the rest the values agree.
+    for (auto v : mesh_legacy.vertices()) {
+        int j = maps.v_idx[v];
+        double u_legacy = (j >= 0) ? nr.x[static_cast<std::size_t>(j)] : 0.0;
+        EXPECT_NEAR(result_new.u_per_vertex[v.idx()], u_legacy, 1e-10)
+            << "Wrapper diverges from legacy for vertex " << v.idx();
+    }
+}

code/tests/cgal/test_conformal_mesh.cpp

@@ -0,0 +1,243 @@
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
+// test_conformal_mesh.cpp
+//
+// Phase 3a — CGAL Surface_mesh infrastructure tests.
+//
+// Verifies that ConformalMesh (CGAL::Surface_mesh<Point3>) and the
+// mesh_builder factories behave correctly before we build the functionals
+// on top of them (Phase 3b).
+//
+// Test groups
+// ───────────
+//   Topology     – vertex/edge/face counts, Euler characteristic
+//   Traversal    – halfedge iteration around vertex / face / edge
+//   PropertyMaps – read/write of lambda, theta, idx, alpha, f:type
+//   Validity     – all make_* factories produce valid, consistent meshes
+
+#include "conformal_mesh.hpp"
+#include "mesh_builder.hpp"
+#include <CGAL/boost/graph/iterator.h>
+#include <gtest/gtest.h>
+#include <cmath>
+
+using namespace conformallab;
+
+// ════════════════════════════════════════════════════════════
+// Topology
+// ════════════════════════════════════════════════════════════
+
+// Single triangle: 3 vertices, 1 face, 3 edges.
+TEST(ConformalMeshTopology, SingleTriangle)
+{
+    auto mesh = make_triangle();
+    EXPECT_EQ(3u, mesh.number_of_vertices());
+    EXPECT_EQ(1u, mesh.number_of_faces());
+    EXPECT_EQ(3u, mesh.number_of_edges());
+}
+
+// Tetrahedron: V=4, E=6, F=4  →  Euler = 2 (sphere topology).
+TEST(ConformalMeshTopology, TetrahedronEuler)
+{
+    auto mesh = make_tetrahedron();
+    EXPECT_EQ(4u, mesh.number_of_vertices());
+    EXPECT_EQ(6u, mesh.number_of_edges());
+    EXPECT_EQ(4u, mesh.number_of_faces());
+
+    int euler = (int)mesh.number_of_vertices()
+              - (int)mesh.number_of_edges()
+              + (int)mesh.number_of_faces();
+    EXPECT_EQ(2, euler) << "Euler characteristic of closed sphere must be 2";
+}
+
+// Two-triangle strip: V=4, E=5, F=2.
+// The interior edge (shared diagonal) has no border halfedge.
+TEST(ConformalMeshTopology, QuadStrip)
+{
+    auto mesh = make_quad_strip();
+    EXPECT_EQ(4u, mesh.number_of_vertices());
+    EXPECT_EQ(5u, mesh.number_of_edges());
+    EXPECT_EQ(2u, mesh.number_of_faces());
+
+    // Count interior (non-boundary) edges
+    int interior = 0;
+    for (auto e : mesh.edges())
+        if (!mesh.is_border(e)) ++interior;
+    EXPECT_EQ(1, interior) << "Only the shared diagonal should be interior";
+}
+
+// Fan with n triangles: V=n+1, E=2n, F=n.
+TEST(ConformalMeshTopology, FanCounts)
+{
+    for (int n : {3, 4, 6, 8}) {
+        auto mesh = make_fan(n);
+        EXPECT_EQ((std::size_t)(n + 1), mesh.number_of_vertices());
+        EXPECT_EQ((std::size_t)(2 * n),  mesh.number_of_edges());
+        EXPECT_EQ((std::size_t)(n),       mesh.number_of_faces());
+    }
+}
+
+// ════════════════════════════════════════════════════════════
+// Halfedge Traversal
+// ════════════════════════════════════════════════════════════
+
+// For a regular tetrahedron every vertex has valence 3.
+TEST(ConformalMeshTraversal, TetrahedronVertexValence)
+{
+    auto mesh = make_tetrahedron();
+    for (auto v : mesh.vertices()) {
+        int degree = 0;
+        for (auto h : CGAL::halfedges_around_target(v, mesh))
+            { (void)h; ++degree; }
+        EXPECT_EQ(3, degree) << "Each tetrahedron vertex has degree 3";
+    }
+}
+
+// For a fan with n triangles the center vertex has valence n.
+TEST(ConformalMeshTraversal, FanCenterValence)
+{
+    for (int n : {3, 5, 7}) {
+        auto mesh = make_fan(n);
+
+        // Center vertex is always the first one added (index 0).
+        auto center = *mesh.vertices().begin();
+        int degree = 0;
+        for (auto h : CGAL::halfedges_around_target(center, mesh))
+            { (void)h; ++degree; }
+        EXPECT_EQ(n, degree)
+            << "Fan center vertex must have valence == n=" << n;
+    }
+}
+
+// Every face of the tetrahedron has exactly 3 halfedges.
+TEST(ConformalMeshTraversal, FaceHalfedgeCount)
+{
+    auto mesh = make_tetrahedron();
+    for (auto f : mesh.faces()) {
+        int count = 0;
+        for (auto h : CGAL::halfedges_around_face(mesh.halfedge(f), mesh))
+            { (void)h; ++count; }
+        EXPECT_EQ(3, count) << "Each triangular face must have exactly 3 halfedges";
+    }
+}
+
+// opposite(h) and h share the same edge; opposite(opposite(h)) == h.
+TEST(ConformalMeshTraversal, OppositeHalfedgeConsistency)
+{
+    auto mesh = make_tetrahedron();
+    for (auto h : mesh.halfedges()) {
+        auto opp = mesh.opposite(h);
+        EXPECT_EQ(mesh.edge(h), mesh.edge(opp))
+            << "h and opposite(h) must share the same edge";
+        EXPECT_EQ(h, mesh.opposite(opp))
+            << "opposite(opposite(h)) must equal h";
+    }
+}
+
+// ════════════════════════════════════════════════════════════
+// Property Maps
+// ════════════════════════════════════════════════════════════
+
+// The conformal variable lambda can be written and read back per vertex.
+TEST(ConformalMeshProperties, VertexLambdaReadWrite)
+{
+    auto mesh = make_tetrahedron();
+    auto [lambda, theta, idx] = add_vertex_properties(mesh);
+
+    double value = 0.0;
+    for (auto v : mesh.vertices()) {
+        lambda[v] = value;
+        value    += 1.0;
+    }
+
+    value = 0.0;
+    for (auto v : mesh.vertices()) {
+        EXPECT_DOUBLE_EQ(value, lambda[v]);
+        value += 1.0;
+    }
+}
+
+// Default solver index is -1 (pinned); can be overwritten.
+TEST(ConformalMeshProperties, VertexSolverIndex)
+{
+    auto mesh = make_tetrahedron();
+    auto [lambda, theta, idx] = add_vertex_properties(mesh);
+
+    // All vertices start at -1 (pinned / boundary)
+    for (auto v : mesh.vertices())
+        EXPECT_EQ(-1, idx[v]) << "Default solver index must be -1";
+
+    // Assign sequential indices
+    int i = 0;
+    for (auto v : mesh.vertices())
+        idx[v] = i++;
+
+    i = 0;
+    for (auto v : mesh.vertices())
+        EXPECT_EQ(i++, idx[v]);
+}
+
+// Edge alpha (intersection angle): set and retrieve per edge.
+TEST(ConformalMeshProperties, EdgeAlpha)
+{
+    auto mesh = make_quad_strip();
+    auto alpha = add_edge_properties(mesh);
+
+    const double kAlpha = M_PI / 3.0;   // 60°
+    for (auto e : mesh.edges())
+        alpha[e] = kAlpha;
+
+    for (auto e : mesh.edges())
+        EXPECT_DOUBLE_EQ(kAlpha, alpha[e]);
+
+    EXPECT_EQ(5u, mesh.number_of_edges());
+}
+
+// Face geometry type: Euclidean by default, switchable to Hyperbolic.
+TEST(ConformalMeshProperties, FaceGeometryType)
+{
+    auto mesh = make_tetrahedron();
+    auto ftype = add_face_properties(mesh);
+
+    // Default: Euclidean
+    for (auto f : mesh.faces())
+        EXPECT_EQ(static_cast<int>(GeometryType::Euclidean), ftype[f]);
+
+    // Switch all to Hyperbolic
+    for (auto f : mesh.faces())
+        ftype[f] = static_cast<int>(GeometryType::Hyperbolic);
+
+    for (auto f : mesh.faces())
+        EXPECT_EQ(static_cast<int>(GeometryType::Hyperbolic), ftype[f]);
+}
+
+// Adding the same named property map twice: second call returns ok=false
+// and both handles alias the same storage.
+TEST(ConformalMeshProperties, PropertyMapIdempotent)
+{
+    auto mesh = make_triangle();
+    auto [pm1, ok1] = mesh.add_property_map<Vertex_index, double>("v:lambda", 0.0);
+    auto [pm2, ok2] = mesh.add_property_map<Vertex_index, double>("v:lambda", 0.0);
+
+    EXPECT_TRUE(ok1)  << "First add_property_map must succeed";
+    EXPECT_FALSE(ok2) << "Second add_property_map on existing name must return ok=false";
+
+    // Both handles must alias the same storage
+    auto v = *mesh.vertices().begin();
+    pm1[v] = 42.0;
+    EXPECT_DOUBLE_EQ(42.0, pm2[v]) << "Both handles must alias the same storage";
+}
+
+// ════════════════════════════════════════════════════════════
+// Mesh validity
+// ════════════════════════════════════════════════════════════
+
+// CGAL's built-in validity check must pass for all factory meshes.
+TEST(ConformalMeshValidity, AllBuilders)
+{
+    EXPECT_TRUE(make_triangle().is_valid())    << "triangle mesh invalid";
+    EXPECT_TRUE(make_tetrahedron().is_valid()) << "tetrahedron mesh invalid";
+    EXPECT_TRUE(make_quad_strip().is_valid())  << "quad strip mesh invalid";
+    EXPECT_TRUE(make_fan(6).is_valid())        << "fan-6 mesh invalid";
+}

code/tests/cgal/test_cp_euclidean_functional.cpp

@@ -0,0 +1,270 @@
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
+// test_cp_euclidean_functional.cpp
+//
+// Phase 9a.1 — CPEuclideanFunctional (BPS 2010) tests.
+//
+// Replicates de.varylab.discreteconformal.functional.CPEuclideanFunctionalTest
+// (88 lines) and adds boundary-edge coverage plus a closed-mesh case.
+//
+// Java test pattern (lines 50-87):
+//   1. Build dodecahedron via HalfEdgeUtils.addDodecahedron.
+//   2. Remove face 0 to produce an open mesh.
+//   3. theta_e = π/2 for every edge.       (orthogonal circle packing)
+//   4. phi_f   = 2π for every face.        (flat target)
+//   5. Random ρ ∈ [−0.5, 0.5] (seed 1).
+//   6. FunctionalTest.setXGradient(ρ) → FD-vs-analytic gradient check.
+//   7. FunctionalTest.setXHessian(ρ)  → FD-vs-analytic Hessian  check.
+//
+// C++ port uses the tetrahedron (4 faces) instead of the dodecahedron (12 faces)
+// because the analytic structure is identical and the smaller mesh keeps the
+// test fast and human-inspectable.  We exercise the boundary-edge code path
+// by additionally testing a tetrahedron with one face removed (3 faces, 3
+// boundary edges, 3 interior edges).
+
+#include "cp_euclidean_functional.hpp"
+#include "mesh_builder.hpp"
+#include "conformal_mesh.hpp"
+
+#include <Eigen/Eigenvalues>
+#include <gtest/gtest.h>
+#include <vector>
+#include <random>
+
+using namespace conformallab;
+
+// ════════════════════════════════════════════════════════════════════════════
+// 1.  Helper: explicit values for p(θ*, Δρ) at known inputs
+//
+// p(θ*, 0) = 0          (tanh 0 = 0)
+// p(π, Δρ) = π·sign(Δρ) (tan(π/2) = ∞, atan saturates to ±π/2)
+// p(0, Δρ) = 0          (tan(0) = 0)
+// p odd in Δρ (tanh is odd).
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(CPEuclideanFunctional, PFunctionKnownValues)
+{
+    using cp_detail::p_function;
+    constexpr double PI_ = 3.14159265358979323846;
+
+    // p(any, 0) = 0
+    EXPECT_NEAR(p_function(PI_ / 4,  0.0), 0.0, 1e-15);
+    EXPECT_NEAR(p_function(PI_ / 2,  0.0), 0.0, 1e-15);
+
+    // Odd in Δρ
+    const double thStar = PI_ / 3;
+    for (double dr : {0.1, 0.5, 1.0, 2.0}) {
+        EXPECT_NEAR(p_function(thStar, dr) + p_function(thStar, -dr), 0.0, 1e-12)
+            << "p(θ*, Δρ) should be odd in Δρ";
+    }
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// 2.  Property-map setup defaults
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(CPEuclideanFunctional, SetupDefaults)
+{
+    auto mesh = make_tetrahedron();
+    auto m = setup_cp_euclidean_maps(mesh);
+
+    constexpr double PI_ = 3.14159265358979323846;
+    for (auto e : mesh.edges()) EXPECT_NEAR(m.theta_e[e], PI_ / 2, 1e-15);
+    for (auto f : mesh.faces()) EXPECT_NEAR(m.phi_f[f],   2.0 * PI_, 1e-15);
+    for (auto f : mesh.faces()) EXPECT_EQ(m.f_idx[f], -1) << "all faces start pinned";
+}
+
+TEST(CPEuclideanFunctional, AssignDofIndices_PinsOneFace)
+{
+    auto mesh = make_tetrahedron();
+    auto m    = setup_cp_euclidean_maps(mesh);
+    const int n = assign_cp_euclidean_face_dof_indices(mesh, m);
+
+    EXPECT_EQ(n, 3) << "tetrahedron has 4 faces; 1 pinned ⇒ 3 free DOFs";
+
+    int pinned_count = 0;
+    int max_idx = -1;
+    for (auto f : mesh.faces()) {
+        if (m.f_idx[f] == -1) ++pinned_count;
+        else max_idx = std::max(max_idx, m.f_idx[f]);
+    }
+    EXPECT_EQ(pinned_count, 1);
+    EXPECT_EQ(max_idx, 2);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// 3.  Tangential limit (θ = 0): p = 0, energy collapses, gradient = φ_f
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(CPEuclideanFunctional, TangentialLimitGradientEqualsPhi)
+{
+    auto mesh = make_tetrahedron();
+    auto m    = setup_cp_euclidean_maps(mesh);
+    for (auto e : mesh.edges()) m.theta_e[e] = 0.0;   // tangential limit
+    const int n = assign_cp_euclidean_face_dof_indices(mesh, m);
+
+    // At θ = 0: θ* = π.  Interior edge contribution: −(p+θ*) where p = π·sign(Δρ).
+    // Boundary contribution: −2π.  At ρ = 0, Δρ = 0 so p = 0; each interior face
+    // contributes −π per incident interior halfedge; for a tetrahedron each face
+    // has 3 interior halfedges ⇒ −3π.  Net gradient: 2π − 3π = −π per free face.
+    std::vector<double> x(static_cast<std::size_t>(n), 0.0);
+    auto G = cp_euclidean_gradient(mesh, x, m);
+
+    constexpr double PI_ = 3.14159265358979323846;
+    for (double g : G) EXPECT_NEAR(g, -PI_, 1e-10);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// 4.  FD gradient check on closed tetrahedron at random ρ
+//
+// Java parity: this is exactly the structure of CPEuclideanFunctionalTest.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(CPEuclideanFunctional, FDGradientCheck_ClosedTetrahedron_RandomRho)
+{
+    auto mesh = make_tetrahedron();
+    auto m    = setup_cp_euclidean_maps(mesh);
+    const int n = assign_cp_euclidean_face_dof_indices(mesh, m);
+
+    // Java: rnd.setSeed(1);  rho_i = rnd.nextDouble() − 0.5
+    std::mt19937 rng(1);
+    std::uniform_real_distribution<double> u(-0.5, 0.5);
+    std::vector<double> rho(static_cast<std::size_t>(n));
+    for (auto& r : rho) r = u(rng);
+
+    EXPECT_TRUE(gradient_check_cp_euclidean(mesh, rho, m))
+        << "FD vs analytic gradient mismatch on closed tetrahedron";
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// 5.  FD Hessian check on closed tetrahedron at random ρ
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(CPEuclideanFunctional, FDHessianCheck_ClosedTetrahedron_RandomRho)
+{
+    auto mesh = make_tetrahedron();
+    auto m    = setup_cp_euclidean_maps(mesh);
+    const int n = assign_cp_euclidean_face_dof_indices(mesh, m);
+
+    std::mt19937 rng(1);
+    std::uniform_real_distribution<double> u(-0.5, 0.5);
+    std::vector<double> rho(static_cast<std::size_t>(n));
+    for (auto& r : rho) r = u(rng);
+
+    EXPECT_TRUE(hessian_check_cp_euclidean(mesh, rho, m))
+        << "FD vs analytic Hessian mismatch on closed tetrahedron";
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// 6.  Boundary-edge coverage: open mesh (tetrahedron with one face removed)
+//
+// Java test does this via `hds.removeFace(hds.getFace(0))`.  In CGAL we get
+// an equivalent open mesh by skipping the construction of one face.
+// ════════════════════════════════════════════════════════════════════════════
+
+inline ConformalMesh make_open_tetrahedron()
+{
+    ConformalMesh mesh;
+    auto v0 = mesh.add_vertex(Point3( 1,  1,  1));
+    auto v1 = mesh.add_vertex(Point3( 1, -1, -1));
+    auto v2 = mesh.add_vertex(Point3(-1,  1, -1));
+    auto v3 = mesh.add_vertex(Point3(-1, -1,  1));
+    // Three faces (omit the one opposite v0):
+    mesh.add_face(v0, v2, v1);
+    mesh.add_face(v0, v1, v3);
+    mesh.add_face(v0, v3, v2);
+    return mesh;
+}
+
+TEST(CPEuclideanFunctional, FDGradientCheck_OpenTetrahedron_RandomRho)
+{
+    auto mesh = make_open_tetrahedron();
+    auto m    = setup_cp_euclidean_maps(mesh);
+    const int n = assign_cp_euclidean_face_dof_indices(mesh, m);
+
+    EXPECT_EQ(n, 2);   // 3 faces, 1 pinned ⇒ 2 free DOFs
+
+    std::mt19937 rng(1);
+    std::uniform_real_distribution<double> u(-0.5, 0.5);
+    std::vector<double> rho(static_cast<std::size_t>(n));
+    for (auto& r : rho) r = u(rng);
+
+    EXPECT_TRUE(gradient_check_cp_euclidean(mesh, rho, m))
+        << "FD vs analytic gradient mismatch on open tetrahedron";
+}
+
+TEST(CPEuclideanFunctional, FDHessianCheck_OpenTetrahedron_RandomRho)
+{
+    auto mesh = make_open_tetrahedron();
+    auto m    = setup_cp_euclidean_maps(mesh);
+    const int n = assign_cp_euclidean_face_dof_indices(mesh, m);
+
+    std::mt19937 rng(1);
+    std::uniform_real_distribution<double> u(-0.5, 0.5);
+    std::vector<double> rho(static_cast<std::size_t>(n));
+    for (auto& r : rho) r = u(rng);
+
+    EXPECT_TRUE(hessian_check_cp_euclidean(mesh, rho, m))
+        << "FD vs analytic Hessian mismatch on open tetrahedron";
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// 7.  Hessian is symmetric positive-semidefinite (BPS-2010 §6 convexity)
+//
+// The energy is convex in ρ on its domain of validity.  Hence H is PSD with
+// a 1-dim null space (constant shift of all ρ, removed by gauge pin).
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(CPEuclideanFunctional, HessianIsPSD)
+{
+    auto mesh = make_tetrahedron();
+    auto m    = setup_cp_euclidean_maps(mesh);
+    const int n = assign_cp_euclidean_face_dof_indices(mesh, m);
+
+    std::vector<double> rho(static_cast<std::size_t>(n), 0.1);
+    auto H = cp_euclidean_hessian(mesh, rho, m);
+
+    // Symmetry
+    Eigen::MatrixXd Hd(H);
+    EXPECT_NEAR((Hd - Hd.transpose()).cwiseAbs().maxCoeff(), 0.0, 1e-15);
+
+    // Smallest eigenvalue ≥ 0 (PSD)
+    Eigen::SelfAdjointEigenSolver<Eigen::MatrixXd> es(Hd);
+    EXPECT_GE(es.eigenvalues().minCoeff(), -1e-12)
+        << "Hessian must be PSD (BPS-2010 §6)";
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// 8.  At equilibrium (Newton-converged ρ*), the gradient is zero by construction
+//
+// We do not run a full Newton solver here; we set up the "natural-theta" trick:
+// adjust φ_f so that ρ = 0 is the equilibrium.  This is the analog of the
+// natural-theta convention already used in euclidean_functional tests
+// (see test_euclidean_functional.cpp lines 159-189).
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(CPEuclideanFunctional, NaturalPhiMakesZeroTheEquilibrium)
+{
+    auto mesh = make_tetrahedron();
+    auto m    = setup_cp_euclidean_maps(mesh);
+    const int n = assign_cp_euclidean_face_dof_indices(mesh, m);
+
+    std::vector<double> rho(static_cast<std::size_t>(n), 0.0);
+
+    // Step 1: gradient at ρ = 0 with default φ.
+    auto G0 = cp_euclidean_gradient(mesh, rho, m);
+
+    // Step 2: adjust φ_f so the new gradient at ρ = 0 is zero.
+    //   ∂E/∂ρ_f = φ_f − (sum of edge contributions)
+    //   To zero G_f: subtract G_f from φ_f.
+    for (auto f : mesh.faces()) {
+        int i = m.f_idx[f];
+        if (i < 0) continue;
+        m.phi_f[f] -= G0[static_cast<std::size_t>(i)];
+    }
+
+    // Step 3: gradient at ρ = 0 should now be ~zero.
+    auto G_eq = cp_euclidean_gradient(mesh, rho, m);
+    for (double g : G_eq) EXPECT_NEAR(g, 0.0, 1e-13);
+}

code/tests/cgal/test_euclidean_functional.cpp

@@ -0,0 +1,292 @@
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
+// test_euclidean_functional.cpp
+//
+// Phase 3d — EuclideanCyclicFunctional ported to ConformalMesh.
+//
+// Corresponds to de.varylab.discreteconformal.functional.EuclideanCyclicFunctionalTest.
+//
+// Test map (Java → C++)
+// ──────────────────────
+//   testHessian (Ignored)       → GradientCheck_Hessian           (ported)
+//   testGradient…Triangle       → GradientCheck_TriangleVertex     (ported)
+//   testGradient…QuadStrip      → GradientCheck_QuadStripVertex    (ported)
+//   testGradient…Tetrahedron    → GradientCheck_TetrahedronVertex  (ported)
+//   testGradient…AllDofs        → GradientCheck_TetrahedronAllDofs (ported)
+//   testFunctionalAtNaNValue    → AnglesFiniteAtKnownPoint          (ported)
+//
+// Energy model
+// ────────────
+//   Uses the Schläfli path integral E(x) = ∫₀¹⟨G(tx),x⟩dt (10-point GL).
+//   The gradient check verifies G is curl-free.
+
+#include "conformal_mesh.hpp"
+#include "mesh_builder.hpp"
+#include "euclidean_geometry.hpp"
+#include "euclidean_functional.hpp"
+#include "euclidean_hessian.hpp"
+#include <gtest/gtest.h>
+#include <cmath>
+#include <vector>
+
+using namespace conformallab;
+
+// ════════════════════════════════════════════════════════════════════════════
+// Cross-module Hessian check: euclidean_gradient() ↔ euclidean_hessian()
+//
+// Java @Ignore reason: "no Hessian implemented yet" — the Java functional
+// test was written before the Hessian existed.  In C++ the analytic
+// cotangent-Laplace Hessian (euclidean_hessian.hpp, Phase 3f) is complete.
+//
+// This test verifies cross-module consistency:
+//   H[i,j] ≈ (G_i(x+ε·eⱼ) − G_i(x−ε·eⱼ)) / (2ε)
+// using the gradient from euclidean_functional.hpp and the Hessian from
+// euclidean_hessian.hpp.  A bug in DOF-index mapping or sign convention
+// that affects both modules independently would only be caught here.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(EuclideanFunctional, GradientCheck_Hessian)
+{
+    auto mesh = make_triangle();
+    auto maps = setup_euclidean_maps(mesh);
+    compute_euclidean_lambda0_from_mesh(mesh, maps);
+    int n = assign_euclidean_vertex_dof_indices(mesh, maps);
+
+    std::vector<double> x(static_cast<std::size_t>(n), -0.1);
+
+    // hessian_check_euclidean: H[i,j] ≈ FD(G)[i,j] using euclidean_gradient()
+    EXPECT_TRUE(hessian_check_euclidean(mesh, x, maps))
+        << "Cross-module: euclidean_gradient() and euclidean_hessian() are inconsistent";
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Angle formula: equilateral triangle → all angles = π/3
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(EuclideanFunctional, EquilateralTriangleAnglesArePiOver3)
+{
+    // All sides equal: l = 1.0, log-length = 0.
+    auto fa = euclidean_angles(0.0, 0.0, 0.0);
+
+    ASSERT_TRUE(fa.valid) << "Equilateral triangle must be valid";
+    constexpr double PI_3 = 3.14159265358979323846 / 3.0;
+    EXPECT_NEAR(fa.alpha1, PI_3, 1e-12);
+    EXPECT_NEAR(fa.alpha2, PI_3, 1e-12);
+    EXPECT_NEAR(fa.alpha3, PI_3, 1e-12);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Angle formula: right isosceles triangle (legs 1, hypotenuse √2)
+//
+// For the 45-45-90 triangle: angles are π/4, π/4, π/2.
+// From make_triangle default (v0=(0,0), v1=(1,0), v2=(0,1)):
+//   e01: l=1, λ°=0
+//   e12: l=√2, λ°=log(2)
+//   e02: l=1, λ°=0
+// Angle at v0 (opposite e12) = π/2.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(EuclideanFunctional, RightIsoscelesTriangleAnglesCorrect)
+{
+    const double log2 = std::log(2.0);
+    // lam12 = 0 (v0-v1, length 1), lam23 = log(2) (v1-v2, length √2), lam31 = 0 (v2-v0, length 1)
+    // v1 = v0 in our ordering → remap: l01=1, l12=√2, l20=1
+    // Using euclidean_angles(lam_v1v2, lam_v2v3, lam_v3v1):
+    //   v1=(0,0), v2=(1,0), v3=(0,1)
+    //   lam12 = log(1²) = 0, lam23 = log(√2 ²) = log2, lam31 = log(1²) = 0
+    auto fa = euclidean_angles(0.0, log2, 0.0);
+
+    ASSERT_TRUE(fa.valid);
+    constexpr double PI   = 3.14159265358979323846;
+    // v1=(0,0) is at the right-angle corner (opposite the hypotenuse l23=√2) → α1 = 90°.
+    // v2=(1,0) and v3=(0,1) are the 45° corners (each opposite a leg of length 1).
+    EXPECT_NEAR(fa.alpha1, PI / 2.0, 1e-12);   // angle at v1 (opposite l23=√2): 90°
+    EXPECT_NEAR(fa.alpha2, PI / 4.0, 1e-12);   // angle at v2 (opposite l31=1):  45°
+    EXPECT_NEAR(fa.alpha3, PI / 4.0, 1e-12);   // angle at v3 (opposite l12=1):  45°
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Angle sum = π for any valid Euclidean triangle
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(EuclideanFunctional, AngleSumEqualsPi)
+{
+    // Scalene triangle with log-lengths (0, 0.5, -0.3).
+    auto fa = euclidean_angles(0.0, 0.5, -0.3);
+
+    ASSERT_TRUE(fa.valid);
+    constexpr double PI = 3.14159265358979323846;
+    EXPECT_NEAR(fa.alpha1 + fa.alpha2 + fa.alpha3, PI, 1e-12);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Degenerate triangle → valid = false
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(EuclideanFunctional, DegenerateTriangleReturnsFalse)
+{
+    // l12 = l23 = 1, l31 = 3 → violates triangle inequality.
+    auto fa = euclidean_angles_from_lengths(1.0, 1.0, 3.0);
+    EXPECT_FALSE(fa.valid);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Gradient check: default right-isosceles triangle, vertex DOFs only
+//
+// Mirrors Java testGradient…SingleTriangle.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(EuclideanFunctional, GradientCheck_TriangleVertex)
+{
+    auto mesh = make_triangle();   // (0,0)–(1,0)–(0,1)
+    auto maps = setup_euclidean_maps(mesh);
+    compute_euclidean_lambda0_from_mesh(mesh, maps);
+    int n = assign_euclidean_vertex_dof_indices(mesh, maps);
+
+    // Small uniform conformal perturbation.
+    std::vector<double> x(static_cast<std::size_t>(n), -0.1);
+
+    EXPECT_TRUE(gradient_check_euclidean(mesh, x, maps))
+        << "Gradient check failed on right-isosceles triangle (vertex DOFs)";
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Gradient check: quad strip (2 triangles, 1 interior edge), vertex DOFs only
+//
+// Mirrors Java testGradient…QuadStrip / testGradientInExtendedDomain.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(EuclideanFunctional, GradientCheck_QuadStripVertex)
+{
+    auto mesh = make_quad_strip();
+    auto maps = setup_euclidean_maps(mesh);
+    compute_euclidean_lambda0_from_mesh(mesh, maps);
+    int n = assign_euclidean_vertex_dof_indices(mesh, maps);
+
+    std::vector<double> x(static_cast<std::size_t>(n), -0.2);
+
+    EXPECT_TRUE(gradient_check_euclidean(mesh, x, maps))
+        << "Gradient check failed on quad strip (vertex DOFs)";
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Gradient check: regular tetrahedron, vertex DOFs only
+//
+// Closed surface (4 faces, 4 vertices, 6 interior edges).
+// Exercises per-vertex angle-sum accumulation on multiple faces.
+// Mirrors Java testGradient…Tetrahedron / testGradientWithHyperIdeal…
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(EuclideanFunctional, GradientCheck_TetrahedronVertex)
+{
+    auto mesh = make_tetrahedron();
+    auto maps = setup_euclidean_maps(mesh);
+    compute_euclidean_lambda0_from_mesh(mesh, maps);
+    int n = assign_euclidean_vertex_dof_indices(mesh, maps);
+
+    std::vector<double> x(static_cast<std::size_t>(n), -0.15);
+
+    EXPECT_TRUE(gradient_check_euclidean(mesh, x, maps))
+        << "Gradient check failed on regular tetrahedron (vertex DOFs)";
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Gradient check: tetrahedron, all DOFs (vertex + edge)
+//
+// Exercises the edge-gradient branch G_e = α_opp⁺ + α_opp⁻ − π.
+// Mirrors Java testGradientWithHyperellipticCurve.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(EuclideanFunctional, GradientCheck_TetrahedronAllDofs)
+{
+    auto mesh = make_tetrahedron();
+    auto maps = setup_euclidean_maps(mesh);
+    compute_euclidean_lambda0_from_mesh(mesh, maps);
+    int n = assign_euclidean_all_dof_indices(mesh, maps);
+
+    // 4 vertex DOFs + 6 edge DOFs = 10 total.
+    std::vector<double> x(static_cast<std::size_t>(n), 0.0);
+    // Set vertex DOFs slightly negative to keep triangles non-degenerate.
+    for (int i = 0; i < 4; ++i)
+        x[static_cast<std::size_t>(i)] = -0.15;
+
+    EXPECT_TRUE(gradient_check_euclidean(mesh, x, maps))
+        << "Gradient check failed on regular tetrahedron (all DOFs)";
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Angles are finite at a known interior point
+//
+// Mirrors Java testFunctionalAtNaNValue: stress-test the angle formula with
+// large negative conformal factors (compressed triangle) to ensure no NaN/Inf.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(EuclideanFunctional, AnglesFiniteAtKnownPoint)
+{
+    auto mesh = make_tetrahedron();
+    auto maps = setup_euclidean_maps(mesh);
+    compute_euclidean_lambda0_from_mesh(mesh, maps);
+    int n = assign_euclidean_vertex_dof_indices(mesh, maps);
+
+    // Very compressed: u_i = -3 (all sides shrunk by exp(-3) ≈ 0.05).
+    // Triangle stays well-formed (equilateral shrinks uniformly).
+    std::vector<double> x(static_cast<std::size_t>(n), -3.0);
+    auto G = euclidean_gradient(mesh, x, maps);
+
+    for (std::size_t i = 0; i < G.size(); ++i) {
+        EXPECT_FALSE(std::isnan(G[i])) << "Gradient component " << i << " is NaN";
+        EXPECT_FALSE(std::isinf(G[i])) << "Gradient component " << i << " is Inf";
+    }
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Gradient check: fan of 5 flat triangles, vertex DOFs only
+//
+// High-valence central vertex: exercises per-vertex angle accumulation
+// across 5 incident faces.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(EuclideanFunctional, GradientCheck_Fan5Vertex)
+{
+    auto mesh = make_fan(5);
+    auto maps = setup_euclidean_maps(mesh);
+    compute_euclidean_lambda0_from_mesh(mesh, maps);
+    int n = assign_euclidean_vertex_dof_indices(mesh, maps);
+
+    std::vector<double> x(static_cast<std::size_t>(n), -0.05);
+
+    EXPECT_TRUE(gradient_check_euclidean(mesh, x, maps))
+        << "Gradient check failed on flat fan-5 mesh";
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Gradient check: mixed pinned/variable vertices
+//
+// Pins the first vertex (u_v0 = 0 fixed), lets the rest be variable.
+// Verifies that the gradient accumulator skips pinned vertices correctly.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(EuclideanFunctional, GradientCheck_MixedPinnedVertices)
+{
+    auto mesh = make_quad_strip();
+    auto maps = setup_euclidean_maps(mesh);
+    compute_euclidean_lambda0_from_mesh(mesh, maps);
+
+    // Manually pin v0; assign v1, v2, v3 as DOFs 0, 1, 2.
+    auto vit = mesh.vertices().begin();
+    Vertex_index v0 = *vit++;
+    Vertex_index v1 = *vit++;
+    Vertex_index v2 = *vit++;
+    Vertex_index v3 = *vit;
+
+    maps.v_idx[v0] = -1;   // pinned
+    maps.v_idx[v1] = 0;
+    maps.v_idx[v2] = 1;
+    maps.v_idx[v3] = 2;
+
+    std::vector<double> x = {-0.1, -0.3, -0.2};
+
+    EXPECT_TRUE(gradient_check_euclidean(mesh, x, maps))
+        << "Gradient check failed for mixed pinned/variable vertices";
+}

code/tests/cgal/test_euclidean_hessian.cpp

@@ -0,0 +1,216 @@
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
+// test_euclidean_hessian.cpp
+//
+// Phase 3f — Euclidean cotangent-Laplace Hessian.
+//
+// The Hessian of the Euclidean discrete conformal energy is the well-known
+// cotangent-Laplace operator (Pinkall–Polthier 1993, Springborn 2008).
+//
+// Tests:
+//   1. Cotangent weights are analytically correct for simple triangles.
+//   2. Hessian is symmetric.
+//   3. Hessian has the null-space property H·1 = 0 (uniform-shift mode).
+//   4. Hessian is positive semi-definite (all eigenvalues ≥ 0).
+//   5. Finite-difference check H[i,j] ≈ (G_i(x+ε·eⱼ)−G_i(x−ε·eⱼ))/(2ε).
+//
+// All tests use meshes and maps built with Phase-3d infrastructure.
+
+#include "conformal_mesh.hpp"
+#include "mesh_builder.hpp"
+#include "euclidean_hessian.hpp"
+#include <gtest/gtest.h>
+#include <Eigen/Dense>   // for dense conversion and eigenvalue solver
+#include <cmath>
+#include <vector>
+
+using namespace conformallab;
+
+// ════════════════════════════════════════════════════════════════════════════
+// Cotangent weight: equilateral triangle → all cots = 1/√3 = cot(60°)
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(EuclideanHessian, CotWeights_EquilateralTriangle)
+{
+    // Equilateral triangle with l = 1 (all log-lengths = 0).
+    auto cw = euclidean_cot_weights(1.0, 1.0, 1.0);
+
+    ASSERT_TRUE(cw.valid);
+    const double expected = 1.0 / std::sqrt(3.0);   // cot(60°)
+    EXPECT_NEAR(cw.cot1, expected, 1e-12);
+    EXPECT_NEAR(cw.cot2, expected, 1e-12);
+    EXPECT_NEAR(cw.cot3, expected, 1e-12);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Cotangent weight: right-isosceles triangle (legs 1, hypotenuse √2)
+//
+//   v1=(0,0): right angle → cot(90°) = 0
+//   v2=(1,0), v3=(0,1): 45° angles → cot(45°) = 1
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(EuclideanHessian, CotWeights_RightIsoscelesTriangle)
+{
+    // l12=1, l23=√2, l31=1
+    auto cw = euclidean_cot_weights(1.0, std::sqrt(2.0), 1.0);
+
+    ASSERT_TRUE(cw.valid);
+    EXPECT_NEAR(cw.cot1, 0.0, 1e-12);   // right angle at v1
+    EXPECT_NEAR(cw.cot2, 1.0, 1e-12);   // 45° at v2
+    EXPECT_NEAR(cw.cot3, 1.0, 1e-12);   // 45° at v3
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Hessian is symmetric: H[i,j] == H[j,i]
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(EuclideanHessian, HessianIsSymmetric)
+{
+    auto mesh = make_quad_strip();
+    auto maps = setup_euclidean_maps(mesh);
+    compute_euclidean_lambda0_from_mesh(mesh, maps);
+    int n = assign_euclidean_vertex_dof_indices(mesh, maps);
+
+    std::vector<double> x(static_cast<std::size_t>(n), -0.1);
+    auto H = euclidean_hessian(mesh, x, maps);
+
+    Eigen::MatrixXd Hd = Eigen::MatrixXd(H);
+    EXPECT_NEAR((Hd - Hd.transpose()).norm(), 0.0, 1e-12)
+        << "Hessian must be symmetric";
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Null-space property: H·1 = 0 for a closed surface (regular tetrahedron)
+//
+// The cotangent Laplacian on a closed mesh has the constant vector in its
+// null space (each row sums to zero).
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(EuclideanHessian, NullSpaceIsConstantVector_ClosedMesh)
+{
+    auto mesh = make_tetrahedron();
+    auto maps = setup_euclidean_maps(mesh);
+    compute_euclidean_lambda0_from_mesh(mesh, maps);
+    int n = assign_euclidean_vertex_dof_indices(mesh, maps);
+
+    std::vector<double> x(static_cast<std::size_t>(n), 0.0);
+    auto H = euclidean_hessian(mesh, x, maps);
+
+    // 1-vector
+    Eigen::VectorXd ones = Eigen::VectorXd::Ones(n);
+    Eigen::VectorXd Hones = H * ones;
+
+    EXPECT_NEAR(Hones.norm(), 0.0, 1e-10)
+        << "H·1 must be zero on a closed mesh (cotangent Laplacian null-space)";
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Hessian is positive semi-definite: all eigenvalues ≥ 0
+//
+// Checked on a small mesh (regular tetrahedron, 4 vertices) using dense
+// self-adjoint eigenvalue decomposition (only feasible for small n).
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(EuclideanHessian, HessianIsPositiveSemiDefinite)
+{
+    auto mesh = make_tetrahedron();
+    auto maps = setup_euclidean_maps(mesh);
+    compute_euclidean_lambda0_from_mesh(mesh, maps);
+    int n = assign_euclidean_vertex_dof_indices(mesh, maps);
+
+    std::vector<double> x(static_cast<std::size_t>(n), 0.0);
+    auto H = euclidean_hessian(mesh, x, maps);
+
+    Eigen::MatrixXd Hd = Eigen::MatrixXd(H);
+    Eigen::SelfAdjointEigenSolver<Eigen::MatrixXd> es(Hd);
+    double min_ev = es.eigenvalues().minCoeff();
+
+    EXPECT_GE(min_ev, -1e-10)
+        << "All eigenvalues of the cotangent Laplacian must be ≥ 0; "
+           "smallest = " << min_ev;
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Finite-difference Hessian check: single right-isosceles triangle
+//
+// H[i,j] ≈ (G_i(x+ε·eⱼ) − G_i(x−ε·eⱼ)) / (2ε)
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(EuclideanHessian, FDCheck_Triangle)
+{
+    auto mesh = make_triangle();
+    auto maps = setup_euclidean_maps(mesh);
+    compute_euclidean_lambda0_from_mesh(mesh, maps);
+    int n = assign_euclidean_vertex_dof_indices(mesh, maps);
+
+    std::vector<double> x(static_cast<std::size_t>(n), -0.1);
+
+    EXPECT_TRUE(hessian_check_euclidean(mesh, x, maps))
+        << "FD Hessian check failed on right-isosceles triangle";
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Finite-difference Hessian check: quad strip (2 triangles, 1 interior edge)
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(EuclideanHessian, FDCheck_QuadStrip)
+{
+    auto mesh = make_quad_strip();
+    auto maps = setup_euclidean_maps(mesh);
+    compute_euclidean_lambda0_from_mesh(mesh, maps);
+    int n = assign_euclidean_vertex_dof_indices(mesh, maps);
+
+    std::vector<double> x(static_cast<std::size_t>(n), -0.1);
+
+    EXPECT_TRUE(hessian_check_euclidean(mesh, x, maps))
+        << "FD Hessian check failed on quad strip";
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Finite-difference Hessian check: regular tetrahedron (closed, 4 faces)
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(EuclideanHessian, FDCheck_Tetrahedron)
+{
+    auto mesh = make_tetrahedron();
+    auto maps = setup_euclidean_maps(mesh);
+    compute_euclidean_lambda0_from_mesh(mesh, maps);
+    int n = assign_euclidean_vertex_dof_indices(mesh, maps);
+
+    std::vector<double> x(static_cast<std::size_t>(n), -0.15);
+
+    EXPECT_TRUE(hessian_check_euclidean(mesh, x, maps))
+        << "FD Hessian check failed on regular tetrahedron";
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Finite-difference Hessian check: with mixed pinned/variable vertices
+//
+// One vertex pinned: the corresponding row/column must be absent from H
+// while the diagonal of neighbouring variable vertices still gets the full
+// cotangent contribution.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(EuclideanHessian, FDCheck_MixedPinnedVertices)
+{
+    auto mesh = make_quad_strip();
+    auto maps = setup_euclidean_maps(mesh);
+    compute_euclidean_lambda0_from_mesh(mesh, maps);
+
+    auto vit = mesh.vertices().begin();
+    Vertex_index v0 = *vit++;
+    Vertex_index v1 = *vit++;
+    Vertex_index v2 = *vit++;
+    Vertex_index v3 = *vit;
+
+    maps.v_idx[v0] = -1;   // pinned
+    maps.v_idx[v1] = 0;
+    maps.v_idx[v2] = 1;
+    maps.v_idx[v3] = 2;
+
+    std::vector<double> x = {-0.1, -0.2, -0.15};
+
+    EXPECT_TRUE(hessian_check_euclidean(mesh, x, maps))
+        << "FD Hessian check failed for mixed pinned/variable vertices";
+}

code/tests/cgal/test_geometry_utils.cpp

@@ -0,0 +1,513 @@
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
+// test_geometry_utils.cpp
+//
+// Port of the Java ConformalLab geometry utility tests.
+//
+// Java source                           Java test method                         Status
+// ─────────────────────────────────────────────────────────────────────────────────────
+// CuttinUtilityTest.java                testIsInConvexTextureFace_False          PORTED
+// CuttinUtilityTest.java                testIsInConvexTextureFace_True           PORTED
+// UnwrapUtilityTest.java                testGetAngleReturnsPI                    PORTED
+// ConvergenceUtilityTests.java          testGetTextureCircumRadius               PORTED
+// ConvergenceUtilityTests.java          testGetTextureTriangleArea               PORTED
+// ConvergenceUtilityTests.java          testScaleInvariantCircumCircleRadius     PORTED
+// HomologyTest.java                     testHomology                             PORTED
+// EuclideanLayoutTest.java              testDoLayout                             PORTED
+// EuclideanCyclicConvergenceTest.java   testEuclideanConvergence                 PORTED
+// SphericalConvergenceTest.java         testSphericalConvergence                 PORTED
+//
+// ─── Geometric background ────────────────────────────────────────────────────────────
+//
+//  Tests 1–2  Point-in-convex-triangle (2D UV space, barycentric sign method)
+//             Java: CuttingUtility.isInConvexTextureFace(pp, face, adapters)
+//             Note: Java test 2 has a 5-element T-array with w=0 (point at
+//             infinity), which is a typo in the original. Equivalent, well-formed
+//             coordinates are used here instead.
+//
+//  Test 3     Corner angle for collinear vertices via the law of cosines.
+//             Java: UnwrapUtility.getAngle(edge, adapters) — returns the angle at
+//             the target vertex. For v0=(-1,0,0), v1=(0,0,0), v2=(1,0,0) the
+//             angle at v1 is exactly π (degenerate triangle inequality).
+//
+//  Tests 4–5  2D circumradius and triangle area.
+//             Java: ConvergenceUtility.getTextureCircumCircleRadius(face)
+//                   ConvergenceUtility.getTextureTriangleArea(face)
+//             Formulas: Area = |det([B-A, C-A])| / 2
+//                       R    = (a·b·c) / (4·Area)
+//
+//  Test 6     Scale-invariant circumradius over a mesh.
+//             Java: ConvergenceUtility.getMaxMeanSumScaleInvariantCircumRadius(hds)
+//             Returns [max, mean, sum] of  R_f / sqrt(total_texture_area).
+//             Invariant under uniform scaling of texture coordinates (tested with
+//             homogeneous weight w: position = (T[0]/w, T[1]/w)).
+//
+//  Test 7     Genus-2 homology generators.
+//             Java: HomologyTest.testHomology (brezel2.obj)
+//             Expected: getGeneratorPaths(root).size() == 4  (2g = 4 for g = 2)
+//             C++: compute_cut_graph(mesh).cut_edge_indices.size() == 4
+//             Mesh: code/data/obj/brezel2.obj (V=2622, F=5248, χ=−2, g=2)
+//             Path set at compile time via CONFORMALLAB_DATA_DIR (CMakeLists.txt).
+//
+//  Tests 8–9  Layout edge-length preservation (tetraflat.obj).
+//             Java: EuclideanLayoutTest.testDoLayout
+//             After layout with u=0, UV edge lengths must equal 3D edge lengths (±1e-10).
+//
+//  Test 10    Euclidean Newton on cathead.obj — convergence + angle deficit.
+//             Java: EuclideanLayoutTest.testLayout02 (130-value array for cathead.heml)
+//             C++: Newton from u=0, checks convergence + Σα_v ≈ 2π for all interior nodes.
+//
+//  Test 11    Spherical Newton on octahedron — convergence + angle deficit.
+//             Java: SphericalConvergenceTest.testSphericalConvergence (octahedron, randomly
+//             perturbed radii, seed=1). C++: constructed regular octahedron, checks
+//             convergence and that Σα_v ≈ 2π (target for sphere after prepareInvariantData).
+//
+// ─────────────────────────────────────────────────────────────────────────────────────
+
+#include "cut_graph.hpp"
+#include "gauss_bonnet.hpp"
+#include "conformal_mesh.hpp"
+#include "mesh_builder.hpp"
+#include "mesh_io.hpp"
+#include "euclidean_functional.hpp"
+#include "spherical_functional.hpp"
+#include "newton_solver.hpp"
+#include "layout.hpp"
+#include <gtest/gtest.h>
+#include <Eigen/Dense>
+#include <array>
+#include <cmath>
+#include <string>
+#include <vector>
+
+using namespace conformallab;
+
+// ─────────────────────────────────────────────────────────────────────────────
+// Local geometry helper functions
+// (ported from Java CuttingUtility / ConvergenceUtility)
+// ─────────────────────────────────────────────────────────────────────────────
+
+/// Point-in-triangle test (2D, barycentric sign method).
+/// Returns true if p lies strictly inside or on the boundary of v0-v1-v2.
+/// Java: CuttingUtility.isInConvexTextureFace
+static bool point_in_triangle_2d(
+    Eigen::Vector2d p,
+    Eigen::Vector2d v0, Eigen::Vector2d v1, Eigen::Vector2d v2)
+{
+    auto cross2d = [](Eigen::Vector2d a, Eigen::Vector2d b) -> double {
+        return a.x() * b.y() - a.y() * b.x();
+    };
+    double d0 = cross2d(v1 - v0, p - v0);
+    double d1 = cross2d(v2 - v1, p - v1);
+    double d2 = cross2d(v0 - v2, p - v2);
+    bool has_neg = (d0 < 0.0) || (d1 < 0.0) || (d2 < 0.0);
+    bool has_pos = (d0 > 0.0) || (d1 > 0.0) || (d2 > 0.0);
+    return !(has_neg && has_pos);
+}
+
+/// 2D triangle area (half cross product).
+/// Java: ConvergenceUtility.getTextureTriangleArea
+static double triangle_area_2d(
+    Eigen::Vector2d A, Eigen::Vector2d B, Eigen::Vector2d C)
+{
+    return std::abs((B - A).x() * (C - A).y()
+                  - (B - A).y() * (C - A).x()) * 0.5;
+}
+
+/// 2D circumradius: R = (a·b·c) / (4·Area).
+/// Java: ConvergenceUtility.getTextureCircumCircleRadius
+static double circumradius_2d(
+    Eigen::Vector2d A, Eigen::Vector2d B, Eigen::Vector2d C)
+{
+    double a    = (B - C).norm();
+    double b    = (A - C).norm();
+    double c    = (A - B).norm();
+    double area = triangle_area_2d(A, B, C);
+    if (area < 1e-14) return 0.0;
+    return (a * b * c) / (4.0 * area);
+}
+
+/// Scale-invariant circumradius for a mesh:
+///   scale_R_f = R_f / sqrt(total_area)
+/// Returns {max, mean, sum} over all faces.
+/// Java: ConvergenceUtility.getMaxMeanSumScaleInvariantCircumRadius
+///
+/// Homogeneous coordinates: position = (x/w, y/w).
+static std::array<double, 3> scale_invariant_circumradius_stats(
+    const std::vector<Eigen::Vector2d>& verts,
+    const std::vector<std::array<int, 3>>& faces)
+{
+    // Total area
+    double total_area = 0.0;
+    for (auto& f : faces)
+        total_area += triangle_area_2d(verts[f[0]], verts[f[1]], verts[f[2]]);
+    if (total_area < 1e-14) return {0, 0, 0};
+
+    double sqrt_total = std::sqrt(total_area);
+    double max_r = 0.0, sum_r = 0.0;
+    for (auto& f : faces) {
+        double R = circumradius_2d(verts[f[0]], verts[f[1]], verts[f[2]]);
+        double sr = R / sqrt_total;
+        max_r  = std::max(max_r, sr);
+        sum_r += sr;
+    }
+    double mean_r = sum_r / static_cast<double>(faces.size());
+    return {max_r, mean_r, sum_r};
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Tests 1–2 — CuttingUtility: point-in-convex-triangle (2D UV space)
+//             Java: CuttinUtilityTest.testIsInConvexTextureFace_False / _True
+// ════════════════════════════════════════════════════════════════════════════
+
+// Test 1: point lies far outside — exact Java coordinates
+TEST(CuttingUtility, IsInConvexTextureFace_False)
+{
+    // Tiny triangle around (0.7488, 0.0629) — Java test coordinates (T[3]=1, w=1)
+    Eigen::Vector2d v0(0.7488102998904661,  0.06293998610761144);
+    Eigen::Vector2d v1(0.7487811940754379,  0.06289451051246124);
+    Eigen::Vector2d v2(0.7487254625255592,  0.06291429499873116);
+    // Test point far away at (0.447, 0.000228)
+    Eigen::Vector2d pp(0.44661534423161037, 2.2808373704822393e-4);
+
+    EXPECT_FALSE(point_in_triangle_2d(pp, v0, v1, v2));
+}
+
+// Test 2: point lies inside
+// Note: the original Java array p2 has 5 elements with w=0 (typo in the
+// Java original). Equivalent, well-formed coordinates are used here
+// that represent the same geometric scenario.
+TEST(CuttingUtility, IsInConvexTextureFace_True)
+{
+    // Triangle: (0,0) — (1e-8, 0) — (0, 1e-8)
+    Eigen::Vector2d v0(0.0,   0.0);
+    Eigen::Vector2d v1(1e-8,  0.0);
+    Eigen::Vector2d v2(0.0,   1e-8);
+    // Centroid of the triangle — always lies inside
+    Eigen::Vector2d pp(1e-8 / 3.0, 1e-8 / 3.0);
+
+    EXPECT_TRUE(point_in_triangle_2d(pp, v0, v1, v2));
+}
+
+// Additional: simple unit triangle for clarity
+TEST(CuttingUtility, IsInConvexTextureFace_UnitTriangle_InAndOut)
+{
+    Eigen::Vector2d v0(0.0, 0.0), v1(1.0, 0.0), v2(0.0, 1.0);
+    EXPECT_TRUE( point_in_triangle_2d(Eigen::Vector2d(0.25, 0.25), v0, v1, v2));
+    EXPECT_FALSE(point_in_triangle_2d(Eigen::Vector2d(2.0,  2.0),  v0, v1, v2));
+    EXPECT_FALSE(point_in_triangle_2d(Eigen::Vector2d(0.6,  0.6),  v0, v1, v2));  // beyond hypotenuse
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Test 3 — UnwrapUtility: corner angle = π for collinear vertices
+//          Java: UnwrapUtilityTest.testGetAngleReturnsPI
+// ════════════════════════════════════════════════════════════════════════════
+
+// Java: v0=(-1,0,0), v1=(0,0,0), v2=(1,0,0) collinear.
+// Edge e from v2 to v1. getAngle(e) = angle at v1 = π.
+//
+// C++: law of cosines with edge lengths a=|v0-v1|=1, b=|v1-v2|=1, c=|v0-v2|=2.
+//   cos(γ_v1) = (a² + b² − c²) / (2ab) = (1 + 1 − 4) / 2 = −1  →  γ = π
+TEST(UnwrapUtility, GetAngle_CollinearVertices_ReturnsPI)
+{
+    const double a = 1.0;   // |v0 − v1|
+    const double b = 1.0;   // |v1 − v2|
+    const double c = 2.0;   // |v0 − v2|   (= a + b, degenerate)
+    double cos_angle = (a*a + b*b - c*c) / (2.0 * a * b);
+    cos_angle = std::max(-1.0, std::min(1.0, cos_angle));   // numeric clamp
+    double angle = std::acos(cos_angle);
+    EXPECT_NEAR(M_PI, angle, 1e-15);
+}
+
+// Counter-check: equilateral triangle → angle = π/3
+TEST(UnwrapUtility, GetAngle_EquilateralTriangle_ReturnsPiOver3)
+{
+    const double s = 1.0;
+    double cos_angle = (s*s + s*s - s*s) / (2.0 * s * s);   // = 0.5
+    double angle = std::acos(cos_angle);
+    EXPECT_NEAR(M_PI / 3.0, angle, 1e-15);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Test 4 — ConvergenceUtility: 2D circumradius
+//          Java: ConvergenceUtilityTests.testGetTextureCircumRadius
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(ConvergenceUtility, TextureCircumRadius_RightTriangle)
+{
+    // A=(0,0), B=(1,0), C=(0,1): right isosceles triangle
+    // Sides: 1, 1, √2.  R = √2 / (4 · 0.5) = √2/2
+    Eigen::Vector2d A(0.0, 0.0), B(1.0, 0.0), C(0.0, 1.0);
+    EXPECT_NEAR(std::sqrt(2.0) / 2.0, circumradius_2d(A, B, C), 1e-10);
+}
+
+TEST(ConvergenceUtility, TextureCircumRadius_SmallerTriangle)
+{
+    // A=(0,0), B=(0.5,0.5), C=(0,1): Java variant with B.T={0.5,0.5,0,1}
+    // Sides: √0.5, √0.5, 1.  Area = 0.25.  R = (√0.5·√0.5·1)/(4·0.25) = 0.5
+    Eigen::Vector2d A(0.0, 0.0), B(0.5, 0.5), C(0.0, 1.0);
+    EXPECT_NEAR(0.5, circumradius_2d(A, B, C), 1e-10);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Test 5 — ConvergenceUtility: 2D triangle area
+//          Java: ConvergenceUtilityTests.testGetTextureTriangleArea
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(ConvergenceUtility, TextureTriangleArea_RightTriangle)
+{
+    // A=(0,0), B=(1,0), C=(0,1) → area = 0.5
+    Eigen::Vector2d A(0.0, 0.0), B(1.0, 0.0), C(0.0, 1.0);
+    EXPECT_NEAR(0.5, triangle_area_2d(A, B, C), 1e-10);
+}
+
+TEST(ConvergenceUtility, TextureTriangleArea_SmallerTriangle)
+{
+    // A=(0,0), B=(0.5,0.5), C=(0,1) → area = 0.25
+    Eigen::Vector2d A(0.0, 0.0), B(0.5, 0.5), C(0.0, 1.0);
+    EXPECT_NEAR(0.25, triangle_area_2d(A, B, C), 1e-10);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Test 6 — ConvergenceUtility: scale-invariant circumradius
+//          Java: ConvergenceUtilityTests.testScaleInvariantCircumCircleRadius
+//
+// Mesh: 4 vertices (v1..v4), 2 faces (f1: v1-v2-v3, f2: v1-v3-v4).
+// Scale-invariant quantity: R_f / sqrt(total_area) — invariant under
+// uniform scaling (homogeneous weight w: pos = (x/w, y/w)).
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(ConvergenceUtility, ScaleInvariantCircumRadius_BaseScale)
+{
+    // Positions at w=1 (T[3]=1): v1=(0,0), v2=(1,0), v3=(0,1), v4=(-1,0)
+    std::vector<Eigen::Vector2d> verts = {
+        {0.0,  0.0},   // v1
+        {1.0,  0.0},   // v2
+        {0.0,  1.0},   // v3
+        {-1.0, 0.0},   // v4
+    };
+    // f1: v1-v2-v3,  f2: v1-v3-v4
+    std::vector<std::array<int, 3>> faces = { {0, 1, 2}, {0, 2, 3} };
+
+    // Per-face check (Java testGetTextureTriangleArea requirement)
+    EXPECT_NEAR(0.5, triangle_area_2d(verts[0], verts[1], verts[2]), 1e-10);
+    EXPECT_NEAR(0.5, triangle_area_2d(verts[0], verts[2], verts[3]), 1e-10);
+
+    auto [max_r, mean_r, sum_r] = scale_invariant_circumradius_stats(verts, faces);
+
+    // Expected: sin(π/4) = √2/2 for max and mean (both triangles identical)
+    EXPECT_NEAR(std::sin(M_PI / 4.0), max_r,  1e-10);
+    EXPECT_NEAR(std::sin(M_PI / 4.0), mean_r, 1e-10);
+    EXPECT_NEAR(2.0 * std::sin(M_PI / 4.0), sum_r, 1e-10);
+}
+
+TEST(ConvergenceUtility, ScaleInvariantCircumRadius_HalvedByW2_SameResult)
+{
+    // Scaling by w=2: all positions halved (homogeneous coordinates)
+    // pos_scaled = (T[0]/2, T[1]/2)
+    std::vector<Eigen::Vector2d> verts = {
+        {0.0,   0.0},   // v1/2
+        {0.5,   0.0},   // v2/2
+        {0.0,   0.5},   // v3/2
+        {-0.5,  0.0},   // v4/2
+    };
+    std::vector<std::array<int, 3>> faces = { {0, 1, 2}, {0, 2, 3} };
+
+    // Areas are one quarter of the original (lengths halved → Area / 4)
+    EXPECT_NEAR(0.125, triangle_area_2d(verts[0], verts[1], verts[2]), 1e-10);
+    EXPECT_NEAR(0.125, triangle_area_2d(verts[0], verts[2], verts[3]), 1e-10);
+
+    auto [max_r, mean_r, sum_r] = scale_invariant_circumradius_stats(verts, faces);
+
+    // Scale-invariant quantity must be identical to the w=1 case
+    EXPECT_NEAR(std::sin(M_PI / 4.0), max_r,  1e-10);
+    EXPECT_NEAR(std::sin(M_PI / 4.0), mean_r, 1e-10);
+    EXPECT_NEAR(2.0 * std::sin(M_PI / 4.0), sum_r, 1e-10);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Test 7 — HomologyTest: genus-2 homology generators
+//          Java: HomologyTest.testHomology
+//
+// Java test:
+//   CoHDS hds = TestUtility.readOBJ("brezel2.obj");  // genus-2 pretzel surface
+//   List<Set<CoEdge>> paths = getGeneratorPaths(hds.getVertex(0), weightAdapter);
+//   Assert.assertEquals(4, paths.size());             // 2g = 4 for g = 2
+//
+// C++ equivalent:
+//   ConformalMesh mesh = load_mesh("code/data/obj/brezel2.obj");
+//   CutGraph cg = compute_cut_graph(mesh);
+//   EXPECT_EQ(4u, cg.cut_edge_indices.size());        // 2g = 4
+//   EXPECT_EQ(2,  cg.genus);
+//
+// Mesh: V=2622, F=5248, E=7872, χ=−2, genus=2.
+// Path via CONFORMALLAB_DATA_DIR (CMakeLists.txt: ${CMAKE_SOURCE_DIR}/data).
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(HomologyGenerators, Genus2_FourCutEdges)
+{
+    const std::string path = std::string(CONFORMALLAB_DATA_DIR) + "/obj/brezel2.obj";
+    ConformalMesh mesh;
+    ASSERT_NO_THROW(mesh = load_mesh(path)) << "brezel2.obj not found at: " << path;
+
+    // Topology check: genus-2 surface has χ = -2.
+    EXPECT_EQ(-2, euler_characteristic(mesh));
+
+    // Tree-cotree algorithm must produce exactly 2g = 4 cut edges.
+    CutGraph cg = compute_cut_graph(mesh);
+    EXPECT_EQ(4u, cg.cut_edge_indices.size())
+        << "Genus-2 surface must have 2g = 4 cut edges (homology generators).";
+    EXPECT_EQ(2, cg.genus);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Tests 8–9 — EuclideanLayoutTest: edge-length preservation on tetraflat.obj
+//             Java: EuclideanLayoutTest.testDoLayout
+//
+// Java test:
+//   Vector u = new SparseVector(n);   // u = 0 (no conformal factor)
+//   EuclideanLayout.doLayout(hds, fun, u);
+//   for (CoEdge e : hds.getEdges())
+//       assertEquals(Pn.distanceBetween(s.P, t.P), Pn.distanceBetween(s.T, t.T), 1E-11);
+//
+// Meaning: with u=0 the conformal factor is 0, so ℓ̃ = ℓ (no deformation).
+//          The layout must reproduce the original 3D edge lengths exactly.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(EuclideanLayout, DoLayout_TetraFlat_EdgeLengthsPreserved)
+{
+    const std::string path = std::string(CONFORMALLAB_DATA_DIR) + "/obj/tetraflat.obj";
+    ConformalMesh mesh;
+    ASSERT_NO_THROW(mesh = load_mesh(path)) << "tetraflat.obj not found at: " << path;
+
+    auto maps = setup_euclidean_maps(mesh);
+    compute_euclidean_lambda0_from_mesh(mesh, maps);
+
+    // u = 0: no conformal deformation — layout must preserve 3D edge lengths exactly.
+    // tetraflat.obj is an open mesh; pin boundary vertices, sequential DOFs interior.
+    int idx = 0;
+    for (auto v : mesh.vertices())
+        maps.v_idx[v] = mesh.is_border(v) ? -1 : idx++;
+    const int n = idx;
+    std::vector<double> x(static_cast<std::size_t>(n), 0.0);
+
+    Layout2D layout = euclidean_layout(mesh, x, maps);
+
+    // For every edge: UV length must equal 3D length within 1e-10.
+    for (auto e : mesh.edges()) {
+        auto h  = mesh.halfedge(e);
+        auto vs = mesh.source(h);
+        auto vt = mesh.target(h);
+
+        auto ps = mesh.point(vs);
+        auto pt = mesh.point(vt);
+        double l3d = std::sqrt(
+            (pt.x()-ps.x())*(pt.x()-ps.x()) +
+            (pt.y()-ps.y())*(pt.y()-ps.y()) +
+            (pt.z()-ps.z())*(pt.z()-ps.z()));
+
+        auto us = layout.uv[vs.idx()];
+        auto ut = layout.uv[vt.idx()];
+        double luv = (ut - us).norm();
+
+        EXPECT_NEAR(l3d, luv, 1e-10)
+            << "Edge " << e.idx() << ": 3D=" << l3d << " UV=" << luv;
+    }
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Test 10 — EuclideanCyclicConvergenceTest: Newton on cathead.obj
+//           Java: EuclideanLayoutTest.testLayout02 (130-value regression on cathead.heml)
+//                 EuclideanCyclicConvergenceTest.testEuclideanConvergence
+//
+// Java test:
+//   EuclideanLayout.doLayout(hdsCat, fun, uCat);
+//   for (CoVertex v : interior vertices)
+//       assertEquals(2*PI, calculateAngleSum(v), 1E-6);
+//   for (CoEdge e : positiveEdges)
+//       assertEquals(fun.getNewLength(e, u), tLength, 1E-6);
+//
+// C++ equivalent: Newton converges on cathead.obj; interior angle sums ≈ 2π.
+// The 130-value u-vector from the Java test is cathead-topology-specific and
+// depends on vertex ordering in the Java CoHDS — not portable directly.
+// Instead we verify the same mathematical invariant: convergence + angle sums.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(EuclideanLayout, CatHead_NewtonConverges_AngleSumsTwoPi)
+{
+    const std::string path = std::string(CONFORMALLAB_DATA_DIR) + "/obj/cathead.obj";
+    ConformalMesh mesh;
+    ASSERT_NO_THROW(mesh = load_mesh(path)) << "cathead.obj not found at: " << path;
+
+    auto maps = setup_euclidean_maps(mesh);
+    compute_euclidean_lambda0_from_mesh(mesh, maps);
+
+    // cathead.obj is an open mesh (boundary present).
+    // Pin boundary vertices (v_idx = -1), assign sequential DOFs to interior.
+    int idx = 0;
+    for (auto v : mesh.vertices())
+        maps.v_idx[v] = mesh.is_border(v) ? -1 : idx++;
+    const int n = idx;
+    ASSERT_GT(n, 0) << "No interior vertices found in cathead.obj";
+
+    enforce_gauss_bonnet(mesh, maps);
+
+    std::vector<double> x0(static_cast<std::size_t>(n), 0.0);
+
+    auto res = newton_euclidean(mesh, x0, maps, 1e-8, 200);
+    EXPECT_TRUE(res.converged)
+        << "Newton did not converge on cathead.obj (iterations=" << res.iterations
+        << ", |G|inf=" << res.grad_inf_norm << ")";
+    EXPECT_LT(res.grad_inf_norm, 1e-8);
+    EXPECT_LT(res.iterations, 200);
+
+    // After convergence: all interior vertex angle sums must equal θ_v (2π for flat).
+    // Matches Java: assertEquals(2*PI, calculateAngleSum(v), 1E-6) for interior v.
+    auto G_final = euclidean_gradient(mesh, res.x, maps);
+    for (std::size_t i = 0; i < G_final.size(); ++i)
+        EXPECT_NEAR(0.0, G_final[i], 1e-6)
+            << "Angle sum residual at DOF " << i << " = " << G_final[i];
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Test 11 — SphericalConvergenceTest: Newton on octahedron
+//           Java: SphericalConvergenceTest.testSphericalConvergence
+//
+// Java test:
+//   FunctionalTest.createOctahedron(hds, aSet);
+//   // randomly perturb vertex radii (seed=1)
+//   prepareInvariantDataHyperbolicAndSpherical(functional, hds, aSet, u);
+//   optimizer.minimize(u, opt);
+//   for (CoVertex v) assertEquals(2*PI, sum of angles at v, 1E-8);
+//
+// C++: regular octahedron (all vertices on S², no perturbation), spherical Newton,
+// checks convergence + residual gradients (≡ angle deficit = 0 after convergence).
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(SphericalLayout, SphericalTetrahedron_NewtonConverges_AngleSumsTwoPi)
+{
+    // Build a spherical tetrahedron (genus 0, 4 vertices, 4 faces).
+    // Java uses a randomly-perturbed octahedron; we use the canonical
+    // spherical tetrahedron from mesh_builder.hpp for reproducibility.
+    ConformalMesh mesh = make_spherical_tetrahedron();
+
+    auto maps = setup_spherical_maps(mesh);
+    compute_lambda0_from_mesh(mesh, maps);  // SphericalMaps version
+    int n = assign_vertex_dof_indices(mesh, maps);  // pins gauge_vertex, assigns DOFs
+    // Note: enforce_gauss_bonnet not needed — natural theta from mesh satisfies Σ(2π-Θ)>0.
+
+    std::vector<double> x0(static_cast<std::size_t>(n), 0.0);
+
+    auto res = newton_spherical(mesh, x0, maps, 1e-8, 200);
+    EXPECT_TRUE(res.converged)
+        << "Spherical Newton did not converge (iterations=" << res.iterations
+        << ", |G|inf=" << res.grad_inf_norm << ")";
+    EXPECT_LT(res.grad_inf_norm, 1e-8);
+
+    // Angle sum residual = 0 after convergence (≡ each interior vertex has Σα = θ_v).
+    auto G_final = spherical_gradient(mesh, res.x, maps);
+    for (std::size_t i = 0; i < G_final.size(); ++i)
+        EXPECT_NEAR(0.0, G_final[i], 1e-6)
+            << "Spherical angle sum residual at DOF " << i << " = " << G_final[i];
+}

code/tests/cgal/test_hyper_ideal_functional.cpp

@@ -0,0 +1,199 @@
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
+// test_hyper_ideal_functional.cpp
+//
+// Phase 3b — HyperIdealFunctional ported to ConformalMesh.
+//
+// Corresponds to de.varylab.discreteconformal.functional.HyperIdealFunctionalTest.
+//
+// The Java tests use CoHDS + HyperIdealGenerator to build complex meshes and
+// then verify correctness via a finite-difference gradient check (FunctionalTest).
+// Here we apply the same gradient-check strategy on simple hand-crafted meshes
+// (triangle, tetrahedron) to validate the port independently of the generator.
+//
+// Test map (Java → C++)
+// ──────────────────────
+//   testHessian (Ignored)         → GradientCheck_Hessian (SKIPPED, not implemented)
+//   testGradientWithHyperIdeal…   → GradientCheck_AllHyperIdealTriangle  (ported)
+//   testGradientInExtendedDomain  → GradientCheck_ExtendedDomain         (ported)
+//   testGradientWithHyperelliptic → GradientCheck_TetrahedronAllVariable  (ported)
+//   testFunctionalAtNaNValue      → EnergyFiniteAtTestPoint               (ported)
+
+#include "conformal_mesh.hpp"
+#include "mesh_builder.hpp"
+#include "hyper_ideal_functional.hpp"
+#include "hyper_ideal_hessian.hpp"
+#include <gtest/gtest.h>
+#include <Eigen/Dense>
+#include <cmath>
+#include <vector>
+
+using namespace conformallab;
+
+// ════════════════════════════════════════════════════════════════════════════
+// Helpers
+// ════════════════════════════════════════════════════════════════════════════
+
+// Build a mesh with all vertices and edges variable, and set reasonable
+// DOF values: b_i = b_val, a_e = a_val.
+static std::vector<double> make_x_all_variable(
+    ConformalMesh& mesh, HyperIdealMaps& maps,
+    double b_val, double a_val)
+{
+    int n = assign_all_dof_indices(mesh, maps);
+    std::vector<double> x(static_cast<std::size_t>(n));
+
+    // Vertices first, then edges (matching assign_all_dof_indices order)
+    for (auto v : mesh.vertices())
+        x[static_cast<std::size_t>(maps.v_idx[v])] = b_val;
+    for (auto e : mesh.edges())
+        x[static_cast<std::size_t>(maps.e_idx[e])] = a_val;
+
+    return x;
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// @Ignore in Java: no Hessian implemented
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(HyperIdealFunctional, HessianSymmetryCheck)
+{
+    // Hessian is now implemented (numerical FD).  Verify it is symmetric.
+    auto mesh = make_triangle();
+    auto maps = setup_hyper_ideal_maps(mesh);
+    int n = assign_all_dof_indices(mesh, maps);
+
+    std::vector<double> x(static_cast<std::size_t>(n), 0.5);
+    auto H = hyper_ideal_hessian_sym(mesh, x, maps);
+
+    Eigen::MatrixXd Hd(H);
+    EXPECT_NEAR((Hd - Hd.transpose()).norm(), 0.0, 1e-8)
+        << "HyperIdeal Hessian must be symmetric";
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Gradient check: single triangle, all vertices hyper-ideal
+//
+// Mirrors testGradientWithHyperIdealAndIdealPoints (simplified to single face).
+// DOFs: 3 vertex b-values + 3 edge a-values = 6 total.
+// Point: b_i = 1.0, a_e = 0.5.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(HyperIdealFunctional, GradientCheck_AllHyperIdealTriangle)
+{
+    auto mesh = make_triangle();
+    auto maps = setup_hyper_ideal_maps(mesh);
+    auto x    = make_x_all_variable(mesh, maps, /*b=*/1.0, /*a=*/0.5);
+
+    EXPECT_TRUE(gradient_check(mesh, x, maps))
+        << "Finite-difference gradient check failed on all-hyper-ideal triangle";
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Gradient check in the "extended domain"
+//
+// Mirrors testGradientInTheExtendedDomain: larger DOF values
+// (Java: x_i = 1.2 + |rnd|, so typically > 1.2).
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(HyperIdealFunctional, GradientCheck_ExtendedDomain)
+{
+    auto mesh = make_triangle();
+    auto maps = setup_hyper_ideal_maps(mesh);
+    auto x    = make_x_all_variable(mesh, maps, /*b=*/2.0, /*a=*/1.5);
+
+    EXPECT_TRUE(gradient_check(mesh, x, maps))
+        << "Finite-difference gradient check failed in extended domain";
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Gradient check: tetrahedron (4 faces), all vertices and edges variable
+//
+// Mirrors testGradientWithHyperellipticCurve — larger mesh, closed surface.
+// DOFs: 4 vertex b-values + 6 edge a-values = 10 total.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(HyperIdealFunctional, GradientCheck_TetrahedronAllVariable)
+{
+    auto mesh = make_tetrahedron();
+    auto maps = setup_hyper_ideal_maps(mesh);
+    auto x    = make_x_all_variable(mesh, maps, /*b=*/1.0, /*a=*/0.5);
+
+    EXPECT_TRUE(gradient_check(mesh, x, maps))
+        << "Finite-difference gradient check failed on all-variable tetrahedron";
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Energy is finite (not NaN / Inf)
+//
+// Mirrors testFunctionalAtNaNValue: evaluates at a test point and checks
+// the energy is a real number.  Uses a quad-strip to exercise the interior
+// edge path (adjacent faces sharing one non-border edge).
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(HyperIdealFunctional, EnergyFiniteAtTestPoint)
+{
+    auto mesh = make_quad_strip();
+    auto maps = setup_hyper_ideal_maps(mesh);
+    int  n    = assign_all_dof_indices(mesh, maps);
+
+    // Values taken from the Java testFunctionalAtNaNValue spirit:
+    // large-ish but positive DOF values that could expose degenerate paths.
+    std::vector<double> x(static_cast<std::size_t>(n), 1.5);
+
+    auto res = evaluate_hyper_ideal(mesh, x, maps, true, false);
+
+    EXPECT_FALSE(std::isnan(res.energy))  << "Energy must not be NaN";
+    EXPECT_FALSE(std::isinf(res.energy))  << "Energy must not be Inf";
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Gradient check: mixed vertices (some ideal = pinned, some hyper-ideal)
+//
+// One vertex is ideal (v_idx = -1, b = 0), the other two are hyper-ideal.
+// This exercises the lij / αij branches for the ideal-vertex case.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(HyperIdealFunctional, GradientCheck_MixedIdealHyperIdeal)
+{
+    auto mesh = make_triangle();
+    auto maps = setup_hyper_ideal_maps(mesh);
+
+    // Make v0 ideal (pinned), v1 and v2 hyper-ideal.
+    auto vit = mesh.vertices().begin();
+    Vertex_index v0 = *vit++;
+    Vertex_index v1 = *vit++;
+    Vertex_index v2 = *vit;
+
+    maps.v_idx[v0] = -1;   // ideal: b_0 = 0 (fixed)
+    maps.v_idx[v1] = 0;
+    maps.v_idx[v2] = 1;
+
+    // All edges variable: indices 2, 3, 4
+    int eidx = 2;
+    for (auto e : mesh.edges()) maps.e_idx[e] = eidx++;
+
+    // DOF vector: [b1, b2, a_e0, a_e1, a_e2]
+    std::vector<double> x = {1.0, 1.0, 0.5, 0.5, 0.5};
+
+    EXPECT_TRUE(gradient_check(mesh, x, maps))
+        << "Finite-difference gradient check failed for mixed ideal / hyper-ideal";
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Gradient check: fan mesh (n=6), all variable
+//
+// Exercises the full halfedge-around-vertex traversal for a high-valence
+// interior vertex.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(HyperIdealFunctional, GradientCheck_Fan6AllVariable)
+{
+    auto mesh = make_fan(6);
+    auto maps = setup_hyper_ideal_maps(mesh);
+    auto x    = make_x_all_variable(mesh, maps, /*b=*/1.0, /*a=*/0.5);
+
+    EXPECT_TRUE(gradient_check(mesh, x, maps))
+        << "Finite-difference gradient check failed on fan-6 mesh";
+}

code/tests/cgal/test_hyper_ideal_hessian.cpp

@@ -0,0 +1,340 @@
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
+// test_hyper_ideal_hessian.cpp
+//
+// Phase 9b — Hyper-ideal Hessian: block-FD vs full-FD cross-validation.
+//
+// The block-FD Hessian (Phase 9b) exploits the per-face locality of the
+// hyper-ideal functional to compute the Hessian as a sum of 6×6 per-face
+// blocks.  This file verifies:
+//
+//   1. Block-FD reproduces the full-FD Hessian to machine precision
+//      on tetrahedron (closed) and a 3-face open mesh.
+//   2. The result is symmetric and positive-semi-definite (Springborn
+//      2020 strict-convexity result).
+//   3. The kernel `face_angles_from_local_dofs` matches the existing
+//      `compute_face_angles` at the same DOFs — sanity that the pure
+//      refactor is non-regressing.
+//   4. Both Hessians agree with a from-scratch FD-of-energy reference
+//      at the same x.  (This is the highest-confidence cross-check.)
+
+#include "hyper_ideal_functional.hpp"
+#include "hyper_ideal_hessian.hpp"
+#include "mesh_builder.hpp"
+
+#include <Eigen/Eigenvalues>
+#include <gtest/gtest.h>
+#include <chrono>
+#include <iostream>
+#include <vector>
+
+using namespace conformallab;
+
+namespace {
+
+// Open 3-face mesh (tetrahedron minus one face) — exercises boundary edges.
+inline ConformalMesh make_open_3face_mesh()
+{
+    ConformalMesh mesh;
+    auto v0 = mesh.add_vertex(Point3( 1,  1,  1));
+    auto v1 = mesh.add_vertex(Point3( 1, -1, -1));
+    auto v2 = mesh.add_vertex(Point3(-1,  1, -1));
+    auto v3 = mesh.add_vertex(Point3(-1, -1,  1));
+    mesh.add_face(v0, v2, v1);
+    mesh.add_face(v0, v1, v3);
+    mesh.add_face(v0, v3, v2);
+    return mesh;
+}
+
+// Construct an x ≈ "natural" hyper-ideal initialisation:
+//   b_v  = 1 (positive log scale)
+//   a_e  = 0.5 (moderate intersection angle)
+inline std::vector<double> natural_x(const ConformalMesh& mesh,
+                                       const HyperIdealMaps& m)
+{
+    const int n = hyper_ideal_dimension(mesh, m);
+    std::vector<double> x(static_cast<std::size_t>(n), 0.0);
+    for (auto v : mesh.vertices()) {
+        int i = m.v_idx[v];
+        if (i >= 0) x[static_cast<std::size_t>(i)] = 1.0;
+    }
+    for (auto e : mesh.edges()) {
+        int i = m.e_idx[e];
+        if (i >= 0) x[static_cast<std::size_t>(i)] = 0.5;
+    }
+    return x;
+}
+
+} // anonymous namespace
+
+// ════════════════════════════════════════════════════════════════════════════
+// 1.  Pure helper face_angles_from_local_dofs reproduces compute_face_angles
+//
+// Refactor sanity check: the new pure 6→6 function must produce identical
+// (β₁,β₂,β₃,α₁₂,α₂₃,α₃₁) to the existing mesh-reading compute_face_angles.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(HyperIdealHessian, PureHelperMatchesMeshHelper)
+{
+    auto mesh = make_tetrahedron();
+    auto m    = setup_hyper_ideal_maps(mesh);
+    const int n = assign_all_dof_indices(mesh, m);
+    auto x = natural_x(mesh, m);
+
+    for (auto f : mesh.faces()) {
+        FaceAngles fa = compute_face_angles(mesh, f, x, m);
+
+        // Read the 6 local DOFs the same way Block-FD does.
+        Halfedge_index h0 = mesh.halfedge(f);
+        Halfedge_index h1 = mesh.next(h0);
+        Halfedge_index h2 = mesh.next(h1);
+        Vertex_index v1 = mesh.source(h0);
+        Vertex_index v2 = mesh.source(h1);
+        Vertex_index v3 = mesh.source(h2);
+        Edge_index e12 = mesh.edge(h0);
+        Edge_index e23 = mesh.edge(h1);
+        Edge_index e31 = mesh.edge(h2);
+
+        FaceAngleOutputs o = face_angles_from_local_dofs(
+            dof_val(m.v_idx[v1], x), dof_val(m.v_idx[v2], x), dof_val(m.v_idx[v3], x),
+            dof_val(m.e_idx[e12], x), dof_val(m.e_idx[e23], x), dof_val(m.e_idx[e31], x),
+            m.v_idx[v1] >= 0, m.v_idx[v2] >= 0, m.v_idx[v3] >= 0);
+
+        EXPECT_NEAR(o.beta1, fa.beta1, 1e-14);
+        EXPECT_NEAR(o.beta2, fa.beta2, 1e-14);
+        EXPECT_NEAR(o.beta3, fa.beta3, 1e-14);
+        EXPECT_NEAR(o.alpha12, fa.alpha12, 1e-14);
+        EXPECT_NEAR(o.alpha23, fa.alpha23, 1e-14);
+        EXPECT_NEAR(o.alpha31, fa.alpha31, 1e-14);
+    }
+    (void)n;
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// 2.  Block-FD ≡ Full-FD  on closed tetrahedron
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(HyperIdealHessian, BlockFD_MatchesFullFD_ClosedTetrahedron)
+{
+    auto mesh = make_tetrahedron();
+    auto m    = setup_hyper_ideal_maps(mesh);
+    const int n = assign_all_dof_indices(mesh, m);
+    auto x = natural_x(mesh, m);
+
+    auto H_full  = hyper_ideal_hessian_sym         (mesh, x, m);
+    auto H_block = hyper_ideal_hessian_block_fd_sym(mesh, x, m);
+
+    Eigen::MatrixXd Df(H_full), Db(H_block);
+    const double diff = (Df - Db).cwiseAbs().maxCoeff();
+    EXPECT_LT(diff, 1e-8)
+        << "Block-FD diverges from Full-FD by " << diff << " on tetrahedron";
+
+    (void)n;
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// 3.  Block-FD ≡ Full-FD  on open 3-face mesh  (boundary code path)
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(HyperIdealHessian, BlockFD_MatchesFullFD_Open3FaceMesh)
+{
+    auto mesh = make_open_3face_mesh();
+    auto m    = setup_hyper_ideal_maps(mesh);
+    const int n = assign_all_dof_indices(mesh, m);
+    auto x = natural_x(mesh, m);
+
+    auto H_full  = hyper_ideal_hessian_sym         (mesh, x, m);
+    auto H_block = hyper_ideal_hessian_block_fd_sym(mesh, x, m);
+
+    Eigen::MatrixXd Df(H_full), Db(H_block);
+    const double diff = (Df - Db).cwiseAbs().maxCoeff();
+    EXPECT_LT(diff, 1e-8)
+        << "Block-FD diverges from Full-FD by " << diff << " on open 3-face mesh";
+
+    (void)n;
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// 4.  Block-FD ≡ Full-FD  with pinned DOFs  (partial-DOF code path)
+//
+// Tests the case where some DOFs are pinned (v_idx = -1).  Block-FD must
+// skip pinned columns/rows just like Full-FD does.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(HyperIdealHessian, BlockFD_MatchesFullFD_PinnedDOFs)
+{
+    auto mesh = make_tetrahedron();
+    auto m    = setup_hyper_ideal_maps(mesh);
+
+    // Assign vertex DOFs only; leave edges pinned (a_e fixed at 0).
+    int idx = 0;
+    for (auto v : mesh.vertices()) m.v_idx[v] = idx++;
+    for (auto e : mesh.edges())    m.e_idx[e] = -1;
+    const int n = hyper_ideal_dimension(mesh, m);
+    ASSERT_EQ(n, 4);   // tetrahedron: 4 vertex DOFs, 0 edge DOFs
+
+    std::vector<double> x(static_cast<std::size_t>(n), 1.0);
+
+    auto H_full  = hyper_ideal_hessian_sym         (mesh, x, m);
+    auto H_block = hyper_ideal_hessian_block_fd_sym(mesh, x, m);
+
+    Eigen::MatrixXd Df(H_full), Db(H_block);
+    const double diff = (Df - Db).cwiseAbs().maxCoeff();
+    EXPECT_LT(diff, 1e-8) << "Block-FD diverges by " << diff << " with pinned edges";
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// 5.  PSD property (Springborn 2020 strict convexity)
+//
+// The hyper-ideal energy is strictly convex on its domain of validity, so
+// the Hessian is PSD at every interior point.  Both block-FD and full-FD
+// must report this consistently.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(HyperIdealHessian, BlockFD_IsPSD)
+{
+    auto mesh = make_tetrahedron();
+    auto m    = setup_hyper_ideal_maps(mesh);
+    const int n = assign_all_dof_indices(mesh, m);
+    auto x = natural_x(mesh, m);
+
+    auto H = hyper_ideal_hessian_block_fd_sym(mesh, x, m);
+    Eigen::MatrixXd Hd(H);
+
+    // Symmetry to FD rounding tolerance.
+    EXPECT_LT((Hd - Hd.transpose()).cwiseAbs().maxCoeff(), 1e-10)
+        << "Block-FD Hessian should be symmetric after _sym normalisation";
+
+    // PSD via smallest eigenvalue.
+    Eigen::SelfAdjointEigenSolver<Eigen::MatrixXd> es(Hd);
+    EXPECT_GE(es.eigenvalues().minCoeff(), -1e-8)
+        << "Hyper-ideal Hessian must be PSD (Springborn 2020)";
+
+    (void)n;
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// 6.  Sparsity:  block-FD respects the 6-DOF-per-face locality
+//
+// Each non-zero (i,j) entry must correspond to a pair of DOFs that share at
+// least one face.  This is a structural correctness test independent of the
+// numerical values.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(HyperIdealHessian, BlockFD_SparsityMatchesFaceAdjacency)
+{
+    auto mesh = make_open_3face_mesh();
+    auto m    = setup_hyper_ideal_maps(mesh);
+    const int n = assign_all_dof_indices(mesh, m);
+    auto x = natural_x(mesh, m);
+
+    auto H = hyper_ideal_hessian_block_fd(mesh, x, m);
+
+    // Build the "should-be-nonzero" mask from face adjacency.
+    std::vector<std::vector<bool>> face_pair(n, std::vector<bool>(n, false));
+    for (auto f : mesh.faces()) {
+        auto h0 = mesh.halfedge(f);
+        auto h1 = mesh.next(h0);
+        auto h2 = mesh.next(h1);
+        int idx[6] = {
+            m.v_idx[mesh.source(h0)],
+            m.v_idx[mesh.source(h1)],
+            m.v_idx[mesh.source(h2)],
+            m.e_idx[mesh.edge(h0)],
+            m.e_idx[mesh.edge(h1)],
+            m.e_idx[mesh.edge(h2)],
+        };
+        for (int i = 0; i < 6; ++i) {
+            if (idx[i] < 0) continue;
+            for (int j = 0; j < 6; ++j) {
+                if (idx[j] < 0) continue;
+                face_pair[idx[i]][idx[j]] = true;
+            }
+        }
+    }
+
+    // Every non-zero entry must come from a face-adjacent pair.
+    for (int k = 0; k < H.outerSize(); ++k) {
+        for (Eigen::SparseMatrix<double>::InnerIterator it(H, k); it; ++it) {
+            EXPECT_TRUE(face_pair[it.row()][it.col()])
+                << "Hessian nonzero at (" << it.row() << "," << it.col()
+                << ") between DOFs that share no face";
+        }
+    }
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// 7.  Performance: measure block-FD vs full-FD on a moderately-sized mesh
+//
+// Builds a "long" tetrahedron-strip mesh: V tetrahedron-cells joined along
+// shared faces.  Asserts the block-FD Hessian computes ≥ 3× faster than
+// the full-FD baseline.  This is the operational case for the Phase 9b
+// optimisation (the asymptotic ratio is ~ n/36, which grows linearly in
+// mesh size).  Wall-clock is printed for the record but the assertion
+// uses a conservative ratio so the test stays stable on slow CI hardware.
+// ════════════════════════════════════════════════════════════════════════════
+
+namespace {
+
+// Build a strip of `n_cells` connected tetrahedra (subdivision-like).
+// The resulting mesh has ~ 2*n_cells + 2 vertices, 4*n_cells faces.
+// (Approximation; the exact count depends on shared-vertex handling.)
+inline ConformalMesh make_tet_strip(int n_cells)
+{
+    ConformalMesh mesh;
+    // Lay out vertex chain at z=0 / z=1 alternating.
+    std::vector<Vertex_index> top, bot;
+    for (int i = 0; i <= n_cells; ++i) {
+        top.push_back(mesh.add_vertex(Point3(i,   0, 0)));
+        bot.push_back(mesh.add_vertex(Point3(i, 0.7, 0.5 * std::sin(0.3*i))));
+    }
+    // Add two triangles per cell (one row of "zig-zag" triangles).
+    for (int i = 0; i < n_cells; ++i) {
+        mesh.add_face(top[i],   bot[i],   top[i+1]);
+        mesh.add_face(bot[i],   bot[i+1], top[i+1]);
+    }
+    return mesh;
+}
+
+} // anonymous namespace
+
+TEST(HyperIdealHessian, BlockFD_FasterThanFullFD)
+{
+    // 100 cells → ~200 faces, ~200 vertex DOFs + ~300 edge DOFs ≈ 500 DOFs.
+    // Full-FD: 500 × 200 ≈ 100 k face evaluations
+    // Block-FD: 200 × 12 ≈ 2.4 k face evaluations
+    // Theoretical ratio: ~42×.  We assert ≥ 3× to leave wide CI tolerance.
+    auto mesh = make_tet_strip(100);
+    auto m    = setup_hyper_ideal_maps(mesh);
+    const int n = assign_all_dof_indices(mesh, m);
+    auto x = natural_x(mesh, m);
+
+    using clk = std::chrono::steady_clock;
+
+    auto t1 = clk::now();
+    auto H_full  = hyper_ideal_hessian         (mesh, x, m);
+    auto t2 = clk::now();
+    auto H_block = hyper_ideal_hessian_block_fd(mesh, x, m);
+    auto t3 = clk::now();
+
+    auto ms_full  = std::chrono::duration_cast<std::chrono::microseconds>(t2-t1).count();
+    auto ms_block = std::chrono::duration_cast<std::chrono::microseconds>(t3-t2).count();
+
+    std::cerr << "[HyperIdealHessian.BlockFD_FasterThanFullFD]"
+              << "  V=" << mesh.number_of_vertices()
+              << "  F=" << mesh.number_of_faces()
+              << "  DOFs=" << n
+              << "  full-FD: " << ms_full  << " µs"
+              << "  block-FD: " << ms_block << " µs"
+              << "  speed-up: " << (ms_block > 0 ? (double)ms_full / (double)ms_block : 0.0)
+              << "×\n";
+
+    // Both must report identical Hessians (within FD rounding).
+    Eigen::MatrixXd Df(H_full), Db(H_block);
+    EXPECT_LT((Df - Db).cwiseAbs().maxCoeff(), 1e-8);
+
+    // Conservative speed-up assertion — typically observe ~30×, accept ≥ 3×.
+    EXPECT_GE(ms_full, 3 * ms_block)
+        << "Block-FD should be at least 3× faster than full-FD on this mesh";
+}

code/tests/cgal/test_inversive_distance_functional.cpp

@@ -0,0 +1,265 @@
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
+// test_inversive_distance_functional.cpp
+//
+// Phase 9a.2 — Inversive-distance functional (Luo 2004) tests.
+//
+// Validation against three mathematical references:
+//
+//   [Luo 2004]        ℓ_ij² = exp(2u_i) + exp(2u_j) + 2 I_ij exp(u_i+u_j)
+//                     ∂E/∂u_v = Θ_v − Σ α_v   (Lemma 3.1)
+//
+//   [BS 2004]         I_ij = (ℓ² − r_i² − r_j²) / (2 r_i r_j)
+//                     I = 1 ⇒ tangential circles
+//                     I = 0 ⇒ orthogonal circles
+//
+//   [Glickenstein 2011 §5]
+//                     correspondence to BPS-2010 face-based CP:
+//                     I_ij = cos θ_e on the face-dual mesh
+//
+// No Java reference exists for this functional in
+// de.varylab.discreteconformal.  Cross-validation is done via:
+//   1. FD-vs-analytic gradient check (numerical),
+//   2. Luo's edge-length identity check (mathematical),
+//   3. Tangential-limit identity I=1 ⇒ ℓ = r_i+r_j (geometric).
+
+#include "inversive_distance_functional.hpp"
+#include "euclidean_functional.hpp"
+#include "mesh_builder.hpp"
+#include "conformal_mesh.hpp"
+
+#include <gtest/gtest.h>
+#include <vector>
+#include <random>
+#include <cmath>
+
+using namespace conformallab;
+
+// ════════════════════════════════════════════════════════════════════════════
+// 1.  Edge-length formula  (Luo 2004 §3)
+//
+//   ℓ² = exp(2u_i) + exp(2u_j) + 2 I exp(u_i+u_j)
+//      = r_i² + r_j² + 2 I r_i r_j
+//
+// Special cases:
+//   I = 1  ⇒ ℓ² = (r_i + r_j)²  ⇒  ℓ = r_i + r_j     (tangential)
+//   I = 0  ⇒ ℓ² = r_i² + r_j²    (orthogonal — circles meet at 90°)
+//   I = −1 ⇒ ℓ² = (r_i − r_j)²  ⇒  ℓ = |r_i − r_j|  (inside-tangent)
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(InversiveDistanceFunctional, EdgeLengthFormula_TangentialLimit)
+{
+    // ui = 0 ⇒ ri = 1;  uj = log(2) ⇒ rj = 2;  I = 1 (tangential):
+    //   ℓ² = 1 + 4 + 2·1·1·2 = 9 ⇒ ℓ = 3 = r_i + r_j ✓
+    double l2 = id_detail::edge_length_squared(0.0, std::log(2.0), 1.0);
+    EXPECT_NEAR(std::sqrt(l2), 3.0, 1e-12);
+}
+
+TEST(InversiveDistanceFunctional, EdgeLengthFormula_OrthogonalLimit)
+{
+    // r_i = 3, r_j = 4, I = 0:  ℓ² = 9 + 16 = 25 ⇒ ℓ = 5  (Pythagorean)
+    double l2 = id_detail::edge_length_squared(std::log(3.0), std::log(4.0), 0.0);
+    EXPECT_NEAR(std::sqrt(l2), 5.0, 1e-12);
+}
+
+TEST(InversiveDistanceFunctional, EdgeLengthFormula_InsideTangentLimit)
+{
+    // r_i = 2, r_j = 5, I = −1:  ℓ² = (5 − 2)² = 9 ⇒ ℓ = 3
+    double l2 = id_detail::edge_length_squared(std::log(2.0), std::log(5.0), -1.0);
+    EXPECT_NEAR(std::sqrt(l2), 3.0, 1e-12);
+}
+
+TEST(InversiveDistanceFunctional, EdgeLengthFormula_DegenerateReturnsMinusOne)
+{
+    // r_i = r_j = 1, I = −2:  ℓ² = 1 + 1 − 4 = −2  (impossible packing)
+    double l2 = id_detail::edge_length_squared(0.0, 0.0, -2.0);
+    EXPECT_EQ(l2, -1.0) << "should signal degenerate packing";
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// 2.  Bowers-Stephenson identity round-trip
+//
+//   Given (ℓ, r_i, r_j), the I_ij that compute_init produces must satisfy
+//   Luo's edge-length formula exactly: ℓ²(I_ij, r_i, r_j) = ℓ².
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(InversiveDistanceFunctional, BowersStephensonRoundTrip)
+{
+    auto mesh = make_triangle();   // (0,0,0)-(1,0,0)-(0,1,0)
+    auto m    = setup_inversive_distance_maps(mesh);
+    compute_inversive_distance_init_from_mesh(mesh, m);
+
+    // At u = 0, exp(u) = r0.  Reconstruct ℓ from (r_i, r_j, I_ij) and compare
+    // to the 3-D Euclidean edge length from the mesh.
+    for (auto e : mesh.edges()) {
+        auto h = mesh.halfedge(e);
+        auto p1 = mesh.point(mesh.source(h));
+        auto p2 = mesh.point(mesh.target(h));
+        double dx = p1.x() - p2.x();
+        double dy = p1.y() - p2.y();
+        double dz = p1.z() - p2.z();
+        double l_3d = std::sqrt(dx*dx + dy*dy + dz*dz);
+
+        double ri = m.r0[mesh.source(h)];
+        double rj = m.r0[mesh.target(h)];
+        double l2_reconstructed = ri*ri + rj*rj + 2.0 * m.I_e[e] * ri * rj;
+        EXPECT_NEAR(std::sqrt(l2_reconstructed), l_3d, 1e-12)
+            << "Bowers-Stephenson round-trip failed for an edge";
+    }
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// 3.  Properties of the init step
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(InversiveDistanceFunctional, InitProducesValidPositiveRadii)
+{
+    auto mesh = make_tetrahedron();
+    auto m    = setup_inversive_distance_maps(mesh);
+    compute_inversive_distance_init_from_mesh(mesh, m);
+
+    for (auto v : mesh.vertices()) {
+        EXPECT_GT(m.r0[v], 0.0) << "init radius must be positive";
+        EXPECT_TRUE(std::isfinite(m.r0[v]));
+    }
+    for (auto e : mesh.edges()) {
+        EXPECT_TRUE(std::isfinite(m.I_e[e]));
+        // I > −1 is required for any valid inversive-distance packing.
+        EXPECT_GT(m.I_e[e], -1.0);
+    }
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// 4.  Gradient at the "natural equilibrium" is zero by construction
+//
+// Same trick as in test_euclidean_functional.cpp:
+//   • Set u = 0  ⇒  r = r0  ⇒  ℓ = ℓ_3d  (Bowers-Stephenson round-trip)
+//   • Compute G(0) — that's the angle defect Θ − Σ_actual.
+//   • Subtract G(0) from Θ → new G(0) is zero.
+// This means u = 0 is now the Newton equilibrium of the functional, just
+// like in the euclidean functional natural-theta trick.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(InversiveDistanceFunctional, NaturalThetaGivesZeroGradientAtU0)
+{
+    auto mesh = make_triangle();
+    auto m    = setup_inversive_distance_maps(mesh);
+    compute_inversive_distance_init_from_mesh(mesh, m);
+
+    // Assign DOFs to all vertices.
+    int n = 0;
+    for (auto v : mesh.vertices()) m.v_idx[v] = n++;
+    std::vector<double> x(static_cast<std::size_t>(n), 0.0);
+
+    auto G0 = inversive_distance_gradient(mesh, x, m);
+    for (auto v : mesh.vertices()) {
+        int i = m.v_idx[v];
+        m.theta_v[v] -= G0[static_cast<std::size_t>(i)];
+    }
+
+    auto G_eq = inversive_distance_gradient(mesh, x, m);
+    for (double g : G_eq) EXPECT_NEAR(g, 0.0, 1e-13);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// 5.  FD-vs-analytic gradient check (the main acceptance test for the port)
+//
+// Pattern: identical to test_euclidean_functional.cpp's
+// GradientCheck_TriangleVertex (lines 137-149).  The energy is the path
+// integral of the gradient (by construction); a consistent FD-vs-analytic
+// match validates both energy and gradient implementations together.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(InversiveDistanceFunctional, FDGradientCheck_Triangle)
+{
+    auto mesh = make_triangle();
+    auto m    = setup_inversive_distance_maps(mesh);
+    compute_inversive_distance_init_from_mesh(mesh, m);
+
+    int n = 0;
+    for (auto v : mesh.vertices()) m.v_idx[v] = n++;
+
+    // Small perturbation u_v ≈ −0.1 keeps every triangle valid.
+    std::vector<double> x(static_cast<std::size_t>(n), -0.1);
+    EXPECT_TRUE(gradient_check_inversive_distance(mesh, x, m))
+        << "FD gradient mismatch on single triangle (u = −0.1)";
+}
+
+TEST(InversiveDistanceFunctional, FDGradientCheck_QuadStrip)
+{
+    auto mesh = make_quad_strip();
+    auto m    = setup_inversive_distance_maps(mesh);
+    compute_inversive_distance_init_from_mesh(mesh, m);
+
+    int n = 0;
+    for (auto v : mesh.vertices()) m.v_idx[v] = n++;
+
+    std::vector<double> x(static_cast<std::size_t>(n), -0.15);
+    EXPECT_TRUE(gradient_check_inversive_distance(mesh, x, m))
+        << "FD gradient mismatch on quad strip";
+}
+
+TEST(InversiveDistanceFunctional, FDGradientCheck_Tetrahedron)
+{
+    auto mesh = make_tetrahedron();
+    auto m    = setup_inversive_distance_maps(mesh);
+    compute_inversive_distance_init_from_mesh(mesh, m);
+
+    int n = 0;
+    for (auto v : mesh.vertices()) m.v_idx[v] = n++;
+
+    std::vector<double> x(static_cast<std::size_t>(n), -0.2);
+    EXPECT_TRUE(gradient_check_inversive_distance(mesh, x, m))
+        << "FD gradient mismatch on regular tetrahedron";
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// 6.  Cross-validation with euclidean_functional.hpp
+//
+// The two functionals are DIFFERENT geometric models.  At u = 0 with their
+// natural inits both produce a valid triangulation, but the per-edge length
+// is different:
+//   • Euclidean:           ℓ = ℓ_3d                  (exact, by lambda0 init)
+//   • Inversive distance:  ℓ = ℓ_3d                  (exact, by BS round-trip)
+//
+// HOWEVER the GRADIENT at u = 0 differs because the chain rule ∂ℓ/∂u is
+// different.  Specifically:
+//   • Euclidean:           ∂(2 log ℓ)/∂u_i = 1
+//   • Inversive distance:  ∂(2 log ℓ)/∂u_i = (r_i² + I r_i r_j) / ℓ²
+//
+// This test pins one quantitative consequence: at u = 0 both gradients have
+// the SAME angle-defect structure Θ − Σ_actual.  After applying the natural-
+// theta trick on each, both must be at equilibrium with G(0) = 0.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(InversiveDistanceFunctional, AngleDefectAtU0_AgreesWithEuclideanAtU0)
+{
+    auto mesh = make_quad_strip();
+
+    // ── Inversive distance side ────────────────────────────────────────────
+    auto m_id = setup_inversive_distance_maps(mesh);
+    compute_inversive_distance_init_from_mesh(mesh, m_id);
+    int n_id = 0;
+    for (auto v : mesh.vertices()) m_id.v_idx[v] = n_id++;
+    std::vector<double> x_id(static_cast<std::size_t>(n_id), 0.0);
+    auto G_id = inversive_distance_gradient(mesh, x_id, m_id);
+
+    // ── Euclidean side (same mesh, same DOF order) ─────────────────────────
+    auto m_eu = setup_euclidean_maps(mesh);
+    compute_euclidean_lambda0_from_mesh(mesh, m_eu);
+    int n_eu = 0;
+    for (auto v : mesh.vertices()) m_eu.v_idx[v] = n_eu++;
+    std::vector<double> x_eu(static_cast<std::size_t>(n_eu), 0.0);
+    auto G_eu = euclidean_gradient(const_cast<ConformalMesh&>(mesh), x_eu, m_eu);
+
+    // Both should report the same actual angle sum per vertex at u = 0
+    // (since both reproduce ℓ = ℓ_3d at u = 0).  Therefore  Θ − Σ_actual
+    // is identical for the two functionals (Θ default 2π in both).
+    ASSERT_EQ(G_id.size(), G_eu.size());
+    for (std::size_t i = 0; i < G_id.size(); ++i) {
+        EXPECT_NEAR(G_id[i], G_eu[i], 1e-10)
+            << "angle-defect mismatch at u=0, DOF " << i
+            << ": id=" << G_id[i] << "  eu=" << G_eu[i];
+    }
+}

code/tests/cgal/test_layout.cpp

@@ -0,0 +1,372 @@
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
+// test_layout.cpp
+//
+// Phase 5 — Layout / embedding tests.
+//
+// Strategy: a conformal map that is the identity (natural equilibrium x*=0)
+// should reproduce the mesh's own edge lengths.  We verify:
+//   1. Euclidean layout preserves edge lengths (to solver tolerance).
+//   2. Spherical layout preserves arc-lengths on S².
+//   3. Layout2D has correct size (one entry per vertex).
+//   4. Serialisation round-trip: save JSON → load → DOF vector unchanged.
+//   5. Serialisation round-trip: save XML  → load → DOF vector unchanged.
+//   6. HyperIdeal layout returns success and finite positions.
+
+#include "conformal_mesh.hpp"
+#include "mesh_builder.hpp"
+#include "euclidean_functional.hpp"
+#include "spherical_functional.hpp"
+#include "hyper_ideal_functional.hpp"
+#include "newton_solver.hpp"
+#include "layout.hpp"
+#include "serialization.hpp"
+#include <gtest/gtest.h>
+#include <cmath>
+#include <vector>
+#include <filesystem>
+
+using namespace conformallab;
+
+// ────────────────────────────────────────────────────────────────────────────
+// Helpers
+// ────────────────────────────────────────────────────────────────────────────
+
+// Euclidean edge length from layout (2D)
+static double layout_edge_len(const Layout2D& lay,
+                               ConformalMesh& mesh, Halfedge_index h)
+{
+    auto vi = mesh.source(h).idx();
+    auto vj = mesh.target(h).idx();
+    return (lay.uv[vi] - lay.uv[vj]).norm();
+}
+
+// Spherical arc-length from layout (3D)
+static double layout_arc_len(const Layout3D& lay,
+                              ConformalMesh& mesh, Halfedge_index h)
+{
+    auto& a = lay.pos[mesh.source(h).idx()];
+    auto& b = lay.pos[mesh.target(h).idx()];
+    double c = std::max(-1.0, std::min(1.0, a.dot(b)));
+    return std::acos(c);
+}
+
+// Euclidean edge length from maps + x (the "true" target)
+static double maps_edge_len(const EuclideanMaps& m,
+                             ConformalMesh& mesh,
+                             Halfedge_index h,
+                             const std::vector<double>& x)
+{
+    auto get_u = [&](Vertex_index v) -> double {
+        int iv = m.v_idx[v]; return (iv >= 0) ? x[static_cast<std::size_t>(iv)] : 0.0;
+    };
+    Edge_index e = mesh.edge(h);
+    double lam = m.lambda0[e] + get_u(mesh.source(h)) + get_u(mesh.target(h));
+    return std::exp(lam * 0.5);
+}
+
+// Spherical arc-length from maps + x
+static double maps_arc_len(const SphericalMaps& m,
+                            ConformalMesh& mesh,
+                            Halfedge_index h,
+                            const std::vector<double>& x)
+{
+    auto get_u = [&](Vertex_index v) -> double {
+        int iv = m.v_idx[v]; return (iv >= 0) ? x[static_cast<std::size_t>(iv)] : 0.0;
+    };
+    Edge_index e = mesh.edge(h);
+    double lam = m.lambda0[e] + get_u(mesh.source(h)) + get_u(mesh.target(h));
+    return 2.0 * std::asin(std::min(std::exp(lam * 0.5), 1.0));
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Test 1 — Euclidean layout preserves edge lengths (identity map)
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(Layout, Euclidean_PreservesEdgeLengths)
+{
+    auto mesh = make_quad_strip();
+    auto maps = setup_euclidean_maps(mesh);
+    compute_euclidean_lambda0_from_mesh(mesh, maps);
+
+    // Pin first vertex; natural equilibrium so x* = 0
+    auto vit = mesh.vertices().begin();
+    maps.v_idx[*vit++] = -1;
+    int idx = 0;
+    for (; vit != mesh.vertices().end(); ++vit) maps.v_idx[*vit] = idx++;
+    const int n = idx;
+
+    std::vector<double> x0(static_cast<std::size_t>(n), 0.0);
+    auto G0 = euclidean_gradient(mesh, x0, maps);
+    for (auto v : mesh.vertices()) {
+        int iv = maps.v_idx[v];
+        if (iv >= 0) maps.theta_v[v] -= G0[static_cast<std::size_t>(iv)];
+    }
+
+    // Solve (identity map)
+    auto res = newton_euclidean(mesh, x0, maps, 1e-10, 100);
+    ASSERT_TRUE(res.converged);
+
+    auto layout = euclidean_layout(mesh, res.x, maps);
+    ASSERT_TRUE(layout.success);
+    EXPECT_EQ(layout.uv.size(), mesh.number_of_vertices());
+
+    // Every edge: |layout_len - expected_len| < 1e-8
+    for (auto e : mesh.edges()) {
+        Halfedge_index h = mesh.halfedge(e, 0);
+        double expected = maps_edge_len(maps, mesh, h, res.x);
+        double actual   = layout_edge_len(layout, mesh, h);
+        EXPECT_NEAR(actual, expected, 1e-7)
+            << "Edge " << e << ": expected " << expected << " got " << actual;
+    }
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Test 2 — Euclidean layout: correct vertex count
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(Layout, Euclidean_CorrectVertexCount)
+{
+    auto mesh = make_triangle();
+    auto maps = setup_euclidean_maps(mesh);
+    compute_euclidean_lambda0_from_mesh(mesh, maps);
+
+    auto vit = mesh.vertices().begin();
+    maps.v_idx[*vit++] = -1;
+    int idx = 0;
+    for (; vit != mesh.vertices().end(); ++vit) maps.v_idx[*vit] = idx++;
+    const int n = idx;
+
+    std::vector<double> x(static_cast<std::size_t>(n), 0.0);
+    auto layout = euclidean_layout(mesh, x, maps);
+
+    EXPECT_TRUE(layout.success);
+    EXPECT_EQ(static_cast<int>(layout.uv.size()),
+              static_cast<int>(mesh.number_of_vertices()));
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Test 3 — Euclidean triangle layout: root face is a valid triangle
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(Layout, Euclidean_TriangleIsNonDegenerate)
+{
+    auto mesh = make_triangle();
+    auto maps = setup_euclidean_maps(mesh);
+    compute_euclidean_lambda0_from_mesh(mesh, maps);
+    for (auto v : mesh.vertices()) maps.v_idx[v] = static_cast<int>(v.idx());
+
+    std::vector<double> x(mesh.number_of_vertices(), 0.0);
+    auto layout = euclidean_layout(mesh, x, maps);
+    ASSERT_TRUE(layout.success);
+
+    // Area of layout triangle > 0
+    const auto& A = layout.uv[0];
+    const auto& B = layout.uv[1];
+    const auto& C = layout.uv[2];
+    double area = std::abs((B - A).x() * (C - A).y()
+                         - (C - A).x() * (B - A).y()) * 0.5;
+    EXPECT_GT(area, 1e-10) << "Layout triangle must be non-degenerate";
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Test 4 — Spherical layout: arc-lengths preserved (identity map)
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(Layout, Spherical_PreservesArcLengths)
+{
+    auto mesh = make_spherical_tetrahedron();
+    auto maps = setup_spherical_maps(mesh);
+    compute_lambda0_from_mesh(mesh, maps);
+    int n = assign_vertex_dof_indices(mesh, maps);
+
+    // Solve to identity
+    std::vector<double> x0(static_cast<std::size_t>(n), 0.0);
+    auto G0 = spherical_gradient(mesh, x0, maps);
+    for (auto v : mesh.vertices()) {
+        int iv = maps.v_idx[v];
+        if (iv >= 0) maps.theta_v[v] -= G0[static_cast<std::size_t>(iv)];
+    }
+    auto res = newton_spherical(mesh, x0, maps, 1e-10, 100);
+    ASSERT_TRUE(res.converged);
+
+    auto layout = spherical_layout(mesh, res.x, maps);
+    ASSERT_TRUE(layout.success);
+    EXPECT_EQ(layout.pos.size(), mesh.number_of_vertices());
+
+    for (auto e : mesh.edges()) {
+        Halfedge_index h = mesh.halfedge(e, 0);
+        double expected = maps_arc_len(maps, mesh, h, res.x);
+        double actual   = layout_arc_len(layout, mesh, h);
+        EXPECT_NEAR(actual, expected, 1e-6)
+            << "Spherical edge " << e << ": expected " << expected << " got " << actual;
+    }
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Test 5 — Spherical layout: all positions on unit sphere
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(Layout, Spherical_PositionsOnUnitSphere)
+{
+    auto mesh = make_spherical_tetrahedron();
+    auto maps = setup_spherical_maps(mesh);
+    compute_lambda0_from_mesh(mesh, maps);
+    int n = assign_vertex_dof_indices(mesh, maps);
+
+    std::vector<double> x(static_cast<std::size_t>(n), 0.0);
+    auto layout = spherical_layout(mesh, x, maps);
+    ASSERT_TRUE(layout.success);
+
+    for (auto v : mesh.vertices()) {
+        double r = layout.pos[v.idx()].norm();
+        EXPECT_NEAR(r, 1.0, 1e-12) << "Vertex " << v << " not on unit sphere, |p| = " << r;
+    }
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Test 6 — HyperIdeal layout returns success and finite positions
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(Layout, HyperIdeal_SuccessAndFinitePositions)
+{
+    auto mesh = make_triangle();
+    auto maps = setup_hyper_ideal_maps(mesh);
+    int n = assign_all_dof_indices(mesh, maps);
+
+    // Natural equilibrium at (b=1, a=0.5)
+    std::vector<double> xbase(static_cast<std::size_t>(n));
+    for (auto v : mesh.vertices()) {
+        int iv = maps.v_idx[v]; if (iv >= 0) xbase[iv] = 1.0;
+    }
+    for (auto e : mesh.edges()) {
+        int ie = maps.e_idx[e]; if (ie >= 0) xbase[ie] = 0.5;
+    }
+    auto G0 = evaluate_hyper_ideal(mesh, xbase, maps, false).gradient;
+    for (auto v : mesh.vertices()) {
+        int iv = maps.v_idx[v]; if (iv >= 0) maps.theta_v[v] += G0[iv];
+    }
+    for (auto e : mesh.edges()) {
+        int ie = maps.e_idx[e]; if (ie >= 0) maps.theta_e[e] += G0[ie];
+    }
+
+    auto res = newton_hyper_ideal(mesh, xbase, maps, 1e-9, 100);
+    ASSERT_TRUE(res.converged);
+
+    auto layout = hyper_ideal_layout(mesh, res.x, maps);
+    EXPECT_TRUE(layout.success);
+    ASSERT_EQ(layout.uv.size(), mesh.number_of_vertices());
+
+    for (auto v : mesh.vertices()) {
+        auto& p = layout.uv[v.idx()];
+        EXPECT_FALSE(std::isnan(p.x())) << "NaN in layout x for v" << v;
+        EXPECT_FALSE(std::isnan(p.y())) << "NaN in layout y for v" << v;
+        // Poincaré disk: all points within the unit disk
+        EXPECT_LE(p.norm(), 1.0 + 1e-9) << "Point outside Poincaré disk for v" << v;
+    }
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Test 7 — JSON serialisation round-trip
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(Serialization, JSON_RoundTrip)
+{
+    auto mesh = make_triangle();
+    auto maps = setup_euclidean_maps(mesh);
+    compute_euclidean_lambda0_from_mesh(mesh, maps);
+    auto vit = mesh.vertices().begin();
+    maps.v_idx[*vit++] = -1;
+    int idx = 0;
+    for (; vit != mesh.vertices().end(); ++vit) maps.v_idx[*vit] = idx++;
+    const int n = idx;
+
+    std::vector<double> x0(static_cast<std::size_t>(n), -0.05);
+    auto G0 = euclidean_gradient(mesh, std::vector<double>(n, 0.0), maps);
+    for (auto v : mesh.vertices()) {
+        int iv = maps.v_idx[v];
+        if (iv >= 0) maps.theta_v[v] -= G0[static_cast<std::size_t>(iv)];
+    }
+    auto res = newton_euclidean(mesh, x0, maps, 1e-10, 100);
+    ASSERT_TRUE(res.converged);
+
+    auto layout = euclidean_layout(mesh, res.x, maps);
+
+    const std::string path = "/tmp/conformallab_test_round_trip.json";
+    ASSERT_NO_THROW(save_result_json(path, res, "euclidean",
+        static_cast<int>(mesh.number_of_vertices()),
+        static_cast<int>(mesh.number_of_faces()),
+        &layout));
+    ASSERT_TRUE(std::filesystem::exists(path));
+
+    NewtonResult res2;
+    std::string geom;
+    Layout2D layout2;
+    auto x2 = load_result_json(path, &res2, &geom, &layout2);
+
+    EXPECT_EQ(geom, "euclidean");
+    EXPECT_EQ(res2.converged,  res.converged);
+    EXPECT_EQ(res2.iterations, res.iterations);
+    ASSERT_EQ(x2.size(), res.x.size());
+    for (std::size_t i = 0; i < x2.size(); ++i)
+        EXPECT_NEAR(x2[i], res.x[i], 1e-14);
+
+    EXPECT_TRUE(layout2.success);
+    ASSERT_EQ(layout2.uv.size(), layout.uv.size());
+    for (std::size_t i = 0; i < layout2.uv.size(); ++i) {
+        EXPECT_NEAR(layout2.uv[i].x(), layout.uv[i].x(), 1e-12);
+        EXPECT_NEAR(layout2.uv[i].y(), layout.uv[i].y(), 1e-12);
+    }
+    std::filesystem::remove(path);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Test 8 — XML serialisation round-trip
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(Serialization, XML_RoundTrip)
+{
+    auto mesh = make_triangle();
+    auto maps = setup_euclidean_maps(mesh);
+    compute_euclidean_lambda0_from_mesh(mesh, maps);
+    auto vit = mesh.vertices().begin();
+    maps.v_idx[*vit++] = -1;
+    int idx = 0;
+    for (; vit != mesh.vertices().end(); ++vit) maps.v_idx[*vit] = idx++;
+    const int n = idx;
+
+    std::vector<double> x0(static_cast<std::size_t>(n), 0.0);
+    auto G0 = euclidean_gradient(mesh, x0, maps);
+    for (auto v : mesh.vertices()) {
+        int iv = maps.v_idx[v];
+        if (iv >= 0) maps.theta_v[v] -= G0[static_cast<std::size_t>(iv)];
+    }
+    auto res = newton_euclidean(mesh, x0, maps, 1e-10, 100);
+    ASSERT_TRUE(res.converged);
+
+    auto layout = euclidean_layout(mesh, res.x, maps);
+
+    const std::string path = "/tmp/conformallab_test_round_trip.xml";
+    ASSERT_NO_THROW(save_result_xml(path, res, "euclidean",
+        static_cast<int>(mesh.number_of_vertices()),
+        static_cast<int>(mesh.number_of_faces()),
+        &layout));
+    ASSERT_TRUE(std::filesystem::exists(path));
+
+    NewtonResult res2;
+    std::string geom;
+    Layout2D layout2;
+    auto x2 = load_result_xml(path, &res2, &geom, &layout2);
+
+    EXPECT_EQ(geom, "euclidean");
+    EXPECT_EQ(res2.converged,  res.converged);
+    ASSERT_EQ(x2.size(), res.x.size());
+    for (std::size_t i = 0; i < x2.size(); ++i)
+        EXPECT_NEAR(x2[i], res.x[i], 1e-12);
+
+    EXPECT_TRUE(layout2.success);
+    ASSERT_EQ(layout2.uv.size(), layout.uv.size());
+
+    std::filesystem::remove(path);
+}

code/tests/cgal/test_mesh_io.cpp

@@ -0,0 +1,173 @@
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
+// test_mesh_io.cpp
+//
+// Phase 4b — CGAL::IO mesh round-trip tests.
+//
+// Tests:
+//   1. OFF write + read round-trip: vertex count, face count, edge count preserved.
+//   2. OBJ write + read round-trip: same topology checks.
+//   3. load_mesh() throws on non-existent file.
+//   4. Round-trip preserves vertex positions (within floating-point precision).
+
+#include "conformal_mesh.hpp"
+#include "mesh_builder.hpp"
+#include "mesh_io.hpp"
+#include <gtest/gtest.h>
+#include <cmath>
+#include <cstdio>
+#include <filesystem>
+#include <stdexcept>
+
+using namespace conformallab;
+
+// Helper: temp file path with given extension
+static std::string tmp_path(const char* ext)
+{
+    return std::string("/tmp/conformallab_test.") + ext;
+}
+
+// Helper: delete file if it exists
+static void rm(const std::string& path)
+{
+    std::filesystem::remove(path);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// OFF round-trip: tetrahedron
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(MeshIO, OFF_RoundTrip_Tetrahedron)
+{
+    auto path = tmp_path("off");
+    rm(path);
+
+    auto mesh_out = make_tetrahedron();
+    ASSERT_TRUE(write_mesh(path, mesh_out))
+        << "write_mesh should succeed for tetrahedron";
+
+    ConformalMesh mesh_in;
+    ASSERT_TRUE(read_mesh(path, mesh_in))
+        << "read_mesh should succeed for the written OFF file";
+
+    EXPECT_EQ(mesh_in.number_of_vertices(), mesh_out.number_of_vertices());
+    EXPECT_EQ(mesh_in.number_of_faces(),    mesh_out.number_of_faces());
+    EXPECT_EQ(mesh_in.number_of_edges(),    mesh_out.number_of_edges());
+
+    rm(path);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// OFF round-trip: quad strip
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(MeshIO, OFF_RoundTrip_QuadStrip)
+{
+    auto path = tmp_path("off");
+    rm(path);
+
+    auto mesh_out = make_quad_strip();
+    ASSERT_TRUE(write_mesh(path, mesh_out));
+
+    ConformalMesh mesh_in;
+    ASSERT_TRUE(read_mesh(path, mesh_in));
+
+    EXPECT_EQ(mesh_in.number_of_vertices(), mesh_out.number_of_vertices());
+    EXPECT_EQ(mesh_in.number_of_faces(),    mesh_out.number_of_faces());
+
+    rm(path);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// OBJ round-trip: tetrahedron
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(MeshIO, OBJ_RoundTrip_Tetrahedron)
+{
+    auto path = tmp_path("obj");
+    rm(path);
+
+    auto mesh_out = make_tetrahedron();
+    ASSERT_TRUE(write_mesh(path, mesh_out))
+        << "write_mesh (OBJ) should succeed";
+
+    ConformalMesh mesh_in;
+    ASSERT_TRUE(read_mesh(path, mesh_in))
+        << "read_mesh (OBJ) should succeed";
+
+    EXPECT_EQ(mesh_in.number_of_vertices(), mesh_out.number_of_vertices());
+    EXPECT_EQ(mesh_in.number_of_faces(),    mesh_out.number_of_faces());
+
+    rm(path);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// load_mesh throws on missing file
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(MeshIO, LoadMeshThrowsOnMissingFile)
+{
+    EXPECT_THROW(load_mesh("/tmp/does_not_exist_conflab.off"), std::runtime_error);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Vertex positions survive a round-trip (OFF)
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(MeshIO, OFF_VertexPositionsPreserved)
+{
+    auto path = tmp_path("off");
+    rm(path);
+
+    auto mesh_out = make_tetrahedron();
+    ASSERT_TRUE(write_mesh(path, mesh_out));
+
+    ConformalMesh mesh_in;
+    ASSERT_TRUE(read_mesh(path, mesh_in));
+
+    // Collect and sort vertex positions from both meshes to compare
+    auto collect_sorted = [](const ConformalMesh& m) {
+        std::vector<std::array<double,3>> pts;
+        for (auto v : m.vertices()) {
+            auto p = m.point(v);
+            pts.push_back({CGAL::to_double(p.x()),
+                           CGAL::to_double(p.y()),
+                           CGAL::to_double(p.z())});
+        }
+        std::sort(pts.begin(), pts.end());
+        return pts;
+    };
+
+    auto pts_out = collect_sorted(mesh_out);
+    auto pts_in  = collect_sorted(mesh_in);
+
+    ASSERT_EQ(pts_out.size(), pts_in.size());
+    for (std::size_t i = 0; i < pts_out.size(); ++i) {
+        EXPECT_NEAR(pts_out[i][0], pts_in[i][0], 1e-10);
+        EXPECT_NEAR(pts_out[i][1], pts_in[i][1], 1e-10);
+        EXPECT_NEAR(pts_out[i][2], pts_in[i][2], 1e-10);
+    }
+
+    rm(path);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// save_mesh / load_mesh convenience wrappers
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(MeshIO, SaveLoadConvenienceWrappers)
+{
+    auto path = tmp_path("off");
+    rm(path);
+
+    auto mesh_out = make_quad_strip();
+    EXPECT_NO_THROW(save_mesh(path, mesh_out));
+
+    ConformalMesh mesh_in;
+    EXPECT_NO_THROW(mesh_in = load_mesh(path));
+
+    EXPECT_EQ(mesh_in.number_of_vertices(), mesh_out.number_of_vertices());
+
+    rm(path);
+}

code/tests/cgal/test_newton_phase9a.cpp

@@ -0,0 +1,267 @@
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
+// test_newton_phase9a.cpp
+//
+// Phase 9a Newton solvers — convergence tests for the two new
+// circle-packing functionals.
+//
+// Validates that:
+//   • newton_cp_euclidean()        — face-based BPS-2010 functional.
+//   • newton_inversive_distance()  — vertex-based Luo-2004 functional.
+// both reach a Newton equilibrium (‖G‖∞ < 1e-8) in < 30 iterations
+// on a range of test meshes, and that the converged solution satisfies
+// the relevant geometric invariants.
+
+#include "newton_solver.hpp"
+#include "cp_euclidean_functional.hpp"
+#include "inversive_distance_functional.hpp"
+#include "mesh_builder.hpp"
+#include "conformal_mesh.hpp"
+
+#include <gtest/gtest.h>
+#include <vector>
+
+using namespace conformallab;
+
+namespace {
+
+// Open 3-face mesh (tetrahedron minus one face) — exercises boundary edges.
+inline ConformalMesh make_open_3face_mesh()
+{
+    ConformalMesh mesh;
+    auto v0 = mesh.add_vertex(Point3( 1,  1,  1));
+    auto v1 = mesh.add_vertex(Point3( 1, -1, -1));
+    auto v2 = mesh.add_vertex(Point3(-1,  1, -1));
+    auto v3 = mesh.add_vertex(Point3(-1, -1,  1));
+    mesh.add_face(v0, v2, v1);
+    mesh.add_face(v0, v1, v3);
+    mesh.add_face(v0, v3, v2);
+    return mesh;
+}
+
+} // anonymous
+
+// ════════════════════════════════════════════════════════════════════════════
+// 1. CP-Euclidean Newton — orthogonal circle packing
+//
+// Setup matches CPEuclideanFunctionalTest.java (Java parity at the
+// solver level): θ_e = π/2 everywhere, φ_f = 2π for all faces.  Use
+// the "natural-phi" trick (analog of natural-theta in Euclidean):
+// adjust φ so that ρ = 0 is the natural equilibrium → Newton must
+// converge in zero iterations.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(NewtonPhase9a, CPEuclidean_NaturalPhi_ClosedTetrahedron_ConvergesInZeroIterations)
+{
+    auto mesh = make_tetrahedron();
+    auto m    = setup_cp_euclidean_maps(mesh);
+    const int n = assign_cp_euclidean_face_dof_indices(mesh, m);
+    ASSERT_EQ(n, 3);
+
+    // Natural-phi: shift φ_f so the gradient at ρ = 0 is zero.
+    std::vector<double> x0(static_cast<std::size_t>(n), 0.0);
+    auto G0 = cp_euclidean_gradient(mesh, x0, m);
+    for (auto f : mesh.faces()) {
+        int i = m.f_idx[f];
+        if (i < 0) continue;
+        m.phi_f[f] -= G0[static_cast<std::size_t>(i)];
+    }
+
+    auto res = newton_cp_euclidean(mesh, x0, m);
+    EXPECT_TRUE(res.converged);
+    EXPECT_EQ(res.iterations, 0)
+        << "natural-phi pre-shift should make x=0 the equilibrium";
+    EXPECT_LT(res.grad_inf_norm, 1e-10);
+    for (double r : res.x) EXPECT_NEAR(r, 0.0, 1e-12);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// 2. CP-Euclidean Newton — perturbed equilibrium converges back to 0
+//
+// Same setup as test 1, but start from a small perturbation.  The
+// strictly-convex BPS-2010 energy means Newton must converge back
+// to the natural-phi equilibrium ρ = 0.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(NewtonPhase9a, CPEuclidean_PerturbedStart_ConvergesBackToEquilibrium)
+{
+    auto mesh = make_tetrahedron();
+    auto m    = setup_cp_euclidean_maps(mesh);
+    const int n = assign_cp_euclidean_face_dof_indices(mesh, m);
+
+    // Apply natural-phi (equilibrium at ρ=0).
+    std::vector<double> x0_zero(static_cast<std::size_t>(n), 0.0);
+    auto G0 = cp_euclidean_gradient(mesh, x0_zero, m);
+    for (auto f : mesh.faces()) {
+        int i = m.f_idx[f];
+        if (i < 0) continue;
+        m.phi_f[f] -= G0[static_cast<std::size_t>(i)];
+    }
+
+    // Start from a perturbation.
+    std::vector<double> x0 = {0.1, -0.2, 0.15};
+    auto res = newton_cp_euclidean(mesh, x0, m);
+
+    EXPECT_TRUE(res.converged);
+    EXPECT_LT(res.iterations, 30);
+    EXPECT_LT(res.grad_inf_norm, 1e-8);
+    // Strictly-convex unique minimum → converges back to ρ=0.
+    for (double r : res.x) EXPECT_NEAR(r, 0.0, 1e-6);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// 3. CP-Euclidean Newton — open mesh (boundary edges)
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(NewtonPhase9a, CPEuclidean_OpenTetrahedron_NaturalPhi_Converges)
+{
+    auto mesh = make_open_3face_mesh();
+    auto m    = setup_cp_euclidean_maps(mesh);
+    const int n = assign_cp_euclidean_face_dof_indices(mesh, m);
+    ASSERT_EQ(n, 2);
+
+    std::vector<double> x0(static_cast<std::size_t>(n), 0.0);
+    auto G0 = cp_euclidean_gradient(mesh, x0, m);
+    for (auto f : mesh.faces()) {
+        int i = m.f_idx[f];
+        if (i < 0) continue;
+        m.phi_f[f] -= G0[static_cast<std::size_t>(i)];
+    }
+
+    auto res = newton_cp_euclidean(mesh, x0, m);
+    EXPECT_TRUE(res.converged);
+    EXPECT_LT(res.iterations, 30);
+    EXPECT_LT(res.grad_inf_norm, 1e-8);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// 4. Inversive-Distance Newton — natural-theta on triangle
+//
+// At u = 0, Bowers-Stephenson init reproduces the input edge lengths
+// exactly.  Natural-theta then shifts Θ so the gradient is zero, making
+// u = 0 the equilibrium.  Newton must converge in zero iterations.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(NewtonPhase9a, InversiveDistance_NaturalTheta_Triangle_ConvergesInZero)
+{
+    auto mesh = make_triangle();
+    auto m    = setup_inversive_distance_maps(mesh);
+    compute_inversive_distance_init_from_mesh(mesh, m);
+
+    int n = 0;
+    for (auto v : mesh.vertices()) m.v_idx[v] = n++;
+
+    std::vector<double> x0(static_cast<std::size_t>(n), 0.0);
+    auto G0 = inversive_distance_gradient(mesh, x0, m);
+    for (auto v : mesh.vertices()) {
+        int i = m.v_idx[v];
+        m.theta_v[v] -= G0[static_cast<std::size_t>(i)];
+    }
+
+    auto res = newton_inversive_distance(mesh, x0, m);
+    EXPECT_TRUE(res.converged);
+    EXPECT_EQ(res.iterations, 0);
+    EXPECT_LT(res.grad_inf_norm, 1e-10);
+    for (double u : res.x) EXPECT_NEAR(u, 0.0, 1e-12);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// 5. Inversive-Distance Newton — perturbed start on quad strip
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(NewtonPhase9a, InversiveDistance_PerturbedQuadStrip_Converges)
+{
+    auto mesh = make_quad_strip();
+    auto m    = setup_inversive_distance_maps(mesh);
+    compute_inversive_distance_init_from_mesh(mesh, m);
+
+    // Pin vertex 0; index the rest.
+    auto vit = mesh.vertices().begin();
+    m.v_idx[*vit++] = -1;
+    int n = 0;
+    for (; vit != mesh.vertices().end(); ++vit) m.v_idx[*vit] = n++;
+
+    // Natural-theta with the pin in place.
+    std::vector<double> x0(static_cast<std::size_t>(n), 0.0);
+    auto G0 = inversive_distance_gradient(mesh, x0, m);
+    for (auto v : mesh.vertices()) {
+        int i = m.v_idx[v];
+        if (i >= 0) m.theta_v[v] -= G0[static_cast<std::size_t>(i)];
+    }
+
+    // Perturb away from the equilibrium and watch it return.
+    std::vector<double> x_pert(static_cast<std::size_t>(n), -0.05);
+    auto res = newton_inversive_distance(mesh, x_pert, m);
+
+    EXPECT_TRUE(res.converged);
+    EXPECT_LT(res.iterations, 30);
+    EXPECT_LT(res.grad_inf_norm, 1e-8);
+    // Strictly-convex unique minimum on the open domain → back to 0.
+    for (double u : res.x) EXPECT_NEAR(u, 0.0, 1e-6);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// 6. Inversive-Distance Newton — tetrahedron (closed mesh)
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(NewtonPhase9a, InversiveDistance_PerturbedTetrahedron_Converges)
+{
+    auto mesh = make_tetrahedron();
+    auto m    = setup_inversive_distance_maps(mesh);
+    compute_inversive_distance_init_from_mesh(mesh, m);
+
+    // Closed mesh — pin one vertex to remove the gauge mode.
+    auto vit = mesh.vertices().begin();
+    m.v_idx[*vit++] = -1;
+    int n = 0;
+    for (; vit != mesh.vertices().end(); ++vit) m.v_idx[*vit] = n++;
+
+    std::vector<double> x0(static_cast<std::size_t>(n), 0.0);
+    auto G0 = inversive_distance_gradient(mesh, x0, m);
+    for (auto v : mesh.vertices()) {
+        int i = m.v_idx[v];
+        if (i >= 0) m.theta_v[v] -= G0[static_cast<std::size_t>(i)];
+    }
+
+    std::vector<double> x_pert(static_cast<std::size_t>(n), -0.1);
+    auto res = newton_inversive_distance(mesh, x_pert, m);
+
+    EXPECT_TRUE(res.converged);
+    EXPECT_LT(res.iterations, 30);
+    EXPECT_LT(res.grad_inf_norm, 1e-8);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// 7. CP-Euclidean Newton — uses analytic Hessian (NOT FD)
+//
+// Regression guard: verify the solver actually calls cp_euclidean_hessian
+// (the analytic 2×2-per-edge formula) rather than degenerating to a
+// per-iteration FD pass.  If iteration count exceeds a tight upper bound
+// for a tiny mesh, that would suggest a slow inner Hessian computation
+// or a wrong-sign mistake.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(NewtonPhase9a, CPEuclidean_UsesAnalyticHessian)
+{
+    auto mesh = make_tetrahedron();
+    auto m    = setup_cp_euclidean_maps(mesh);
+    const int n = assign_cp_euclidean_face_dof_indices(mesh, m);
+
+    std::vector<double> x0(static_cast<std::size_t>(n), 0.0);
+    auto G0 = cp_euclidean_gradient(mesh, x0, m);
+    for (auto f : mesh.faces()) {
+        int i = m.f_idx[f];
+        if (i < 0) continue;
+        m.phi_f[f] -= G0[static_cast<std::size_t>(i)];
+    }
+
+    // Strong perturbation — quadratic Newton with analytic Hessian
+    // should still converge in a handful of iterations.
+    std::vector<double> x_pert = {0.5, -0.4, 0.3};
+    auto res = newton_cp_euclidean(mesh, x_pert, m);
+
+    EXPECT_TRUE(res.converged);
+    EXPECT_LE(res.iterations, 10)
+        << "analytic Hessian: expect very fast convergence on a 3-DOF problem";
+}

code/tests/cgal/test_newton_solver.cpp

@@ -0,0 +1,503 @@
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
+// test_newton_solver.cpp
+//
+// Phase 4 — Newton solver tests.
+//
+// Design principle:
+//   We test convergence to a KNOWN equilibrium.  For the spherical tetrahedron
+//   x* = 0 is built-in (G(0) ≈ 0 by construction).  For Euclidean meshes we
+//   use "natural theta": set theta_v[v] = actual angle sum at x=0, which makes
+//   x* = 0 the exact equilibrium by definition.
+//   For HyperIdeal we use the same "natural target" trick at a valid base point
+//   (b=1.0, a=0.5), since x=0 is degenerate for the HyperIdeal functional.
+//
+// Tests:
+//   Spherical:
+//     1. Converges from x=[-0.2,...] to x*=0 (spherical tetrahedron).
+//     2. Converges in few iterations (quadratic convergence near equilibrium).
+//     3. Converges from a large perturbation x=[-0.5,...].
+//     4. Result fields (x.size, grad_inf_norm) are self-consistent.
+//
+//   Euclidean:
+//     5. Converges (triangle, 1 pinned vertex, natural theta).
+//     6. Converges (quad strip, 1 pinned vertex, natural theta).
+//     7. Converges with explicitly chosen mixed pinned/variable layout.
+//
+//   HyperIdeal:
+//     8.  Converges on triangle (all variable, natural targets).
+//     9.  Result fields self-consistent.
+//    10.  Converges on tetrahedron (10 DOFs, larger mesh).
+//    11.  SparseQR: result consistent (valid starting region).
+//
+//   SparseQR fallback (direct unit tests):
+//    12.  solve_linear_system recovers correct solution on rank-deficient matrix.
+//    13.  solve_linear_system sets fallback_used=true on a singular matrix.
+//    14.  Euclidean Newton on closed tetrahedron (no pinned vertex) converges
+//         via SparseQR gauge-mode handling.
+
+#include "conformal_mesh.hpp"
+#include "mesh_builder.hpp"
+#include "euclidean_functional.hpp"
+#include "spherical_functional.hpp"
+#include "hyper_ideal_functional.hpp"
+#include "newton_solver.hpp"
+#include <Eigen/Dense>
+#include <gtest/gtest.h>
+#include <cmath>
+#include <vector>
+
+using namespace conformallab;
+
+// ────────────────────────────────────────────────────────────────────────────
+// Helper: set theta_v[v] = actual angle sum at x=0 for each variable vertex.
+//
+// Requires that DOF indices have already been assigned (v_idx populated).
+// Uses euclidean_gradient directly so the formula is exact and consistent.
+// ────────────────────────────────────────────────────────────────────────────
+static void set_natural_euclidean_theta(ConformalMesh& mesh, EuclideanMaps& maps, int n)
+{
+    std::vector<double> x0(static_cast<std::size_t>(n), 0.0);
+    // G[iv] = theta_v[v] - sum_alpha(x=0)
+    // so  sum_alpha(x=0) = theta_v[v] - G[iv]
+    auto G = euclidean_gradient(mesh, x0, maps);
+    for (auto v : mesh.vertices()) {
+        int iv = maps.v_idx[v];
+        if (iv < 0) continue;
+        maps.theta_v[v] -= G[static_cast<std::size_t>(iv)];
+    }
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Spherical 1 — Converges from moderate perturbation
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(NewtonSolver, Spherical_ConvergesFromPerturbation)
+{
+    auto mesh = make_spherical_tetrahedron();
+    auto maps = setup_spherical_maps(mesh);
+    compute_lambda0_from_mesh(mesh, maps);
+    int n = assign_vertex_dof_indices(mesh, maps);
+
+    std::vector<double> x0(static_cast<std::size_t>(n), -0.2);
+    auto res = newton_spherical(mesh, x0, maps, /*tol=*/1e-8, /*max_iter=*/50);
+
+    EXPECT_TRUE(res.converged)
+        << "Newton (spherical) should converge; grad_inf_norm = " << res.grad_inf_norm;
+    EXPECT_LT(res.grad_inf_norm, 1e-8);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Spherical 2 — Quadratic convergence: few iterations suffice
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(NewtonSolver, Spherical_FewIterations)
+{
+    auto mesh = make_spherical_tetrahedron();
+    auto maps = setup_spherical_maps(mesh);
+    compute_lambda0_from_mesh(mesh, maps);
+    int n = assign_vertex_dof_indices(mesh, maps);
+
+    std::vector<double> x0(static_cast<std::size_t>(n), -0.2);
+    auto res = newton_spherical(mesh, x0, maps, /*tol=*/1e-8, /*max_iter=*/50);
+
+    EXPECT_LE(res.iterations, 20)
+        << "Newton should converge in ≤ 20 iterations; took " << res.iterations;
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Spherical 3 — Large perturbation: global convergence via line search
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(NewtonSolver, Spherical_ConvergesFromLargePerturbation)
+{
+    auto mesh = make_spherical_tetrahedron();
+    auto maps = setup_spherical_maps(mesh);
+    compute_lambda0_from_mesh(mesh, maps);
+    int n = assign_vertex_dof_indices(mesh, maps);
+
+    std::vector<double> x0(static_cast<std::size_t>(n), -0.5);
+    auto res = newton_spherical(mesh, x0, maps, /*tol=*/1e-8, /*max_iter=*/100);
+
+    EXPECT_TRUE(res.converged)
+        << "Newton (spherical, large perturbation) should converge; "
+           "grad_inf_norm = " << res.grad_inf_norm;
+    EXPECT_LT(res.grad_inf_norm, 1e-8);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Spherical 4 — Result fields are self-consistent
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(NewtonSolver, Spherical_ResultFieldsConsistent)
+{
+    auto mesh = make_spherical_tetrahedron();
+    auto maps = setup_spherical_maps(mesh);
+    compute_lambda0_from_mesh(mesh, maps);
+    int n = assign_vertex_dof_indices(mesh, maps);
+
+    std::vector<double> x0(static_cast<std::size_t>(n), -0.1);
+    auto res = newton_spherical(mesh, x0, maps, /*tol=*/1e-8);
+
+    EXPECT_EQ(static_cast<int>(res.x.size()), n);
+
+    // Reported grad_inf_norm must match re-computed gradient at res.x
+    auto G = spherical_gradient(mesh, res.x, maps);
+    double actual_inf = 0.0;
+    for (double v : G) actual_inf = std::max(actual_inf, std::abs(v));
+    EXPECT_NEAR(actual_inf, res.grad_inf_norm, 1e-10);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Euclidean 1 — Triangle with 1 pinned vertex + natural theta
+//
+// make_triangle(): 3 vertices.  Pin v0 → 2 free DOFs.
+// With natural theta: x* = 0 (G(0) = 0 by construction).
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(NewtonSolver, Euclidean_ConvergesTrianglePinned)
+{
+    auto mesh = make_triangle();
+    auto maps = setup_euclidean_maps(mesh);
+    compute_euclidean_lambda0_from_mesh(mesh, maps);
+
+    // Pin first vertex, assign sequential DOFs to the other two
+    auto vit = mesh.vertices().begin();
+    Vertex_index v0 = *vit++;
+    maps.v_idx[v0] = -1;     // pinned
+    int idx = 0;
+    for (; vit != mesh.vertices().end(); ++vit)
+        maps.v_idx[*vit] = idx++;
+    int n = idx;             // = 2
+
+    set_natural_euclidean_theta(mesh, maps, n);
+
+    std::vector<double> x0(static_cast<std::size_t>(n), -0.1);
+    auto res = newton_euclidean(mesh, x0, maps, /*tol=*/1e-8, /*max_iter=*/50);
+
+    EXPECT_TRUE(res.converged)
+        << "Newton (Euclidean, triangle, pinned) should converge; "
+           "grad_inf_norm = " << res.grad_inf_norm;
+    EXPECT_LT(res.grad_inf_norm, 1e-8);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Euclidean 2 — Quad strip with 1 pinned vertex + natural theta
+//
+// make_quad_strip(): 4 vertices, 2 faces.  Pin v0 → 3 free DOFs.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(NewtonSolver, Euclidean_ConvergesQuadStripPinned)
+{
+    auto mesh = make_quad_strip();
+    auto maps = setup_euclidean_maps(mesh);
+    compute_euclidean_lambda0_from_mesh(mesh, maps);
+
+    // Pin first vertex
+    auto vit = mesh.vertices().begin();
+    Vertex_index v0 = *vit++;
+    maps.v_idx[v0] = -1;
+    int idx = 0;
+    for (; vit != mesh.vertices().end(); ++vit)
+        maps.v_idx[*vit] = idx++;
+    int n = idx;             // = 3
+
+    set_natural_euclidean_theta(mesh, maps, n);
+
+    std::vector<double> x0(static_cast<std::size_t>(n), -0.15);
+    auto res = newton_euclidean(mesh, x0, maps, /*tol=*/1e-8, /*max_iter=*/50);
+
+    EXPECT_TRUE(res.converged)
+        << "Newton (Euclidean, quad strip, pinned) should converge; "
+           "grad_inf_norm = " << res.grad_inf_norm;
+    EXPECT_LT(res.grad_inf_norm, 1e-8);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Euclidean 3 — Mixed pinned layout: explicit vertex assignment
+//
+// Quad strip: v0 pinned, v1/v2/v3 free.
+// Natural theta set AFTER DOF assignment so that x* = 0 is the equilibrium
+// for the free vertices (with v0 fixed at u0=0).
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(NewtonSolver, Euclidean_ConvergesMixedPinned)
+{
+    auto mesh = make_quad_strip();
+    auto maps = setup_euclidean_maps(mesh);
+    compute_euclidean_lambda0_from_mesh(mesh, maps);
+
+    // Explicitly assign DOF indices
+    auto vit = mesh.vertices().begin();
+    Vertex_index v0 = *vit++;
+    Vertex_index v1 = *vit++;
+    Vertex_index v2 = *vit++;
+    Vertex_index v3 = *vit;
+
+    maps.v_idx[v0] = -1;   // pinned at u0 = 0
+    maps.v_idx[v1] =  0;
+    maps.v_idx[v2] =  1;
+    maps.v_idx[v3] =  2;
+    const int n = 3;
+
+    // Set natural theta AFTER pinning so that x* = [0,0,0] is the equilibrium
+    set_natural_euclidean_theta(mesh, maps, n);
+
+    std::vector<double> x0 = {-0.1, -0.15, -0.05};
+    auto res = newton_euclidean(mesh, x0, maps, /*tol=*/1e-8, /*max_iter=*/50);
+
+    EXPECT_TRUE(res.converged)
+        << "Newton (Euclidean, mixed pinned) should converge; "
+           "grad_inf_norm = " << res.grad_inf_norm;
+    EXPECT_LT(res.grad_inf_norm, 1e-8);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Helper: set HyperIdeal target angles to actual sums at a non-degenerate
+// base point (b_base, a_base), making that point the equilibrium x*.
+//
+// Note: x = 0 is degenerate for the HyperIdeal functional (log-space; the
+// functional requires b_i > 0 / a_e > 0).  We therefore choose a valid base
+// point, evaluate G there, and absorb G into the targets so that G(xbase) = 0.
+// Newton tests then start from a perturbation of xbase.
+//
+// Returns xbase so callers can construct a perturbed starting point.
+// ════════════════════════════════════════════════════════════════════════════
+static std::vector<double> set_natural_hyper_ideal_targets(
+    ConformalMesh& mesh, HyperIdealMaps& maps, int n,
+    double b_base = 1.0, double a_base = 0.5)
+{
+    const auto sz = static_cast<std::size_t>(n);
+    std::vector<double> xbase(sz, 0.0);
+
+    for (auto v : mesh.vertices()) {
+        int iv = maps.v_idx[v];
+        if (iv >= 0) xbase[static_cast<std::size_t>(iv)] = b_base;
+    }
+    for (auto e : mesh.edges()) {
+        int ie = maps.e_idx[e];
+        if (ie >= 0) xbase[static_cast<std::size_t>(ie)] = a_base;
+    }
+
+    // G = Σβ - theta_target (initial target = 0 → G = Σβ = "actual" angles)
+    auto G = evaluate_hyper_ideal(mesh, xbase, maps, /*energy=*/false).gradient;
+
+    // Set target := actual so that G(xbase) = actual - target = 0
+    for (auto v : mesh.vertices()) {
+        int iv = maps.v_idx[v];
+        if (iv < 0) continue;
+        maps.theta_v[v] += G[static_cast<std::size_t>(iv)];
+    }
+    for (auto e : mesh.edges()) {
+        int ie = maps.e_idx[e];
+        if (ie < 0) continue;
+        maps.theta_e[e] += G[static_cast<std::size_t>(ie)];
+    }
+
+    return xbase;
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// HyperIdeal 1 — Triangle, all DOFs variable, converges from perturbation
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(NewtonSolver, HyperIdeal_ConvergesTriangleAllVariable)
+{
+    auto mesh = make_triangle();
+    auto maps = setup_hyper_ideal_maps(mesh);
+    int n = assign_all_dof_indices(mesh, maps);
+
+    // xbase = (b=1.0, a=0.5) is the equilibrium after natural-target setup.
+    auto xbase = set_natural_hyper_ideal_targets(mesh, maps, n);
+
+    // Perturb by +0.2 uniformly
+    std::vector<double> x0 = xbase;
+    for (auto& v : x0) v += 0.2;
+
+    auto res = newton_hyper_ideal(mesh, x0, maps, /*tol=*/1e-7, /*max_iter=*/100);
+
+    EXPECT_TRUE(res.converged)
+        << "Newton (HyperIdeal, triangle) should converge; "
+           "grad_inf_norm = " << res.grad_inf_norm;
+    EXPECT_LT(res.grad_inf_norm, 1e-7);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// HyperIdeal 2 — Result fields self-consistent
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(NewtonSolver, HyperIdeal_ResultFieldsConsistent)
+{
+    auto mesh = make_triangle();
+    auto maps = setup_hyper_ideal_maps(mesh);
+    int n = assign_all_dof_indices(mesh, maps);
+
+    auto xbase = set_natural_hyper_ideal_targets(mesh, maps, n);
+
+    std::vector<double> x0 = xbase;
+    for (auto& v : x0) v += 0.1;
+
+    auto res = newton_hyper_ideal(mesh, x0, maps, /*tol=*/1e-7, /*max_iter=*/100);
+
+    EXPECT_EQ(static_cast<int>(res.x.size()), n);
+
+    // Reported grad_inf_norm must match re-computed gradient at res.x
+    auto G = evaluate_hyper_ideal(mesh, res.x, maps, false).gradient;
+    double actual_inf = 0.0;
+    for (double v : G) actual_inf = std::max(actual_inf, std::abs(v));
+    EXPECT_NEAR(actual_inf, res.grad_inf_norm, 1e-9);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// HyperIdeal 3 — Tetrahedron (10 DOFs): 4 vertex b-vals + 6 edge a-vals
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(NewtonSolver, HyperIdeal_ConvergesTetrahedron)
+{
+    auto mesh = make_tetrahedron();
+    auto maps = setup_hyper_ideal_maps(mesh);
+    int n = assign_all_dof_indices(mesh, maps);
+
+    auto xbase = set_natural_hyper_ideal_targets(mesh, maps, n);
+
+    // Perturb by +0.15
+    std::vector<double> x0 = xbase;
+    for (auto& v : x0) v += 0.15;
+
+    auto res = newton_hyper_ideal(mesh, x0, maps, /*tol=*/1e-7, /*max_iter=*/200);
+
+    EXPECT_TRUE(res.converged)
+        << "Newton (HyperIdeal, tetrahedron) should converge; "
+           "grad_inf_norm = " << res.grad_inf_norm;
+    EXPECT_LT(res.grad_inf_norm, 1e-7);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// HyperIdeal 4 — SparseQR fallback: solver returns a result (no crash)
+//
+// With all targets = 0 the equilibrium is not at x=0 but the solver should
+// at minimum not crash and return a consistent result struct.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(NewtonSolver, HyperIdeal_SparseQRFallbackNoCrash)
+{
+    auto mesh = make_triangle();
+    auto maps = setup_hyper_ideal_maps(mesh);
+    int n = assign_all_dof_indices(mesh, maps);
+
+    // Leave targets at their default (0): solver tries to solve but the
+    // "equilibrium" is at some unknown x*.  With valid starting point the
+    // Hessian is positive-definite and the solver should not crash.
+    // We don't assert convergence — just that the result struct is consistent.
+    std::vector<double> x0(static_cast<std::size_t>(n), 1.0);
+    // Mix vertex / edge DOFs: b=1.0, a=0.5 (valid region of the functional)
+    for (auto e : mesh.edges()) {
+        int ie = maps.e_idx[e];
+        if (ie >= 0) x0[static_cast<std::size_t>(ie)] = 0.5;
+    }
+    auto res = newton_hyper_ideal(mesh, x0, maps, /*tol=*/1e-7, /*max_iter=*/50);
+
+    // Struct fields must always be populated
+    EXPECT_EQ(static_cast<int>(res.x.size()), n);
+    EXPECT_GE(res.iterations, 0);
+    EXPECT_FALSE(std::isnan(res.grad_inf_norm));
+    EXPECT_FALSE(std::isinf(res.grad_inf_norm));
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// SparseQR fallback — Test 12: solve_linear_system recovers correct solution
+//
+// The public API solve_linear_system(A, rhs) must return the correct answer
+// for a well-conditioned full-rank system (LDLT path taken).
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(SparseQRFallback, FullRankSystem_CorrectSolution)
+{
+    // Build a simple 3×3 diagonal PD matrix: A = diag(1, 2, 3)
+    Eigen::SparseMatrix<double> A(3, 3);
+    A.insert(0, 0) = 1.0;
+    A.insert(1, 1) = 2.0;
+    A.insert(2, 2) = 3.0;
+    A.makeCompressed();
+
+    Eigen::VectorXd rhs(3);
+    rhs << 1.0, 4.0, 9.0;   // solution = [1, 2, 3]
+
+    bool fallback = true;     // expect it to be set to false (LDLT succeeds)
+    Eigen::VectorXd x = conformallab::solve_linear_system(A, rhs, &fallback);
+
+    EXPECT_FALSE(fallback) << "Full-rank system: LDLT should succeed (no SparseQR needed)";
+    EXPECT_NEAR(x[0], 1.0, 1e-12);
+    EXPECT_NEAR(x[1], 2.0, 1e-12);
+    EXPECT_NEAR(x[2], 3.0, 1e-12);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// SparseQR fallback — Test 13: fallback_used=true on a singular matrix
+//
+// Construct a symmetric 3×3 matrix of rank 1 where LDLT fails (the (2,2)
+// pivot is zero).  SparseQR finds the minimum-norm least-squares solution.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(SparseQRFallback, SingularMatrix_FallbackActivated)
+{
+    // A = [[2, 0, 0],
+    //      [0, 0, 0],   ← zero pivot → LDLT failure
+    //      [0, 0, 3]]
+    // rhs compatible with the row space: [2, 0, 3] → solution [1, 0, 1]
+    Eigen::SparseMatrix<double> A(3, 3);
+    A.insert(0, 0) = 2.0;
+    // row/col 1 deliberately all-zero
+    A.insert(2, 2) = 3.0;
+    A.makeCompressed();
+
+    Eigen::VectorXd rhs(3);
+    rhs << 2.0, 0.0, 3.0;
+
+    bool fallback = false;
+    Eigen::VectorXd x = conformallab::solve_linear_system(A, rhs, &fallback);
+
+    EXPECT_TRUE(fallback) << "Singular matrix: SparseQR fallback must be triggered";
+    // SparseQR min-norm solution: x[0]=1, x[1]=0, x[2]=1
+    EXPECT_NEAR(x[0], 1.0, 1e-10);
+    EXPECT_NEAR(x[1], 0.0, 1e-10);
+    EXPECT_NEAR(x[2], 1.0, 1e-10);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// SparseQR fallback — Test 14: Euclidean Newton on a closed mesh, no pinning
+//
+// make_tetrahedron() is a closed surface (4 vertices, 4 faces).  Without a
+// pinned vertex the Euclidean Hessian has a 1-D null space (uniform scale
+// gauge mode): H·1 = 0.  SimplicialLDLT fails on this rank-deficient H;
+// SparseQR finds the min-norm Newton step orthogonal to the null space.
+//
+// The gradient always lives in the row space of H (Σ G_v = 0 by angle-sum
+// invariance), so the SparseQR step is also the Newton step and the solver
+// converges to the natural equilibrium.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(SparseQRFallback, Euclidean_ClosedMeshNoPinConverges)
+{
+    auto mesh = make_tetrahedron();
+    auto maps = setup_euclidean_maps(mesh);
+    compute_euclidean_lambda0_from_mesh(mesh, maps);
+
+    // Assign all 4 vertices as free DOFs (no pinning).
+    int idx = 0;
+    for (auto v : mesh.vertices())
+        maps.v_idx[v] = idx++;
+    const int n = idx;   // = 4
+
+    // Natural theta: equilibrium at x* = 0.
+    set_natural_euclidean_theta(mesh, maps, n);
+
+    std::vector<double> x0(static_cast<std::size_t>(n), -0.1);
+    auto res = newton_euclidean(mesh, x0, maps, /*tol=*/1e-8, /*max_iter=*/100);
+
+    EXPECT_TRUE(res.converged)
+        << "Euclidean Newton on closed tetrahedron (no pin) must converge via SparseQR; "
+           "grad_inf_norm = " << res.grad_inf_norm;
+    EXPECT_LT(res.grad_inf_norm, 1e-8);
+}

code/tests/cgal/test_phase6.cpp

@@ -0,0 +1,409 @@
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
+// test_phase6.cpp
+//
+// Phase 6 — Tests for:
+//   - gauss_bonnet.hpp  : euler_characteristic, genus, GB sum/rhs, check, enforce
+//   - cut_graph.hpp     : compute_cut_graph (tree-cotree algorithm)
+//   - layout.hpp Phase6 : exact hyperbolic trilateration math
+//                         normalise_euclidean (centroid + PCA)
+//                         normalise_hyperbolic (Möbius centering)
+
+#include "conformal_mesh.hpp"
+#include "mesh_builder.hpp"
+#include "euclidean_functional.hpp"
+#include "spherical_functional.hpp"
+#include "hyper_ideal_functional.hpp"
+#include "newton_solver.hpp"
+#include "gauss_bonnet.hpp"
+#include "cut_graph.hpp"
+#include "layout.hpp"
+#include <gtest/gtest.h>
+#include <cmath>
+#include <complex>
+#include <vector>
+
+using namespace conformallab;
+
+// mesh_builder.hpp provides make_triangle(), make_tetrahedron(), make_quad_strip().
+
+// ════════════════════════════════════════════════════════════════════════════
+// GaussBonnet — topology helpers
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(GaussBonnet, EulerCharacteristic_Triangle)
+{
+    // V=3 E=3 F=1  →  χ = 1
+    auto m = make_triangle();
+    EXPECT_EQ(euler_characteristic(m), 1);
+}
+
+TEST(GaussBonnet, EulerCharacteristic_QuadStrip)
+{
+    // V=4 E=5 F=2  →  χ = 1
+    auto m = make_quad_strip();
+    EXPECT_EQ(euler_characteristic(m), 1);
+}
+
+TEST(GaussBonnet, EulerCharacteristic_Tetrahedron)
+{
+    // V=4 E=6 F=4  →  χ = 2
+    auto m = make_tetrahedron();
+    EXPECT_EQ(euler_characteristic(m), 2);
+}
+
+TEST(GaussBonnet, Genus_Tetrahedron_IsZero)
+{
+    auto m = make_tetrahedron();
+    EXPECT_EQ(genus(m), 0);
+}
+
+TEST(GaussBonnet, RHS_Triangle)
+{
+    // χ=1  →  2π·χ = 2π
+    auto m = make_triangle();
+    EXPECT_NEAR(gauss_bonnet_rhs(m), 2.0 * M_PI, 1e-12);
+}
+
+TEST(GaussBonnet, RHS_Tetrahedron)
+{
+    // χ=2  →  2π·χ = 4π
+    auto m = make_tetrahedron();
+    EXPECT_NEAR(gauss_bonnet_rhs(m), 4.0 * M_PI, 1e-12);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// GaussBonnet — sum / deficit / check / enforce
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(GaussBonnet, DeficitIsNonZeroWithDefaultTheta)
+{
+    // Default theta_v = 2π at all V=4 vertices of quad-strip  →
+    //   Σ(2π−2π) = 0,  rhs = 2π·1 = 2π  →  deficit = −2π ≠ 0.
+    auto m = make_quad_strip();
+    auto maps = setup_euclidean_maps(m);
+    double def = gauss_bonnet_deficit(m, maps);
+    EXPECT_GT(std::abs(def), 1e-6);
+}
+
+TEST(GaussBonnet, EnforceAdjustsTheta_Deficit_BecomesZero)
+{
+    auto m = make_quad_strip();
+    auto maps = setup_euclidean_maps(m);
+    // Set clearly wrong theta
+    for (auto v : m.vertices()) maps.theta_v[v] = M_PI;
+    EXPECT_GT(std::abs(gauss_bonnet_deficit(m, maps)), 1e-6);
+
+    enforce_gauss_bonnet(m, maps);
+    EXPECT_NEAR(gauss_bonnet_deficit(m, maps), 0.0, 1e-10);
+}
+
+TEST(GaussBonnet, CheckThrowsWhenViolated)
+{
+    auto m = make_triangle();
+    auto maps = setup_euclidean_maps(m);
+    // theta_v = 2π everywhere  →  sum = 0, rhs = 2π  →  |deficit| = 2π
+    EXPECT_THROW(check_gauss_bonnet(m, maps), std::runtime_error);
+}
+
+TEST(GaussBonnet, CheckPassesAfterEnforce)
+{
+    auto m = make_triangle();
+    auto maps = setup_euclidean_maps(m);
+    enforce_gauss_bonnet(m, maps);
+    EXPECT_NO_THROW(check_gauss_bonnet(m, maps));
+}
+
+TEST(GaussBonnet, SumAfterEnforce_EqualsRHS)
+{
+    auto m = make_tetrahedron();
+    auto maps = setup_euclidean_maps(m);
+    // theta_v starts at 2π everywhere; sum = 0, rhs = 4π
+    enforce_gauss_bonnet(m, maps);
+    double lhs = gauss_bonnet_sum(m, maps);
+    double rhs = gauss_bonnet_rhs(m);
+    EXPECT_NEAR(lhs, rhs, 1e-10);
+}
+
+TEST(GaussBonnet, ManuallySetAnalyticalTheta_PassesCheck)
+{
+    // Single triangle (χ=1): need Σ(2π−Θ_v) = 2π.
+    // Set each of the 3 boundary vertices to Θ_v = 4π/3.
+    // Then Σ(2π − 4π/3) = 3·(2π/3) = 2π. ✓
+    auto m = make_triangle();
+    auto maps = setup_euclidean_maps(m);
+    for (auto v : m.vertices()) maps.theta_v[v] = 4.0 * M_PI / 3.0;
+    EXPECT_NO_THROW(check_gauss_bonnet(m, maps));
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// CutGraph — tree-cotree algorithm
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(CutGraph, Triangle_ZeroCutEdges)
+{
+    // Open disk → genus 0 → 0 cut edges.
+    auto m = make_triangle();
+    auto cg = compute_cut_graph(m);
+    EXPECT_EQ(cg.cut_edge_indices.size(), 0u);
+}
+
+TEST(CutGraph, QuadStrip_ZeroCutEdges)
+{
+    // Open disk → genus 0 → 0 cut edges.
+    auto m = make_quad_strip();
+    auto cg = compute_cut_graph(m);
+    EXPECT_EQ(cg.cut_edge_indices.size(), 0u);
+}
+
+TEST(CutGraph, Tetrahedron_ZeroCutEdges_ClosedSphere)
+{
+    // Closed genus-0 surface → 2g = 0 cut edges.
+    auto m = make_tetrahedron();
+    auto cg = compute_cut_graph(m);
+    EXPECT_EQ(cg.cut_edge_indices.size(), 0u);
+    EXPECT_EQ(cg.genus, 0);
+}
+
+TEST(CutGraph, FlagsVsIndicesConsistent)
+{
+    // Every index in cut_edge_indices has flag=true;
+    // count of true flags == number of indices.
+    auto m = make_tetrahedron();
+    auto cg = compute_cut_graph(m);
+    for (std::size_t idx : cg.cut_edge_indices)
+        EXPECT_TRUE(cg.cut_edge_flags[idx]);
+
+    std::size_t count = 0;
+    for (bool b : cg.cut_edge_flags) count += b ? 1u : 0u;
+    EXPECT_EQ(count, cg.cut_edge_indices.size());
+}
+
+TEST(CutGraph, IsCutMethodMatchesFlags)
+{
+    auto m = make_tetrahedron();
+    auto cg = compute_cut_graph(m);
+    for (auto e : m.edges()) {
+        bool flag = cg.cut_edge_flags[static_cast<std::size_t>(e.idx())];
+        EXPECT_EQ(cg.is_cut(e), flag);
+    }
+}
+
+TEST(CutGraph, FlagsVectorHasCorrectSize)
+{
+    auto m = make_quad_strip();
+    auto cg = compute_cut_graph(m);
+    EXPECT_EQ(cg.cut_edge_flags.size(), m.number_of_edges());
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Exact hyperbolic trilateration — Möbius + law of cosines
+// ════════════════════════════════════════════════════════════════════════════
+
+namespace {
+
+/// Poincaré-disk hyperbolic distance.
+double hyp_dist_disk(Eigen::Vector2d a, Eigen::Vector2d b)
+{
+    double num = (a.x()-b.x())*(a.x()-b.x()) + (a.y()-b.y())*(a.y()-b.y());
+    double da  = 1.0 - a.x()*a.x() - a.y()*a.y();
+    double db_  = 1.0 - b.x()*b.x() - b.y()*b.y();
+    double arg = 1.0 + 2.0*num/(da*db_);
+    return std::acosh(std::max(1.0, arg));
+}
+
+/// Reproduce the exact trilateration formula from layout.hpp detail namespace.
+Eigen::Vector2d trilaterate_hyp(Eigen::Vector2d pa, Eigen::Vector2d pb,
+                                  double D, double da, double db)
+{
+    if (D < 1e-12 || da < 1e-12) return pa;
+    if (std::sinh(da) * std::sinh(D) < 1e-14) return pa;
+    using C = std::complex<double>;
+    auto mobius_fwd = [](C z, C center) -> C {
+        return (z - center) / (C(1.0) - std::conj(center) * z);
+    };
+    auto mobius_inv = [](C w, C center) -> C {
+        return (w + center) / (C(1.0) + std::conj(center) * w);
+    };
+    C a(pa.x(), pa.y()), b(pb.x(), pb.y());
+    C b_mapped = mobius_fwd(b, a);
+    double theta_b = std::arg(b_mapped);
+    double cos_alpha = (std::cosh(da) * std::cosh(D) - std::cosh(db))
+                     / (std::sinh(da) * std::sinh(D));
+    cos_alpha = std::max(-1.0, std::min(1.0, cos_alpha));
+    double alpha = std::acos(cos_alpha);
+    C pc_origin = std::tanh(da * 0.5) * std::exp(C(0.0, theta_b + alpha));
+    C pc_c = mobius_inv(pc_origin, a);
+    return Eigen::Vector2d(pc_c.real(), pc_c.imag());
+}
+
+} // namespace
+
+TEST(HyperbolicTrilateration, RecoveredDistances_Small)
+{
+    // pa at origin, pb at tanh(D/2) on x-axis.
+    double D = 0.8, da = 0.5, db = 0.6;
+    Eigen::Vector2d pa(0.0, 0.0);
+    Eigen::Vector2d pb(std::tanh(D * 0.5), 0.0);
+
+    Eigen::Vector2d pc = trilaterate_hyp(pa, pb, D, da, db);
+
+    EXPECT_NEAR(hyp_dist_disk(pa, pb), D,  1e-10);
+    EXPECT_NEAR(hyp_dist_disk(pa, pc), da, 1e-9);
+    EXPECT_NEAR(hyp_dist_disk(pb, pc), db, 1e-9);
+}
+
+TEST(HyperbolicTrilateration, RecoveredDistances_Large)
+{
+    double D = 2.0, da = 1.5, db = 1.0;
+    Eigen::Vector2d pa(0.0, 0.0);
+    Eigen::Vector2d pb(std::tanh(D * 0.5), 0.0);
+
+    Eigen::Vector2d pc = trilaterate_hyp(pa, pb, D, da, db);
+
+    EXPECT_NEAR(hyp_dist_disk(pa, pc), da, 1e-8);
+    EXPECT_NEAR(hyp_dist_disk(pb, pc), db, 1e-8);
+}
+
+TEST(HyperbolicTrilateration, ResultInsidePoincareDisk)
+{
+    double D = 1.2, da = 0.9, db = 0.7;
+    Eigen::Vector2d pa(0.0, 0.0);
+    Eigen::Vector2d pb(std::tanh(D * 0.5), 0.0);
+
+    Eigen::Vector2d pc = trilaterate_hyp(pa, pb, D, da, db);
+    EXPECT_LT(pc.norm(), 1.0);
+}
+
+TEST(HyperbolicTrilateration, OffOrigin_StillCorrect)
+{
+    // Place pa somewhere in the disk (not at origin) to test Möbius map.
+    double D = 0.6, da = 0.4, db = 0.5;
+    // pa at (0.3, 0.0) in Poincaré coords.
+    Eigen::Vector2d pa(0.3, 0.0);
+    // pb at distance D from pa — compute pb in Poincaré disk:
+    // Möbius inverse: tanh(D/2) maps back from origin frame.
+    using C = std::complex<double>;
+    C a_c(pa.x(), pa.y());
+    C pb_c = (std::tanh(D*0.5) + a_c) / (C(1.0) + std::conj(a_c) * std::tanh(D*0.5));
+    Eigen::Vector2d pb(pb_c.real(), pb_c.imag());
+
+    Eigen::Vector2d pc = trilaterate_hyp(pa, pb, D, da, db);
+    EXPECT_LT(pc.norm(), 1.0);
+    EXPECT_NEAR(hyp_dist_disk(pa, pc), da, 1e-8);
+    EXPECT_NEAR(hyp_dist_disk(pb, pc), db, 1e-8);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Normalisation — euclidean_layout (centroid) + hyper_ideal_layout (Möbius)
+// ════════════════════════════════════════════════════════════════════════════
+
+/// Build a solved euclidean layout for the quad-strip at the natural equilibrium.
+static Layout2D make_euclidean_layout_normalised(bool normalise)
+{
+    auto mesh = make_quad_strip();
+    auto maps = setup_euclidean_maps(mesh);
+    compute_euclidean_lambda0_from_mesh(mesh, maps);
+
+    // Pin first vertex; assign free DOF indices to all others.
+    auto vit = mesh.vertices().begin();
+    maps.v_idx[*vit++] = -1;
+    int idx = 0;
+    for (; vit != mesh.vertices().end(); ++vit) maps.v_idx[*vit] = idx++;
+    const int n = idx;
+
+    std::vector<double> x0(static_cast<std::size_t>(n), 0.0);
+    // Adjust theta so G(x=0) = 0 (natural equilibrium).
+    auto G0 = euclidean_gradient(mesh, x0, maps);
+    for (auto v : mesh.vertices()) {
+        int iv = maps.v_idx[v];
+        if (iv >= 0) maps.theta_v[v] -= G0[static_cast<std::size_t>(iv)];
+    }
+
+    auto res = newton_euclidean(mesh, x0, maps, 1e-10, 100);
+    return euclidean_layout(mesh, res.x, maps, nullptr, nullptr, normalise);
+}
+
+TEST(Normalisation, Euclidean_CentroidAtOrigin)
+{
+    auto L = make_euclidean_layout_normalised(true);
+    ASSERT_GT(L.uv.size(), 0u);
+
+    double cx = 0.0, cy = 0.0;
+    for (auto& p : L.uv) { cx += p.x(); cy += p.y(); }
+    int n = static_cast<int>(L.uv.size());
+    EXPECT_NEAR(cx / n, 0.0, 1e-9);
+    EXPECT_NEAR(cy / n, 0.0, 1e-9);
+}
+
+TEST(Normalisation, Euclidean_NormalisePreservesEdgeLengthRatios)
+{
+    auto L_no  = make_euclidean_layout_normalised(false);
+    auto L_yes = make_euclidean_layout_normalised(true);
+
+    // Ratio of the lengths of the first two edges must be preserved.
+    ASSERT_GE(L_no.uv.size(), 4u);
+    // edges 0→1 and 0→2 (the two edges from vertex 0 in the quad-strip)
+    double len0_no  = (L_no.uv[1]  - L_no.uv[0]).norm();
+    double len1_no  = (L_no.uv[2]  - L_no.uv[0]).norm();
+    double len0_yes = (L_yes.uv[1] - L_yes.uv[0]).norm();
+    double len1_yes = (L_yes.uv[2] - L_yes.uv[0]).norm();
+
+    if (len1_no > 1e-12 && len1_yes > 1e-12) {
+        double ratio_no  = len0_no  / len1_no;
+        double ratio_yes = len0_yes / len1_yes;
+        EXPECT_NEAR(ratio_no, ratio_yes, 1e-9);
+    }
+}
+
+/// Build a solved hyper-ideal layout for a triangle at natural equilibrium.
+static Layout2D make_hyper_ideal_layout_normalised(bool normalise)
+{
+    auto mesh = make_triangle();
+    auto maps = setup_hyper_ideal_maps(mesh);
+    int n = assign_all_dof_indices(mesh, maps);
+
+    std::vector<double> xbase(static_cast<std::size_t>(n));
+    for (auto v : mesh.vertices()) {
+        int iv = maps.v_idx[v]; if (iv >= 0) xbase[iv] = 1.0;
+    }
+    for (auto e : mesh.edges()) {
+        int ie = maps.e_idx[e]; if (ie >= 0) xbase[ie] = 0.5;
+    }
+    auto G0 = evaluate_hyper_ideal(mesh, xbase, maps, false).gradient;
+    for (auto v : mesh.vertices()) {
+        int iv = maps.v_idx[v]; if (iv >= 0) maps.theta_v[v] += G0[iv];
+    }
+    for (auto e : mesh.edges()) {
+        int ie = maps.e_idx[e]; if (ie >= 0) maps.theta_e[e] += G0[ie];
+    }
+    auto res = newton_hyper_ideal(mesh, xbase, maps, 1e-9, 100);
+    return hyper_ideal_layout(mesh, res.x, maps, nullptr, nullptr, normalise);
+}
+
+TEST(Normalisation, HyperIdeal_PositionsInsidePoincareDisk)
+{
+    auto L = make_hyper_ideal_layout_normalised(true);
+    ASSERT_GT(L.uv.size(), 0u);
+    for (auto& p : L.uv) {
+        EXPECT_LT(p.norm(), 1.0 + 1e-9)
+            << "Vertex outside Poincaré disk: r=" << p.norm();
+    }
+}
+
+TEST(Normalisation, HyperIdeal_CenteringReducesAverageRadius)
+{
+    // After Möbius centering the average Euclidean radius inside the disk
+    // must be ≤ the unconstrained average.
+    auto L_no  = make_hyper_ideal_layout_normalised(false);
+    auto L_yes = make_hyper_ideal_layout_normalised(true);
+
+    int n = static_cast<int>(L_no.uv.size());
+    double r_no = 0.0, r_yes = 0.0;
+    for (int i = 0; i < n; ++i) {
+        r_no  += L_no.uv[i].norm();
+        r_yes += L_yes.uv[i].norm();
+    }
+    EXPECT_LE(r_yes / n, r_no / n + 1e-9);
+}

code/tests/cgal/test_phase7.cpp

@@ -0,0 +1,488 @@
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
+// test_phase7.cpp
+//
+// Phase 7 — Tests for Java-parity layout features:
+//   - MobiusMap            : identity, inverse, compose, from_three, is_identity
+//   - best_root_face       : selects a valid face; interior bonus
+//   - halfedge_uv          : size, non-seam consistency, seam divergence
+//   - Priority BFS         : vertex ordering / depth correctness
+//   - normalise_euclidean  : halfedge_uv centroid at origin
+//   - period_matrix.hpp    : τ in upper half-plane, SL(2,ℤ) reduction
+//   - fundamental_domain.hpp: parallelogram CCW, generators, tiling_copy
+
+#include "conformal_mesh.hpp"
+#include "mesh_builder.hpp"
+#include "euclidean_functional.hpp"
+#include "hyper_ideal_functional.hpp"
+#include "newton_solver.hpp"
+#include "layout.hpp"
+#include "period_matrix.hpp"
+#include "fundamental_domain.hpp"
+#include <gtest/gtest.h>
+#include <cmath>
+#include <complex>
+#include <vector>
+#include <limits>
+
+using namespace conformallab;
+using C = std::complex<double>;
+
+// ════════════════════════════════════════════════════════════════════════════
+// MobiusMap
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(MobiusMap, Identity_AppliesAsIdentity)
+{
+    MobiusMap id = MobiusMap::identity();
+    C z(0.3, 0.7);
+    C w = id.apply(z);
+    EXPECT_NEAR(w.real(), z.real(), 1e-12);
+    EXPECT_NEAR(w.imag(), z.imag(), 1e-12);
+}
+
+TEST(MobiusMap, Identity_IsIdentity)
+{
+    EXPECT_TRUE(MobiusMap::identity().is_identity());
+}
+
+TEST(MobiusMap, NonIdentity_IsNotIdentity)
+{
+    // T(z) = z + 1 — translation, clearly not identity
+    MobiusMap T{ C(1), C(1), C(0), C(1) };
+    EXPECT_FALSE(T.is_identity());
+}
+
+TEST(MobiusMap, Inverse_ComposeIsIdentity)
+{
+    // T(z) = (2z + 1) / (z + 3)
+    MobiusMap T{ C(2), C(1), C(1), C(3) };
+    MobiusMap TinvT = T.inverse().compose(T);
+    EXPECT_TRUE(TinvT.is_identity(1e-9));
+}
+
+TEST(MobiusMap, Compose_OrderCorrect)
+{
+    // S: z ↦ z + 1,   T: z ↦ 2z
+    // S.compose(T) means S applied after T: z ↦ 2z + 1
+    MobiusMap S{ C(1), C(1), C(0), C(1) };   // z + 1
+    MobiusMap T{ C(2), C(0), C(0), C(1) };   // 2z
+    MobiusMap ST = S.compose(T);
+    C z(1.0, 0.0);
+    // S(T(z)) = S(2) = 3
+    EXPECT_NEAR(ST.apply(z).real(), 3.0, 1e-12);
+    EXPECT_NEAR(ST.apply(z).imag(), 0.0, 1e-12);
+}
+
+TEST(MobiusMap, FromThree_RecoversMap)
+{
+    // Known map T(z) = (z + i) / (1 + 0·z) — translation by i
+    C w1 = C(0, 1) + C(0, 1);   // T(i) = 2i
+    C w2 = C(1, 0) + C(0, 1);   // T(1) = 1 + i
+    C w3 = C(-1, 0) + C(0, 1);  // T(-1) = -1 + i
+    MobiusMap T = MobiusMap::from_three(C(0, 1), w1, C(1, 0), w2, C(-1, 0), w3);
+    // Verify T maps a fourth point correctly: T(0) = i
+    C result = T.apply(C(0, 0));
+    EXPECT_NEAR(result.real(), 0.0, 1e-9);
+    EXPECT_NEAR(result.imag(), 1.0, 1e-9);
+}
+
+TEST(MobiusMap, FromThree_DegenerateReturnsIdentity)
+{
+    // Three coincident points → singular system → identity fallback
+    C z(0.5, 0.5);
+    MobiusMap T = MobiusMap::from_three(z, z, z, z, z, z);
+    // Should not crash; returns identity (or at least a valid map)
+    // We just check the result is finite
+    C w = T.apply(C(0.1, 0.2));
+    EXPECT_FALSE(std::isnan(w.real()));
+    EXPECT_FALSE(std::isnan(w.imag()));
+}
+
+TEST(MobiusMap, Apply_Vector2d)
+{
+    MobiusMap id = MobiusMap::identity();
+    Eigen::Vector2d p(0.4, 0.6);
+    Eigen::Vector2d q = id.apply(p);
+    EXPECT_NEAR(q.x(), p.x(), 1e-12);
+    EXPECT_NEAR(q.y(), p.y(), 1e-12);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// best_root_face
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(BestRootFace, ReturnsValidFace_Triangle)
+{
+    auto mesh = make_triangle();
+    Face_index f = detail::best_root_face(mesh);
+    EXPECT_NE(f, Face_index());
+    EXPECT_GE(f.idx(), 0);
+}
+
+TEST(BestRootFace, ReturnsValidFace_Tetrahedron)
+{
+    auto mesh = make_tetrahedron();
+    Face_index f = detail::best_root_face(mesh);
+    EXPECT_NE(f, Face_index());
+    // Tetrahedron has 4 faces — best is one of them
+    EXPECT_LT(static_cast<std::size_t>(f.idx()), mesh.number_of_faces());
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// halfedge_uv — size and non-seam consistency
+// ════════════════════════════════════════════════════════════════════════════
+
+// Helper: build equilibrium Euclidean layout for a given mesh.
+// Uses x = 0 (identity scale factor) which is the equilibrium for natural edge lengths.
+static Layout2D make_euclidean_layout(ConformalMesh& mesh)
+{
+    EuclideanMaps maps = setup_euclidean_maps(mesh);
+    compute_euclidean_lambda0_from_mesh(mesh, maps);
+    // Pin first vertex (DOF = -1); assign sequential indices to the rest.
+    auto vit = mesh.vertices().begin();
+    maps.v_idx[*vit++] = -1;
+    int idx = 0;
+    for (; vit != mesh.vertices().end(); ++vit) maps.v_idx[*vit] = idx++;
+    std::vector<double> x(static_cast<std::size_t>(idx), 0.0);
+    return euclidean_layout(mesh, x, maps);
+}
+
+TEST(HalfedgeUV, Size_EqualsNumberOfHalfedges_Triangle)
+{
+    auto mesh = make_triangle();
+    auto lay   = make_euclidean_layout(mesh);
+    EXPECT_EQ(lay.halfedge_uv.size(), mesh.number_of_halfedges());
+}
+
+TEST(HalfedgeUV, Size_EqualsNumberOfHalfedges_QuadStrip)
+{
+    auto mesh = make_quad_strip();
+    auto lay   = make_euclidean_layout(mesh);
+    EXPECT_EQ(lay.halfedge_uv.size(), mesh.number_of_halfedges());
+}
+
+TEST(HalfedgeUV, NonBorderHalfedges_MatchUV)
+{
+    // For an open mesh with no cut graph the layout has no seams.
+    // Every non-border halfedge h must satisfy:
+    //   halfedge_uv[h] == uv[source(h)]
+    auto mesh = make_quad_strip();
+    auto lay   = make_euclidean_layout(mesh);
+    for (auto h : mesh.halfedges()) {
+        if (mesh.is_border(h)) continue;
+        std::size_t hi  = static_cast<std::size_t>(h.idx());
+        std::size_t vi  = static_cast<std::size_t>(mesh.source(h).idx());
+        EXPECT_NEAR(lay.halfedge_uv[hi].x(), lay.uv[vi].x(), 1e-10)
+            << "halfedge " << hi << " source vertex " << vi;
+        EXPECT_NEAR(lay.halfedge_uv[hi].y(), lay.uv[vi].y(), 1e-10)
+            << "halfedge " << hi << " source vertex " << vi;
+    }
+}
+
+TEST(HalfedgeUV, BorderHalfedges_AreZero)
+{
+    auto mesh = make_triangle();
+    auto lay   = make_euclidean_layout(mesh);
+    bool found_border = false;
+    for (auto h : mesh.halfedges()) {
+        if (!mesh.is_border(h)) continue;
+        std::size_t hi = static_cast<std::size_t>(h.idx());
+        EXPECT_NEAR(lay.halfedge_uv[hi].x(), 0.0, 1e-12);
+        EXPECT_NEAR(lay.halfedge_uv[hi].y(), 0.0, 1e-12);
+        found_border = true;
+    }
+    EXPECT_TRUE(found_border);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Priority BFS — depth ordering
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(PriorityBFS, Layout_SucceedsOnOpenMesh)
+{
+    auto mesh = make_quad_strip();
+    auto lay   = make_euclidean_layout(mesh);
+    EXPECT_TRUE(lay.success);
+}
+
+TEST(PriorityBFS, Layout_NoSeamOnOpenMesh)
+{
+    auto mesh = make_quad_strip();
+    auto lay   = make_euclidean_layout(mesh);
+    EXPECT_FALSE(lay.has_seam);
+}
+
+TEST(PriorityBFS, AllVerticesPlaced)
+{
+    auto mesh = make_tetrahedron();
+    // Tetrahedron is closed; layout without cut graph will have a seam
+    EuclideanMaps maps = setup_euclidean_maps(mesh);
+    compute_euclidean_lambda0_from_mesh(mesh, maps);
+    auto vit = mesh.vertices().begin();
+    maps.v_idx[*vit++] = -1;
+    int idx = 0;
+    for (; vit != mesh.vertices().end(); ++vit) maps.v_idx[*vit] = idx++;
+    std::vector<double> x(static_cast<std::size_t>(idx), 0.0);
+    auto lay = euclidean_layout(mesh, x, maps);
+    EXPECT_TRUE(lay.success);
+    // All UVs must be finite
+    for (auto& p : lay.uv) {
+        EXPECT_FALSE(std::isnan(p.x()));
+        EXPECT_FALSE(std::isnan(p.y()));
+    }
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// normalise_euclidean — centroid + PCA applied to both uv and halfedge_uv
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(NormaliseEuclidean, UVCentroidAtOrigin)
+{
+    auto mesh = make_quad_strip();
+    auto lay   = make_euclidean_layout(mesh);
+    normalise_euclidean(lay);
+
+    Eigen::Vector2d mean = Eigen::Vector2d::Zero();
+    for (auto& p : lay.uv) mean += p;
+    mean /= static_cast<double>(lay.uv.size());
+    EXPECT_NEAR(mean.x(), 0.0, 1e-10);
+    EXPECT_NEAR(mean.y(), 0.0, 1e-10);
+}
+
+TEST(NormaliseEuclidean, HalfedgeUVCentroidAlsoShifted)
+{
+    // After normalisation: the non-border halfedge_uv entries should also be
+    // centred (since they are shifted by the same mean as uv).
+    // We verify that the mean of non-border halfedge_uv is near (0,0).
+    auto mesh = make_quad_strip();
+    auto lay   = make_euclidean_layout(mesh);
+    normalise_euclidean(lay);
+
+    Eigen::Vector2d mean = Eigen::Vector2d::Zero();
+    int count = 0;
+    for (auto h : mesh.halfedges()) {
+        if (mesh.is_border(h)) continue;
+        mean += lay.halfedge_uv[static_cast<std::size_t>(h.idx())];
+        ++count;
+    }
+    if (count > 0) mean /= static_cast<double>(count);
+    EXPECT_NEAR(mean.x(), 0.0, 1e-9);
+    EXPECT_NEAR(mean.y(), 0.0, 1e-9);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// PeriodMatrix — reduce_to_fundamental_domain
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(PeriodMatrix, ReduceToFD_AlreadyInFD)
+{
+    // τ = i is in F  (|i|=1, Re(i)=0, Im(i)=1>0)
+    C tau(0.0, 1.0);
+    C reduced = reduce_to_fundamental_domain(tau);
+    EXPECT_TRUE(is_in_fundamental_domain(reduced));
+    EXPECT_NEAR(reduced.real(), 0.0, 1e-10);
+    EXPECT_NEAR(reduced.imag(), 1.0, 1e-10);
+}
+
+TEST(PeriodMatrix, ReduceToFD_ShiftsRealPart)
+{
+    // τ = 2 + 3i  →  T step: τ -= 2  →  3i  (|3i|=3≥1, Re=0)
+    C tau(2.0, 3.0);
+    C reduced = reduce_to_fundamental_domain(tau);
+    EXPECT_TRUE(is_in_fundamental_domain(reduced, 1e-9));
+    EXPECT_NEAR(reduced.real(), 0.0, 1e-10);
+    EXPECT_NEAR(reduced.imag(), 3.0, 1e-10);
+}
+
+TEST(PeriodMatrix, ReduceToFD_InvertsSmallTau)
+{
+    // τ = 0.5i  →  |0.5i|=0.5<1  →  S: τ↦-1/(0.5i) = 2i
+    C tau(0.0, 0.5);
+    C reduced = reduce_to_fundamental_domain(tau);
+    EXPECT_TRUE(is_in_fundamental_domain(reduced, 1e-9));
+    EXPECT_NEAR(reduced.real(), 0.0, 1e-10);
+    EXPECT_NEAR(reduced.imag(), 2.0, 1e-10);
+}
+
+TEST(PeriodMatrix, ReduceToFD_ThrowsForNonUpperHalfPlane)
+{
+    C tau(0.5, -1.0);  // Im < 0 → not in upper half-plane
+    EXPECT_THROW(reduce_to_fundamental_domain(tau), std::domain_error);
+}
+
+TEST(PeriodMatrix, IsInFundamentalDomain_Square)
+{
+    EXPECT_TRUE(is_in_fundamental_domain(C(0.0, 1.0)));        // i
+    EXPECT_TRUE(is_in_fundamental_domain(C(0.3, 1.5)));        // inside
+    EXPECT_FALSE(is_in_fundamental_domain(C(0.6, 1.5)));       // Re > 1/2
+    EXPECT_FALSE(is_in_fundamental_domain(C(0.0, 0.5)));       // |τ| < 1
+}
+
+TEST(PeriodMatrix, ComputePeriodMatrix_UnitSquare)
+{
+    // ω_1 = (1, 0), ω_2 = (0, 1)  →  τ = i
+    HolonomyData hol;
+    hol.translations = { Eigen::Vector2d(1.0, 0.0), Eigen::Vector2d(0.0, 1.0) };
+    PeriodData pd = compute_period_matrix(hol, /*reduce=*/false);
+    EXPECT_EQ(pd.genus(), 1);
+    EXPECT_GT(pd.tau.imag(), 0.0);
+    EXPECT_NEAR(pd.tau.real(), 0.0, 1e-10);
+    EXPECT_NEAR(pd.tau.imag(), 1.0, 1e-10);
+}
+
+TEST(PeriodMatrix, ComputePeriodMatrix_ReducedTau_InFD)
+{
+    // ω_1 = (1, 0), ω_2 = (0.5, 0.25)  →  τ = 0.5 + 0.25i
+    // |τ| = sqrt(0.25 + 0.0625) ≈ 0.559 < 1 → needs S step
+    HolonomyData hol;
+    hol.translations = { Eigen::Vector2d(1.0, 0.0), Eigen::Vector2d(0.5, 0.25) };
+    PeriodData pd = compute_period_matrix(hol, /*reduce=*/true);
+    EXPECT_TRUE(pd.in_fundamental_domain);
+    EXPECT_TRUE(is_in_fundamental_domain(pd.tau, 1e-9));
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// FundamentalDomain — genus-1 parallelogram
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(FundamentalDomain, Genus1_HasFourVertices)
+{
+    HolonomyData hol;
+    hol.translations = { Eigen::Vector2d(1.0, 0.0), Eigen::Vector2d(0.0, 1.0) };
+    FundamentalDomain fd = compute_fundamental_domain_genus1(hol);
+    EXPECT_EQ(fd.vertices.size(), 4u);
+    EXPECT_TRUE(fd.is_valid());
+}
+
+TEST(FundamentalDomain, Genus1_VerticesMatchGenerators_UnitSquare)
+{
+    Eigen::Vector2d w1(1.0, 0.0), w2(0.0, 1.0);
+    HolonomyData hol;
+    hol.translations = { w1, w2 };
+    FundamentalDomain fd = compute_fundamental_domain_genus1(hol);
+    // Expected (CCW): origin, w1, w1+w2, w2
+    EXPECT_NEAR(fd.vertices[0].x(), 0.0, 1e-12);
+    EXPECT_NEAR(fd.vertices[0].y(), 0.0, 1e-12);
+    EXPECT_NEAR(fd.vertices[1].x(), w1.x(), 1e-12);
+    EXPECT_NEAR(fd.vertices[1].y(), w1.y(), 1e-12);
+    EXPECT_NEAR(fd.vertices[2].x(), (w1 + w2).x(), 1e-12);
+    EXPECT_NEAR(fd.vertices[2].y(), (w1 + w2).y(), 1e-12);
+    EXPECT_NEAR(fd.vertices[3].x(), w2.x(), 1e-12);
+    EXPECT_NEAR(fd.vertices[3].y(), w2.y(), 1e-12);
+}
+
+TEST(FundamentalDomain, Genus1_CCWOrientation)
+{
+    // After possible swap, the signed area = cross(v1-v0, v3-v0) > 0 (CCW)
+    HolonomyData hol;
+    hol.translations = { Eigen::Vector2d(1.0, 0.0), Eigen::Vector2d(0.0, 1.0) };
+    FundamentalDomain fd = compute_fundamental_domain_genus1(hol);
+    Eigen::Vector2d v0 = fd.vertices[0], v1 = fd.vertices[1], v3 = fd.vertices[3];
+    double cross = (v1 - v0).x() * (v3 - v0).y() - (v1 - v0).y() * (v3 - v0).x();
+    EXPECT_GT(cross, 0.0);
+}
+
+TEST(FundamentalDomain, Genus1_CCWEnforced_WhenInputIsCW)
+{
+    // If we give CW generators (w2 × w1 < 0), the polygon must still be CCW.
+    // w1 = (0,1), w2 = (1,0):  cross w1×w2 = 0*0 - 1*1 = -1 < 0  →  should swap
+    HolonomyData hol;
+    hol.translations = { Eigen::Vector2d(0.0, 1.0), Eigen::Vector2d(1.0, 0.0) };
+    FundamentalDomain fd = compute_fundamental_domain_genus1(hol);
+    Eigen::Vector2d v0 = fd.vertices[0], v1 = fd.vertices[1], v3 = fd.vertices[3];
+    double cross = (v1 - v0).x() * (v3 - v0).y() - (v1 - v0).y() * (v3 - v0).x();
+    EXPECT_GT(cross, 0.0);
+}
+
+TEST(FundamentalDomain, Genus1_EdgeIdentifications)
+{
+    HolonomyData hol;
+    hol.translations = { Eigen::Vector2d(1.0, 0.0), Eigen::Vector2d(0.0, 1.0) };
+    FundamentalDomain fd = compute_fundamental_domain_genus1(hol);
+    EXPECT_EQ(fd.edge_identifications.size(), 2u);
+    // bottom ≡ top: (0,2)
+    EXPECT_EQ(fd.edge_identifications[0].first,  0);
+    EXPECT_EQ(fd.edge_identifications[0].second, 2);
+    // right ≡ left: (1,3)
+    EXPECT_EQ(fd.edge_identifications[1].first,  1);
+    EXPECT_EQ(fd.edge_identifications[1].second, 3);
+}
+
+TEST(FundamentalDomain, Genus1_GeneratorsStored)
+{
+    Eigen::Vector2d w1(2.0, 1.0), w2(-1.0, 3.0);
+    HolonomyData hol;
+    hol.translations = { w1, w2 };
+    FundamentalDomain fd = compute_fundamental_domain_genus1(hol);
+    EXPECT_EQ(fd.generators.size(), 2u);
+    // Generators are w1 and w2 (possibly swapped to ensure CCW)
+    // Their sum of norms matches the originals
+    double norm_gen = fd.generators[0].norm() + fd.generators[1].norm();
+    double norm_in  = w1.norm() + w2.norm();
+    EXPECT_NEAR(norm_gen, norm_in, 1e-10);
+}
+
+TEST(FundamentalDomain, HigherGenus_ReturnsEmpty)
+{
+    HolonomyData hol;
+    hol.translations = {
+        Eigen::Vector2d(1, 0), Eigen::Vector2d(0, 1),
+        Eigen::Vector2d(2, 0), Eigen::Vector2d(0, 2)   // g=2, 4 generators
+    };
+    FundamentalDomain fd = compute_fundamental_domain(hol);
+    // g > 1 returns empty (TODO Phase 8)
+    EXPECT_FALSE(fd.is_valid());
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// tiling_copy / tiling_neighbourhood
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(TilingCopy, ShiftAppliedToAllUV)
+{
+    auto mesh = make_quad_strip();
+    auto lay   = make_euclidean_layout(mesh);
+
+    Eigen::Vector2d w1(3.0, 0.0), w2(0.0, 2.0);
+    // m=1, n=2 → expected shift = w1 + 2*w2 = (3, 4)
+    Layout2D copy = tiling_copy(lay, w1, w2, 1, 2);
+    Eigen::Vector2d expected_shift(3.0, 4.0);
+    for (std::size_t i = 0; i < lay.uv.size(); ++i) {
+        EXPECT_NEAR(copy.uv[i].x(), lay.uv[i].x() + expected_shift.x(), 1e-12);
+        EXPECT_NEAR(copy.uv[i].y(), lay.uv[i].y() + expected_shift.y(), 1e-12);
+    }
+}
+
+TEST(TilingCopy, ZeroShift_IsSameAsCopy)
+{
+    auto mesh = make_quad_strip();
+    auto lay   = make_euclidean_layout(mesh);
+    Eigen::Vector2d w1(1, 0), w2(0, 1);
+    Layout2D copy = tiling_copy(lay, w1, w2, 0, 0);
+    for (std::size_t i = 0; i < lay.uv.size(); ++i) {
+        EXPECT_NEAR(copy.uv[i].x(), lay.uv[i].x(), 1e-12);
+        EXPECT_NEAR(copy.uv[i].y(), lay.uv[i].y(), 1e-12);
+    }
+}
+
+TEST(TilingNeighbourhood, CorrectCount)
+{
+    auto mesh = make_quad_strip();
+    auto lay   = make_euclidean_layout(mesh);
+    HolonomyData hol;
+    hol.translations = { Eigen::Vector2d(1, 0), Eigen::Vector2d(0, 1) };
+    // m_max=1, n_max=1 → (2*1+1) * (2*1+1) = 9 tiles
+    auto tiles = tiling_neighbourhood(lay, hol, 1, 1);
+    EXPECT_EQ(tiles.size(), 9u);
+}
+
+TEST(TilingNeighbourhood, EmptyHolonomy_ReturnsSingleTile)
+{
+    auto mesh = make_quad_strip();
+    auto lay   = make_euclidean_layout(mesh);
+    HolonomyData hol;  // no translations
+    auto tiles = tiling_neighbourhood(lay, hol);
+    EXPECT_EQ(tiles.size(), 1u);
+}

code/tests/cgal/test_pipeline.cpp

@@ -0,0 +1,295 @@
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
+// test_pipeline.cpp
+//
+// Phase 4c — End-to-end pipeline tests and library-user examples.
+//
+// These tests exercise the full conformallab++ pipeline as a user would:
+//
+//   1. Build (or load) a mesh
+//   2. Set up maps and assign DOFs
+//   3. Configure target angles / targets
+//   4. Solve with Newton
+//   5. Inspect / export the result
+//
+// Each test mirrors a realistic usage scenario documented in the README.
+//
+// Tests:
+//   1. Pipeline_Euclidean_TriangleToEquilibrium
+//        Read mesh → setup Euclidean maps → solve → verify convergence
+//   2. Pipeline_Spherical_TetrahedronToEquilibrium
+//        Setup spherical tetrahedron → solve → verify angles sum to 4π
+//   3. Pipeline_HyperIdeal_TriangleRoundTrip
+//        Build triangle → setup HyperIdeal → solve → verify G ≈ 0
+//   4. Pipeline_MeshIO_SolveAndExport
+//        Build mesh → solve → write OFF → reload → verify vertex count intact
+//   5. Pipeline_AllThreeGeometries_SameTopology
+//        Same quad-strip mesh solved under all three geometries: all converge
+
+#include "conformal_mesh.hpp"
+#include "mesh_builder.hpp"
+#include "mesh_io.hpp"
+#include "euclidean_functional.hpp"
+#include "spherical_functional.hpp"
+#include "hyper_ideal_functional.hpp"
+#include "newton_solver.hpp"
+#include <gtest/gtest.h>
+#include <cmath>
+#include <vector>
+#include <filesystem>
+
+using namespace conformallab;
+
+// ────────────────────────────────────────────────────────────────────────────
+// Shared helpers
+// ────────────────────────────────────────────────────────────────────────────
+
+static void pin_first_vertex_euclidean(ConformalMesh& mesh, EuclideanMaps& maps, int& n)
+{
+    auto vit = mesh.vertices().begin();
+    Vertex_index v0 = *vit++;
+    maps.v_idx[v0] = -1;
+    int idx = 0;
+    for (; vit != mesh.vertices().end(); ++vit)
+        maps.v_idx[*vit] = idx++;
+    n = idx;
+}
+
+static void set_natural_euclidean_theta(ConformalMesh& mesh, EuclideanMaps& maps, int n)
+{
+    std::vector<double> x0(static_cast<std::size_t>(n), 0.0);
+    auto G = euclidean_gradient(mesh, x0, maps);
+    for (auto v : mesh.vertices()) {
+        int iv = maps.v_idx[v];
+        if (iv < 0) continue;
+        maps.theta_v[v] -= G[static_cast<std::size_t>(iv)];
+    }
+}
+
+static std::vector<double> set_natural_hyper_ideal_targets(
+    ConformalMesh& mesh, HyperIdealMaps& maps, int n,
+    double b_base = 1.0, double a_base = 0.5)
+{
+    const auto sz = static_cast<std::size_t>(n);
+    std::vector<double> xbase(sz, 0.0);
+    for (auto v : mesh.vertices()) {
+        int iv = maps.v_idx[v];
+        if (iv >= 0) xbase[static_cast<std::size_t>(iv)] = b_base;
+    }
+    for (auto e : mesh.edges()) {
+        int ie = maps.e_idx[e];
+        if (ie >= 0) xbase[static_cast<std::size_t>(ie)] = a_base;
+    }
+    auto G = evaluate_hyper_ideal(mesh, xbase, maps, false).gradient;
+    for (auto v : mesh.vertices()) {
+        int iv = maps.v_idx[v];
+        if (iv < 0) continue;
+        maps.theta_v[v] += G[static_cast<std::size_t>(iv)];
+    }
+    for (auto e : mesh.edges()) {
+        int ie = maps.e_idx[e];
+        if (ie < 0) continue;
+        maps.theta_e[e] += G[static_cast<std::size_t>(ie)];
+    }
+    return xbase;
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Test 1 — Euclidean full pipeline: triangle → equilibrium
+//
+// Simulates a user doing:
+//   auto mesh = make_triangle();
+//   auto maps = setup_euclidean_maps(mesh);
+//   compute_euclidean_lambda0_from_mesh(mesh, maps);
+//   // … set target angles and DOF indices …
+//   auto result = newton_euclidean(mesh, x0, maps);
+//   assert(result.converged);
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(Pipeline, Euclidean_TriangleToEquilibrium)
+{
+    // ── Step 1: build mesh ────────────────────────────────────────────────
+    auto mesh = make_triangle();
+
+    // ── Step 2: set up maps ───────────────────────────────────────────────
+    auto maps = setup_euclidean_maps(mesh);
+    compute_euclidean_lambda0_from_mesh(mesh, maps);
+
+    // ── Step 3: assign DOFs (pin v0) ──────────────────────────────────────
+    int n = 0;
+    pin_first_vertex_euclidean(mesh, maps, n);
+    ASSERT_EQ(n, 2);
+
+    // ── Step 4: choose natural target angles → x* = 0 ────────────────────
+    set_natural_euclidean_theta(mesh, maps, n);
+
+    // ── Step 5: solve ─────────────────────────────────────────────────────
+    std::vector<double> x0(static_cast<std::size_t>(n), -0.1);
+    auto result = newton_euclidean(mesh, x0, maps);
+
+    // ── Step 6: verify ────────────────────────────────────────────────────
+    EXPECT_TRUE(result.converged)
+        << "Euclidean pipeline: triangle should converge; "
+           "grad_inf_norm = " << result.grad_inf_norm;
+    EXPECT_LT(result.grad_inf_norm, 1e-8);
+    EXPECT_EQ(static_cast<int>(result.x.size()), n);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Test 2 — Spherical full pipeline: tetrahedron → equilibrium
+//
+// The spherical tetrahedron equilibrium x* = 0 is built into the maps.
+// After solving, the total angle defect Σ(Θ_v − Σα_v) should be ≈ 0.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(Pipeline, Spherical_TetrahedronToEquilibrium)
+{
+    // ── Steps 1–3 ─────────────────────────────────────────────────────────
+    auto mesh = make_spherical_tetrahedron();
+    auto maps = setup_spherical_maps(mesh);
+    compute_lambda0_from_mesh(mesh, maps);
+    int n = assign_vertex_dof_indices(mesh, maps);
+
+    // ── Step 4: solve ─────────────────────────────────────────────────────
+    std::vector<double> x0(static_cast<std::size_t>(n), -0.2);
+    auto result = newton_spherical(mesh, x0, maps);
+
+    // ── Step 5: verify convergence ────────────────────────────────────────
+    EXPECT_TRUE(result.converged)
+        << "Spherical pipeline: tetrahedron should converge; "
+           "grad_inf_norm = " << result.grad_inf_norm;
+    EXPECT_LT(result.grad_inf_norm, 1e-8);
+
+    // ── Step 6: verify geometric invariant — total angle defect ≈ 0 ──────
+    auto G_final = spherical_gradient(mesh, result.x, maps);
+    double total_defect = 0.0;
+    for (double gv : G_final) total_defect += gv;
+    EXPECT_NEAR(total_defect, 0.0, 1e-7)
+        << "Spherical: total angle defect should vanish at equilibrium";
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Test 3 — HyperIdeal full pipeline: triangle → equilibrium
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(Pipeline, HyperIdeal_TriangleRoundTrip)
+{
+    // ── Steps 1–3 ─────────────────────────────────────────────────────────
+    auto mesh = make_triangle();
+    auto maps = setup_hyper_ideal_maps(mesh);
+    int n = assign_all_dof_indices(mesh, maps);
+
+    // ── Step 4: natural targets (equilibrium at b=1.0, a=0.5) ────────────
+    auto xbase = set_natural_hyper_ideal_targets(mesh, maps, n);
+
+    // Verify: gradient at xbase must be ≈ 0 before solving
+    auto G_at_base = evaluate_hyper_ideal(mesh, xbase, maps, false).gradient;
+    double max_g = 0.0;
+    for (double v : G_at_base) max_g = std::max(max_g, std::abs(v));
+    ASSERT_LT(max_g, 1e-10) << "Natural target setup: gradient at base should be ~0";
+
+    // ── Step 5: perturb and solve ─────────────────────────────────────────
+    std::vector<double> x0 = xbase;
+    for (auto& v : x0) v += 0.3;
+
+    auto result = newton_hyper_ideal(mesh, x0, maps);
+
+    // ── Step 6: verify ────────────────────────────────────────────────────
+    EXPECT_TRUE(result.converged)
+        << "HyperIdeal pipeline: triangle should converge; "
+           "grad_inf_norm = " << result.grad_inf_norm;
+    EXPECT_LT(result.grad_inf_norm, 1e-8);
+
+    // Solution should be close to xbase (same equilibrium)
+    for (int i = 0; i < n; ++i) {
+        EXPECT_NEAR(result.x[static_cast<std::size_t>(i)],
+                    xbase[static_cast<std::size_t>(i)], 1e-6)
+            << "DOF " << i << " should recover the equilibrium value";
+    }
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Test 4 — Mesh I/O in the pipeline: solve → write → reload → check
+//
+// Demonstrates: compute a conformal factor on a mesh, write it to OFF, reload
+// and check that the mesh topology is preserved.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(Pipeline, MeshIO_SolveAndExport)
+{
+    // ── Build and solve ───────────────────────────────────────────────────
+    auto mesh = make_quad_strip();
+    auto maps = setup_euclidean_maps(mesh);
+    compute_euclidean_lambda0_from_mesh(mesh, maps);
+
+    int n = 0;
+    pin_first_vertex_euclidean(mesh, maps, n);
+    set_natural_euclidean_theta(mesh, maps, n);
+
+    std::vector<double> x0(static_cast<std::size_t>(n), -0.1);
+    auto result = newton_euclidean(mesh, x0, maps);
+    ASSERT_TRUE(result.converged) << "Solver must converge before export test";
+
+    // ── Write mesh ────────────────────────────────────────────────────────
+    const std::string tmp_path = "/tmp/conformallab_pipeline_test.off";
+    ASSERT_NO_THROW(save_mesh(tmp_path, mesh));
+    ASSERT_TRUE(std::filesystem::exists(tmp_path));
+
+    // ── Reload and verify topology ────────────────────────────────────────
+    ConformalMesh mesh2;
+    ASSERT_NO_THROW(mesh2 = load_mesh(tmp_path));
+
+    EXPECT_EQ(mesh2.number_of_vertices(), mesh.number_of_vertices())
+        << "Vertex count must survive OFF round-trip";
+    EXPECT_EQ(mesh2.number_of_faces(), mesh.number_of_faces())
+        << "Face count must survive OFF round-trip";
+
+    std::filesystem::remove(tmp_path);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Test 5 — All three geometries, same quad-strip topology
+//
+// Validates that the solver infrastructure works uniformly: the same mesh
+// topology is solvable under Euclidean, Spherical, and HyperIdeal geometries.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(Pipeline, AllThreeGeometries_QuadStrip)
+{
+    // ── Euclidean ──────────────────────────────────────────────────────────
+    {
+        auto mesh = make_quad_strip();
+        auto maps = setup_euclidean_maps(mesh);
+        compute_euclidean_lambda0_from_mesh(mesh, maps);
+        int n = 0;
+        pin_first_vertex_euclidean(mesh, maps, n);
+        set_natural_euclidean_theta(mesh, maps, n);
+        std::vector<double> x0(static_cast<std::size_t>(n), -0.1);
+        auto res = newton_euclidean(mesh, x0, maps);
+        EXPECT_TRUE(res.converged) << "Euclidean: quad strip should converge";
+    }
+
+    // ── HyperIdeal ────────────────────────────────────────────────────────
+    {
+        auto mesh = make_quad_strip();
+        auto maps = setup_hyper_ideal_maps(mesh);
+        int n = assign_all_dof_indices(mesh, maps);
+        auto xbase = set_natural_hyper_ideal_targets(mesh, maps, n);
+        std::vector<double> x0 = xbase;
+        for (auto& v : x0) v += 0.1;
+        auto res = newton_hyper_ideal(mesh, x0, maps);
+        EXPECT_TRUE(res.converged) << "HyperIdeal: quad strip should converge";
+    }
+
+    // ── Spherical (tetrahedron: smallest closed mesh with all vertices free) ─
+    {
+        auto mesh = make_spherical_tetrahedron();
+        auto maps = setup_spherical_maps(mesh);
+        compute_lambda0_from_mesh(mesh, maps);
+        int n = assign_vertex_dof_indices(mesh, maps);
+        std::vector<double> x0(static_cast<std::size_t>(n), -0.1);
+        auto res = newton_spherical(mesh, x0, maps);
+        EXPECT_TRUE(res.converged) << "Spherical: tetrahedron should converge";
+    }
+}

code/tests/cgal/test_scalability_smoke.cpp

@@ -0,0 +1,204 @@
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
+// test_scalability_smoke.cpp
+//
+// Scalability smoke tests — convergence on large real-world meshes.
+//
+// PURPOSE
+//   These tests verify that the Newton solver converges correctly on meshes
+//   significantly larger than the unit tests (which use tiny synthetic meshes).
+//   They do NOT assert on wall-clock time — timing is printed for information
+//   only, so the tests remain stable on slow CI hardware (Raspberry Pi ARM64).
+//
+//   If Newton fails to converge here, it is a correctness regression, not a
+//   performance regression.  See doc/math/complexity.md for timing context.
+//
+// MESHES USED
+//   cathead.obj   V=131,  F=248,  genus=0, open  — small, sanity check
+//   brezel.obj    V=6910, F=13824, genus=2, closed — large genus-2 mesh (χ=−2)
+//   brezel2.obj   V=2622, F=5248,  genus=2, closed — smaller genus-2 mesh (χ=−2)
+//
+// NOTE: both brezel meshes are genus-2. The naming follows the Java original
+// where "brezel2" is a different triangulation, not a different genus.
+//
+// EXPECTED RESULTS
+//   Newton converges in < 30 iterations for all Euclidean meshes (strictly
+//   convex energy, quadratic convergence from u=0).
+//   Cut graph produces 2g seam edges: 2 for brezel, 4 for brezel2.
+//
+// Tests:
+//   1. SmokeEuclidean.CatHead_SmallOpen
+//   2. SmokeEuclidean.Brezel_LargeGenus2
+//   3. SmokeEuclidean.Brezel2_Genus2_CutGraph
+
+#include "conformal_mesh.hpp"
+#include "mesh_io.hpp"
+#include "euclidean_functional.hpp"
+#include "gauss_bonnet.hpp"
+#include "newton_solver.hpp"
+#include "cut_graph.hpp"
+#include <gtest/gtest.h>
+#include <chrono>
+#include <iostream>
+#include <vector>
+#include <cmath>
+#include <string>
+
+using namespace conformallab;
+using Clock = std::chrono::steady_clock;
+using Ms    = std::chrono::milliseconds;
+
+// ── Helpers ──────────────────────────────────────────────────────────────────
+
+static int setup_open_mesh_dofs(ConformalMesh& mesh, EuclideanMaps& maps)
+{
+    int idx = 0;
+    for (auto v : mesh.vertices())
+        maps.v_idx[v] = mesh.is_border(v) ? -1 : idx++;
+    return idx;
+}
+
+static int setup_closed_mesh_dofs(ConformalMesh& mesh, EuclideanMaps& maps)
+{
+    auto vit = mesh.vertices().begin();
+    maps.v_idx[*vit++] = -1;
+    int idx = 0;
+    for (; vit != mesh.vertices().end(); ++vit)
+        maps.v_idx[*vit] = idx++;
+    return idx;
+}
+
+static void apply_natural_theta(ConformalMesh& mesh, EuclideanMaps& maps, int n)
+{
+    std::vector<double> x0(static_cast<std::size_t>(n), 0.0);
+    auto G0 = euclidean_gradient(mesh, x0, maps);
+    for (auto v : mesh.vertices()) {
+        int iv = maps.v_idx[v];
+        if (iv >= 0) maps.theta_v[v] -= G0[static_cast<std::size_t>(iv)];
+    }
+}
+
+// ── Test 1 — cathead.obj (V=131, F=248, open) ────────────────────────────────
+
+TEST(SmokeEuclidean, CatHead_SmallOpen)
+{
+    const std::string path = std::string(CONFORMALLAB_DATA_DIR) + "/obj/cathead.obj";
+    ConformalMesh mesh;
+    ASSERT_NO_THROW(mesh = load_mesh(path)) << "cathead.obj not found: " << path;
+
+    EXPECT_EQ(131u, mesh.number_of_vertices());
+    EXPECT_EQ(248u, mesh.number_of_faces());
+
+    auto maps = setup_euclidean_maps(mesh);
+    compute_euclidean_lambda0_from_mesh(mesh, maps);
+    const int n = setup_open_mesh_dofs(mesh, maps);
+    apply_natural_theta(mesh, maps, n);
+
+    // Start from a small perturbation so Newton actually iterates.
+    std::vector<double> x0(static_cast<std::size_t>(n), -0.05);
+
+    auto t0 = Clock::now();
+    auto res = newton_euclidean(mesh, x0, maps, 1e-9, 200);
+    auto dt  = std::chrono::duration_cast<Ms>(Clock::now() - t0).count();
+
+    std::cout << "[SmokeEuclidean.CatHead] V=" << mesh.number_of_vertices()
+              << " F=" << mesh.number_of_faces()
+              << " iter=" << res.iterations
+              << " ||G||=" << res.grad_inf_norm
+              << " time=" << dt << "ms\n";
+
+    EXPECT_TRUE(res.converged)   << "Newton did not converge on cathead.obj";
+    EXPECT_LT(res.iterations, 30) << "Newton took ≥ 30 iterations — unexpected";
+    EXPECT_LT(res.grad_inf_norm, 1e-8);
+}
+
+// ── Test 2 — brezel.obj (V=6910, F=13824, genus=2) ───────────────────────────
+// Primary scalability target: largest mesh in the test suite.
+// Newton is started from a small perturbation (x0 = −0.05) so it must
+// actually iterate rather than exit immediately from the trivial equilibrium.
+
+TEST(SmokeEuclidean, Brezel_LargeGenus2)
+{
+    const std::string path = std::string(CONFORMALLAB_DATA_DIR) + "/obj/brezel.obj";
+    ConformalMesh mesh;
+    ASSERT_NO_THROW(mesh = load_mesh(path)) << "brezel.obj not found: " << path;
+
+    EXPECT_EQ(6910u,  mesh.number_of_vertices());
+    EXPECT_EQ(13824u, mesh.number_of_faces());
+
+    // Euler characteristic: V - E + F = −2 for genus-2 closed surface
+    const int chi = static_cast<int>(mesh.number_of_vertices())
+                  - static_cast<int>(mesh.number_of_edges())
+                  + static_cast<int>(mesh.number_of_faces());
+    EXPECT_EQ(-2, chi) << "brezel.obj must be genus-2 (χ=−2)";
+
+    auto maps = setup_euclidean_maps(mesh);
+    compute_euclidean_lambda0_from_mesh(mesh, maps);
+
+    const int n = setup_closed_mesh_dofs(mesh, maps);
+    enforce_gauss_bonnet(mesh, maps);
+    apply_natural_theta(mesh, maps, n);
+
+    // Start from a small perturbation so Newton actually iterates.
+    std::vector<double> x0(static_cast<std::size_t>(n), -0.05);
+
+    // Newton solve
+    auto t0  = Clock::now();
+    auto res = newton_euclidean(mesh, x0, maps, 1e-9, 200);
+    auto dt_newton = std::chrono::duration_cast<Ms>(Clock::now() - t0).count();
+
+    // Cut graph
+    auto t1  = Clock::now();
+    CutGraph cg = compute_cut_graph(mesh);
+    auto dt_cut = std::chrono::duration_cast<Ms>(Clock::now() - t1).count();
+
+    std::cout << "[SmokeEuclidean.Brezel] V=" << mesh.number_of_vertices()
+              << " F=" << mesh.number_of_faces()
+              << " iter=" << res.iterations
+              << " ||G||=" << res.grad_inf_norm
+              << " newton=" << dt_newton << "ms"
+              << " cut=" << dt_cut << "ms\n";
+
+    EXPECT_TRUE(res.converged)    << "Newton did not converge on brezel.obj";
+    EXPECT_LT(res.iterations, 30) << "Newton took ≥ 30 iterations";
+    EXPECT_LT(res.grad_inf_norm, 1e-8);
+
+    // Genus-2: 2g = 4 seam edges
+    EXPECT_EQ(4u, cg.cut_edge_indices.size())
+        << "brezel.obj (genus 2) must yield 2g=4 cut edges";
+    EXPECT_EQ(2, cg.genus);
+}
+
+// ── Test 3 — brezel2.obj (V=2622, F=5248, genus=2) ───────────────────────────
+
+TEST(SmokeEuclidean, Brezel2_Genus2_CutGraph)
+{
+    const std::string path = std::string(CONFORMALLAB_DATA_DIR) + "/obj/brezel2.obj";
+    ConformalMesh mesh;
+    ASSERT_NO_THROW(mesh = load_mesh(path)) << "brezel2.obj not found: " << path;
+
+    EXPECT_EQ(2622u, mesh.number_of_vertices());
+    EXPECT_EQ(5248u, mesh.number_of_faces());
+
+    const int chi = static_cast<int>(mesh.number_of_vertices())
+                  - static_cast<int>(mesh.number_of_edges())
+                  + static_cast<int>(mesh.number_of_faces());
+    EXPECT_EQ(-2, chi) << "brezel2.obj must be genus-2 (χ=−2)";
+
+    // Cut graph only — Newton on genus-2 requires full DOF setup
+    // (tested separately in test_geometry_utils.cpp HomologyGenerators suite)
+    auto t0  = Clock::now();
+    CutGraph cg = compute_cut_graph(mesh);
+    auto dt_cut = std::chrono::duration_cast<Ms>(Clock::now() - t0).count();
+
+    std::cout << "[SmokeEuclidean.Brezel2] V=" << mesh.number_of_vertices()
+              << " F=" << mesh.number_of_faces()
+              << " cut=" << dt_cut << "ms"
+              << " seams=" << cg.cut_edge_indices.size() << "\n";
+
+    // Genus-2: 2g = 4 seam edges
+    EXPECT_EQ(4u, cg.cut_edge_indices.size())
+        << "brezel2.obj (genus 2) must yield 2g=4 cut edges";
+    EXPECT_EQ(2, cg.genus);
+}

code/tests/cgal/test_spherical_functional.cpp

@@ -0,0 +1,361 @@
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
+// test_spherical_functional.cpp  (Phase 3c + 3e)
+//
+// Phase 3c — SphericalFunctional ported to ConformalMesh.
+//
+// Corresponds to de.varylab.discreteconformal.functional.SphericalFunctionalTest.
+//
+// Test map (Java → C++)
+// ──────────────────────
+//   testHessian (Ignored)         → GradientCheck_Hessian     (ported)
+//   testGradientWithHyperIdeal…   → GradientCheck_OctaFaceVertex   (ported)
+//   testGradientInExtendedDomain  → GradientCheck_SpherTetVertex    (ported)
+//   testGradientWithHyperelliptic → GradientCheck_SpherTetAllDofs   (ported)
+//   testFunctionalAtNaNValue      → AnglesFiniteAtKnownPoint         (ported)
+//
+// Energy model
+// ────────────
+//   The energy is computed as the Schläfli path integral E(x) = ∫₀¹⟨G(tx),x⟩dt
+//   using 10-point Gauss-Legendre quadrature. The gradient check therefore
+//   verifies that G is curl-free (the integrability / exactness condition of
+//   the spherical discrete conformal functional). This is equivalent to the
+//   Java FunctionalTest gradient check.
+
+#include "conformal_mesh.hpp"
+#include "mesh_builder.hpp"
+#include "spherical_functional.hpp"
+#include "spherical_hessian.hpp"
+#include <gtest/gtest.h>
+#include <cmath>
+#include <vector>
+
+using namespace conformallab;
+
+// ════════════════════════════════════════════════════════════════════════════
+// Cross-module Hessian check: spherical_gradient() ↔ spherical_hessian()
+//
+// Java @Ignore reason: "no Hessian implemented" — the Java functional test
+// was written before the Hessian existed.  In C++ the analytic spherical
+// Hessian (spherical_hessian.hpp, Phase 3f) is complete.
+//
+// This test verifies cross-module consistency between the functional and
+// the Hessian module.  The spherical Hessian is NSD (negative semi-definite)
+// because the spherical energy is concave — hessian_check_spherical() uses
+// the sign-corrected FD check appropriate for the spherical case.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(SphericalFunctional, GradientCheck_Hessian)
+{
+    auto mesh = make_spherical_tetrahedron();
+    auto maps = setup_spherical_maps(mesh);
+    compute_lambda0_from_mesh(mesh, maps);
+    int n = assign_vertex_dof_indices(mesh, maps);
+
+    std::vector<double> x(static_cast<std::size_t>(n), -0.2);
+
+    EXPECT_TRUE(hessian_check_spherical(mesh, x, maps))
+        << "Cross-module: spherical_gradient() and spherical_hessian() are inconsistent";
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Angle formula: octahedron-face triangle has all angles = π/2
+//
+// The triangle (1,0,0)–(0,1,0)–(0,0,1) has l_ij = π/2 for all edges.
+// Half-angle formula: s = 3π/4, s_ij = π/4 for all three.
+// All angles = π/2 (right-angled spherical triangle).
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(SphericalFunctional, OctaFaceAnglesAreRightAngles)
+{
+    // l_ij = π/2 for all edges (octahedron face on unit sphere)
+    const double l = PI_SPHER / 2.0;
+    auto fa = spherical_angles(l, l, l);
+
+    ASSERT_TRUE(fa.valid) << "Equilateral spherical triangle must be valid";
+    EXPECT_NEAR(PI_SPHER / 2.0, fa.alpha1, 1e-12);
+    EXPECT_NEAR(PI_SPHER / 2.0, fa.alpha2, 1e-12);
+    EXPECT_NEAR(PI_SPHER / 2.0, fa.alpha3, 1e-12);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Angle sum of a spherical triangle exceeds π (positive curvature)
+//
+// For the spherical tetrahedron face (arccos(−1/3) ≈ 1.9106 per edge):
+// The dihedral angle = arccos(1/3) ≈ 70.53°; by symmetry the face angles
+// (vertex angles of the spherical triangle) are all equal.
+// Angle sum must be > π and equal 3·arccos(1/3) ≈ 3·1.2310 ≈ 3.693 rad.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(SphericalFunctional, SpherTetAngleSumExceedsPi)
+{
+    // Edge length of spherical tetrahedron face: arccos(−1/3)
+    const double l = std::acos(-1.0 / 3.0);
+    auto fa = spherical_angles(l, l, l);
+
+    ASSERT_TRUE(fa.valid);
+    EXPECT_GT(fa.alpha1 + fa.alpha2 + fa.alpha3, PI_SPHER)
+        << "Angle sum of spherical triangle must exceed π";
+
+    // By symmetry all three angles must be equal
+    EXPECT_NEAR(fa.alpha1, fa.alpha2, 1e-12);
+    EXPECT_NEAR(fa.alpha2, fa.alpha3, 1e-12);
+
+    // For a regular spherical tetrahedron with edge arccos(−1/3):
+    // half-angle: tan(α/2) = √(sin(l/2)/sin(3l/2)) = √3 → α/2 = π/3 → α = 2π/3.
+    // (arccos(1/3) ≈ 1.231 is the 3D dihedral angle of a Euclidean tetrahedron, not this.)
+    double expected = 2.0 * PI_SPHER / 3.0;   // 120°
+    EXPECT_NEAR(fa.alpha1, expected, 1e-10);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Gradient check: octahedron-face triangle, vertex DOFs only
+//
+// Sets λ° from mesh geometry (unit sphere), all u_i = −0.3 (slightly smaller).
+// Mirrors Java testGradientWithHyperIdeal… on a single-triangle mesh.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(SphericalFunctional, GradientCheck_OctaFaceVertex)
+{
+    auto mesh = make_octahedron_face();
+    auto maps = setup_spherical_maps(mesh);
+    compute_lambda0_from_mesh(mesh, maps);
+    int n = assign_vertex_dof_indices(mesh, maps);
+
+    // Small uniform conformal factor: shrink the triangle slightly.
+    std::vector<double> x(static_cast<std::size_t>(n), -0.3);
+
+    EXPECT_TRUE(gradient_check_spherical(mesh, x, maps))
+        << "Gradient check failed on octahedron-face triangle (vertex DOFs)";
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Gradient check: spherical tetrahedron (4 faces), vertex DOFs only
+//
+// Closed surface; exercises accumulation over multiple faces per vertex.
+// Mirrors Java testGradientInTheExtendedDomain.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(SphericalFunctional, GradientCheck_SpherTetVertex)
+{
+    auto mesh = make_spherical_tetrahedron();
+    auto maps = setup_spherical_maps(mesh);
+    compute_lambda0_from_mesh(mesh, maps);
+    int n = assign_vertex_dof_indices(mesh, maps);
+
+    std::vector<double> x(static_cast<std::size_t>(n), -0.2);
+
+    EXPECT_TRUE(gradient_check_spherical(mesh, x, maps))
+        << "Gradient check failed on spherical tetrahedron (vertex DOFs)";
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Gradient check: spherical tetrahedron, all DOFs (vertex + edge)
+//
+// Exercises the edge-gradient branch: G_e = α_opp⁺ + α_opp⁻ − π.
+// Mirrors Java testGradientWithHyperellipticCurve.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(SphericalFunctional, GradientCheck_SpherTetAllDofs)
+{
+    auto mesh = make_spherical_tetrahedron();
+    auto maps = setup_spherical_maps(mesh);
+    compute_lambda0_from_mesh(mesh, maps);
+    int n = assign_all_spherical_dof_indices(mesh, maps);
+
+    // Small but non-zero values; vertex DOFs negative, edge DOFs zero.
+    // Edge DOF adjusts the effective log-length Λ_ij = λ°_ij + u_i + u_j + λ_e.
+    std::vector<double> x(static_cast<std::size_t>(n), 0.0);
+    // Set vertex DOFs (indices 0..3) to -0.2 to keep triangle well-formed.
+    for (int i = 0; i < 4; ++i) x[static_cast<std::size_t>(i)] = -0.2;
+
+    EXPECT_TRUE(gradient_check_spherical(mesh, x, maps))
+        << "Gradient check failed on spherical tetrahedron (all DOFs)";
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Angles are finite at a known interior point
+//
+// Mirrors Java testFunctionalAtNaNValue: choose DOFs that could hit
+// a degenerate branch (l_ij → 0 or triangle inequality fails) and check
+// the gradient vector is free of NaN/Inf.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(SphericalFunctional, AnglesFiniteAtKnownPoint)
+{
+    auto mesh = make_spherical_tetrahedron();
+    auto maps = setup_spherical_maps(mesh);
+    compute_lambda0_from_mesh(mesh, maps);
+    int n = assign_vertex_dof_indices(mesh, maps);
+
+    // u_i = -1.5: contracts the triangle heavily but stays non-degenerate.
+    std::vector<double> x(static_cast<std::size_t>(n), -1.5);
+    auto G = spherical_gradient(mesh, x, maps);
+
+    for (std::size_t i = 0; i < G.size(); ++i) {
+        EXPECT_FALSE(std::isnan(G[i])) << "Gradient component " << i << " is NaN";
+        EXPECT_FALSE(std::isinf(G[i])) << "Gradient component " << i << " is Inf";
+    }
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Gradient check: fan-4 mesh on unit sphere, vertex DOFs only
+//
+// Make a fan of 4 triangles around the north pole (0,0,1);
+// rim vertices projected onto the equator.
+// Exercises high-valence vertex gradient accumulation.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(SphericalFunctional, GradientCheck_SpherFan4Vertex)
+{
+    // Build a fan with 4 spherical triangles manually (can't use make_fan
+    // directly because those vertices are not on the unit sphere).
+    ConformalMesh mesh;
+    auto center = mesh.add_vertex(Point3(0, 0, 1));  // north pole
+    const int n_rim = 4;
+    std::vector<Vertex_index> rim(n_rim);
+    const double dtheta = 2.0 * PI_SPHER / n_rim;
+    const double phi    = PI_SPHER / 4.0;  // 45° colatitude
+    for (int i = 0; i < n_rim; ++i) {
+        double theta = i * dtheta;
+        rim[i] = mesh.add_vertex(Point3(
+            std::sin(phi) * std::cos(theta),
+            std::sin(phi) * std::sin(theta),
+            std::cos(phi)));
+    }
+    for (int i = 0; i < n_rim; ++i)
+        mesh.add_face(center, rim[i], rim[(i + 1) % n_rim]);
+
+    auto maps = setup_spherical_maps(mesh);
+    compute_lambda0_from_mesh(mesh, maps);
+    int ndof = assign_vertex_dof_indices(mesh, maps);
+
+    std::vector<double> x(static_cast<std::size_t>(ndof), -0.3);
+
+    EXPECT_TRUE(gradient_check_spherical(mesh, x, maps))
+        << "Gradient check failed on spherical fan-4 mesh";
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Gradient check: mixed pinned/variable vertices
+//
+// One vertex pinned (u_v = 0 fixed), others variable.
+// Verifies that the gradient accumulation skips pinned vertices correctly.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(SphericalFunctional, GradientCheck_MixedPinnedVertices)
+{
+    auto mesh = make_octahedron_face();
+    auto maps = setup_spherical_maps(mesh);
+    compute_lambda0_from_mesh(mesh, maps);
+
+    // Pin v0, make v1 and v2 variable.
+    auto vit = mesh.vertices().begin();
+    Vertex_index v0 = *vit++;
+    Vertex_index v1 = *vit++;
+    Vertex_index v2 = *vit;
+
+    maps.v_idx[v0] = -1;   // pinned
+    maps.v_idx[v1] = 0;
+    maps.v_idx[v2] = 1;
+
+    std::vector<double> x = {-0.2, -0.4};
+
+    EXPECT_TRUE(gradient_check_spherical(mesh, x, maps))
+        << "Gradient check failed for mixed pinned/variable vertices";
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Phase 3e — Gauge-fix for closed spherical surfaces
+//
+// On a closed spherical surface, the functional has a gauge mode:
+//   E(u + t·1)  is maximised at some t*.
+//   At t*, the sum of all vertex gradients equals zero: Σ G_v = 0.
+//
+// Test: start from a point with non-zero ΣG_v, apply the gauge shift,
+// and verify ΣG_v(x + t*·1) ≈ 0.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(SphericalFunctional, GaugeFix_SpherTetVertexZerosSumGv)
+{
+    auto mesh = make_spherical_tetrahedron();
+    auto maps = setup_spherical_maps(mesh);
+    compute_lambda0_from_mesh(mesh, maps);
+    int n = assign_vertex_dof_indices(mesh, maps);
+
+    // Off-gauge starting point: all u_i = -0.5
+    std::vector<double> x(static_cast<std::size_t>(n), -0.5);
+
+    // Compute ΣG_v before gauge shift.
+    {
+        auto G = spherical_gradient(mesh, x, maps);
+        double sum = 0.0;
+        for (auto v : mesh.vertices()) {
+            int iv = maps.v_idx[v];
+            if (iv >= 0) sum += G[static_cast<std::size_t>(iv)];
+        }
+        // At -0.5 the surface is compressed; ΣG_v should be non-zero.
+        EXPECT_NE(sum, 0.0) << "Pre-gauge ΣG_v should be non-zero";
+    }
+
+    // Compute gauge shift and apply.
+    double t = spherical_gauge_shift(mesh, x, maps);
+    std::vector<double> x_fixed = x;
+    for (auto v : mesh.vertices()) {
+        int iv = maps.v_idx[v];
+        if (iv >= 0)
+            x_fixed[static_cast<std::size_t>(iv)] += t;
+    }
+
+    // Verify ΣG_v ≈ 0 at the gauge-fixed point.
+    {
+        auto G = spherical_gradient(mesh, x_fixed, maps);
+        double sum = 0.0;
+        for (auto v : mesh.vertices()) {
+            int iv = maps.v_idx[v];
+            if (iv >= 0) sum += G[static_cast<std::size_t>(iv)];
+        }
+        EXPECT_NEAR(sum, 0.0, 1e-6)
+            << "After gauge fix, Σ G_v should vanish; t* = " << t;
+    }
+}
+
+TEST(SphericalFunctional, GaugeFix_ApplyInPlace)
+{
+    auto mesh = make_spherical_tetrahedron();
+    auto maps = setup_spherical_maps(mesh);
+    compute_lambda0_from_mesh(mesh, maps);
+    int n = assign_vertex_dof_indices(mesh, maps);
+
+    // x = -0.3: compressed but inside the valid spherical domain.
+    std::vector<double> x(static_cast<std::size_t>(n), -0.3);
+
+    apply_spherical_gauge(mesh, x, maps);
+
+    // After in-place gauge fix, ΣG_v must be near 0.
+    auto G = spherical_gradient(mesh, x, maps);
+    double sum = 0.0;
+    for (auto v : mesh.vertices()) {
+        int iv = maps.v_idx[v];
+        if (iv >= 0) sum += G[static_cast<std::size_t>(iv)];
+    }
+    EXPECT_NEAR(sum, 0.0, 1e-6)
+        << "apply_spherical_gauge must drive Σ G_v to zero";
+}
+
+TEST(SphericalFunctional, GaugeFix_AlreadyAtGaugeReturnsTNearZero)
+{
+    // A symmetric, equilateral spherical tetrahedron at x=0 is already
+    // at the gauge maximum (by symmetry, ΣG_v = 0).
+    auto mesh = make_spherical_tetrahedron();
+    auto maps = setup_spherical_maps(mesh);
+    compute_lambda0_from_mesh(mesh, maps);
+    int n = assign_vertex_dof_indices(mesh, maps);
+
+    std::vector<double> x(static_cast<std::size_t>(n), 0.0);
+    double t = spherical_gauge_shift(mesh, x, maps);
+    // Symmetric starting point → t* should be very close to 0.
+    EXPECT_NEAR(t, 0.0, 1e-5)
+        << "Gauge shift from the symmetric point should be ~0; got " << t;
+}

code/tests/cgal/test_spherical_hessian.cpp

@@ -0,0 +1,208 @@
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
+// test_spherical_hessian.cpp
+//
+// Phase 3f — Spherical cotangent-Laplace Hessian.
+//
+// The Hessian of the spherical discrete conformal energy is the "spherical
+// cotangent Laplacian" with edge weight
+//   w_k = cot(β_k),  β_k = (π − αi − αj + αk) / 2
+// for the edge (vi, vj) with opposite vertex vk and angles αi, αj, αk.
+//
+// In the flat limit (αi+αj+αk → π) this reduces to the Euclidean cotangent
+// Laplacian (β_k → αk, w_k → cot(αk)).
+//
+// Tests:
+//   1. Spherical cot weights match Euclidean weights in the near-flat limit.
+//   2. Hessian is symmetric.
+//   3. Hessian has H·1 ≈ 0 on the spherical tetrahedron (null-space property).
+//   4. Finite-difference check (the primary correctness criterion).
+
+#include "conformal_mesh.hpp"
+#include "mesh_builder.hpp"
+#include "spherical_hessian.hpp"
+#include "euclidean_hessian.hpp"  // for Euclidean comparison
+#include <gtest/gtest.h>
+#include <Eigen/Dense>
+#include <cmath>
+#include <vector>
+
+using namespace conformallab;
+
+// ════════════════════════════════════════════════════════════════════════════
+// Spherical cot weights reduce to Euclidean cot weights in the flat limit
+//
+// For a nearly-flat equilateral spherical triangle (l → 0, α → 60°):
+//   β_k = (π − 60° − 60° + 60°)/2 = 60°  →  cot(60°) = 1/√3  ✓
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(SphericalHessian, CotWeights_NearFlatEquilateral)
+{
+    // Use a very small equilateral spherical triangle: α1=α2=α3=60°
+    const double alpha = PI / 3.0;
+    auto sw = spherical_cot_weights(alpha, alpha, alpha);
+
+    ASSERT_TRUE(sw.valid);
+
+    const double expected = 1.0 / std::sqrt(3.0);   // = cot(60°)
+    EXPECT_NEAR(sw.w12, expected, 1e-12);
+    EXPECT_NEAR(sw.w23, expected, 1e-12);
+    EXPECT_NEAR(sw.w31, expected, 1e-12);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Spherical cot weights are positive for the regular spherical tetrahedron
+//
+// Each face has α_k = 2π/3 (120°), angle sum = 2π.
+// β_k = (π − 2π/3 − 2π/3 + 2π/3)/2 = (π − 2π/3)/2 = π/6
+// w_k = cot(π/6) = √3
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(SphericalHessian, CotWeights_RegularSphericalTetrahedronFace)
+{
+    const double alpha = 2.0 * PI / 3.0;   // 120°
+    auto sw = spherical_cot_weights(alpha, alpha, alpha);
+
+    ASSERT_TRUE(sw.valid);
+
+    const double expected = std::sqrt(3.0);   // cot(π/6)
+    EXPECT_NEAR(sw.w12, expected, 1e-10);
+    EXPECT_NEAR(sw.w23, expected, 1e-10);
+    EXPECT_NEAR(sw.w31, expected, 1e-10);
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Hessian is symmetric: H[i,j] == H[j,i]
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(SphericalHessian, HessianIsSymmetric)
+{
+    auto mesh = make_spherical_tetrahedron();
+    auto maps = setup_spherical_maps(mesh);
+    compute_lambda0_from_mesh(mesh, maps);
+    int n = assign_vertex_dof_indices(mesh, maps);
+
+    std::vector<double> x(static_cast<std::size_t>(n), -0.2);
+    auto H = spherical_hessian(mesh, x, maps);
+
+    Eigen::MatrixXd Hd = Eigen::MatrixXd(H);
+    EXPECT_NEAR((Hd - Hd.transpose()).norm(), 0.0, 1e-12)
+        << "Spherical Hessian must be symmetric";
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// H·1 is NOT zero for the spherical Hessian (unlike the Euclidean case).
+//
+// In the Euclidean case, a uniform shift u_i → u_i + c scales all edge
+// lengths by e^c, leaving angles unchanged → H·1 = 0 exactly.
+//
+// In the spherical case, l_ij = 2·asin(exp(λ_ij/2)) is NOT a linear
+// function of the DOFs, so a uniform shift DOES change the angles
+// → H·1 ≠ 0 in general.  This test verifies that property.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(SphericalHessian, ConstantVectorNotInNullSpace)
+{
+    auto mesh = make_spherical_tetrahedron();
+    auto maps = setup_spherical_maps(mesh);
+    compute_lambda0_from_mesh(mesh, maps);
+    int n = assign_vertex_dof_indices(mesh, maps);
+
+    std::vector<double> x(static_cast<std::size_t>(n), -0.2);
+    auto H = spherical_hessian(mesh, x, maps);
+
+    Eigen::VectorXd ones = Eigen::VectorXd::Ones(n);
+    Eigen::VectorXd Hones = H * ones;
+
+    // H·1 should be non-trivial (norm well above zero)
+    EXPECT_GT(Hones.norm(), 1e-3)
+        << "Spherical H·1 should be non-zero; norm = " << Hones.norm();
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Hessian is negative semi-definite at the equilibrium point (x = 0)
+//
+// The spherical discrete conformal energy is concave in the vertex DOFs
+// (unlike the Euclidean case which is convex).  At the equilibrium x = 0,
+// the Hessian is NSD: all eigenvalues ≤ 0, with a multi-dimensional null
+// space corresponding to degenerate directions.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(SphericalHessian, HessianIsNegativeSemiDefiniteAtEquilibrium)
+{
+    auto mesh = make_spherical_tetrahedron();
+    auto maps = setup_spherical_maps(mesh);
+    compute_lambda0_from_mesh(mesh, maps);
+    int n = assign_vertex_dof_indices(mesh, maps);
+
+    // x = 0 is the equilibrium for the regular spherical tetrahedron.
+    std::vector<double> x(static_cast<std::size_t>(n), 0.0);
+    auto H = spherical_hessian(mesh, x, maps);
+
+    Eigen::MatrixXd Hd = Eigen::MatrixXd(H);
+    Eigen::SelfAdjointEigenSolver<Eigen::MatrixXd> es(Hd);
+    double max_ev = es.eigenvalues().maxCoeff();
+
+    EXPECT_LE(max_ev, 1e-9)
+        << "Largest eigenvalue of spherical Hessian at equilibrium must be ≤ 0; got " << max_ev;
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Finite-difference Hessian check: octahedron-face triangle (vertex DOFs)
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(SphericalHessian, FDCheck_OctaFaceVertex)
+{
+    auto mesh = make_octahedron_face();
+    auto maps = setup_spherical_maps(mesh);
+    compute_lambda0_from_mesh(mesh, maps);
+    int n = assign_vertex_dof_indices(mesh, maps);
+
+    std::vector<double> x(static_cast<std::size_t>(n), -0.3);
+
+    EXPECT_TRUE(hessian_check_spherical(mesh, x, maps))
+        << "FD Hessian check failed on octahedron-face triangle";
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Finite-difference Hessian check: spherical tetrahedron (vertex DOFs)
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(SphericalHessian, FDCheck_SpherTetVertex)
+{
+    auto mesh = make_spherical_tetrahedron();
+    auto maps = setup_spherical_maps(mesh);
+    compute_lambda0_from_mesh(mesh, maps);
+    int n = assign_vertex_dof_indices(mesh, maps);
+
+    std::vector<double> x(static_cast<std::size_t>(n), -0.2);
+
+    EXPECT_TRUE(hessian_check_spherical(mesh, x, maps))
+        << "FD Hessian check failed on spherical tetrahedron";
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Finite-difference Hessian check: mixed pinned/variable vertices
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(SphericalHessian, FDCheck_MixedPinnedVertices)
+{
+    auto mesh = make_octahedron_face();
+    auto maps = setup_spherical_maps(mesh);
+    compute_lambda0_from_mesh(mesh, maps);
+
+    auto vit = mesh.vertices().begin();
+    Vertex_index v0 = *vit++;
+    Vertex_index v1 = *vit++;
+    Vertex_index v2 = *vit;
+
+    maps.v_idx[v0] = -1;   // pinned
+    maps.v_idx[v1] = 0;
+    maps.v_idx[v2] = 1;
+
+    std::vector<double> x = {-0.2, -0.3};
+
+    EXPECT_TRUE(hessian_check_spherical(mesh, x, maps))
+        << "FD Hessian check failed for mixed pinned/variable vertices";
+}

code/tests/test_clausen.cpp

@@ -1,3 +1,6 @@
+// Copyright (c) 2024-2026 Tarik Moussa.
+// SPDX-License-Identifier: MIT
+
 // Port of de.varylab.discreteconformal.functional.ClausenTest (Java/JUnit).
 // Reference values computed with Mathematica.