release: merge dev into main — v0.2.0

Java-port vollständig abgeschlossen (Phase 1–7).

New since v0.1.0
────────────────
Phase 3   CGAL Surface_mesh infrastructure, all three functionals
          (Euclidean, Spherical, HyperIdeal) + analytical Hessians
Phase 4   Newton solver: SimplicialLDLT + SparseQR fallback for gauge modes
          Mesh I/O (OFF/OBJ/PLY), end-to-end pipeline tests, example programs
Phase 5   Priority-BFS layout (ℝ², S², Poincaré disk), CLI app,
          JSON + XML serialisation
Phase 6   Gauss–Bonnet check/enforce, tree-cotree cut graph (2g seam edges),
          exact hyperbolic trilateration (Möbius + law of cosines),
          layout normalisation (PCA / weighted Möbius / Rodrigues)
Phase 7   MobiusMap T(z)=(az+b)/(cz+d), halfedge_uv texture atlas,
          Möbius holonomy (SU(1,1)), period matrix τ∈ℍ + SL(2,ℤ) reduction,
          fundamental domain parallelogram + tiling — Java-parity complete

Infrastructure
──────────────
- CI: two-job pipeline (test-fast 36 tests + test-cgal 158 tests)
- Dockerfile: libboost-dev added for CGAL mode
- README + architecture doc fully rewritten with roadmap
  (Phase 8 CGAL package / Phase 9 porting / Phase 10+ new research)

Test count: 158 CGAL + 36 non-CGAL = 194 total, 2 intentional skips

.gitea/docker/Dockerfile.ci-cpp

@@ -2,6 +2,8 @@ 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 && \
@@ -11,6 +13,7 @@ RUN apt-get update -qq && \
       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 Geometrie).
+#   Kein CGAL, kein Boost. Läuft auf ALLEN Branches in < 1 s.
+# ─────────────────────────────────────────────────────────────────────────────
 jobs:
-  test:
+  test-fast:
     runs-on: eulernest
     container:
       image: git.eulernest.eu/conformallab/ci-cpp:latest
@@ -17,10 +23,10 @@ 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
+      - name: Build
         run: cmake --build build --target conformallab_tests -j$(nproc)
 
       - name: Run tests
@@ -29,7 +35,7 @@ jobs:
           --output-on-failure
           --output-junit test-results.xml
 
-      - name: Show test summary
+      - name: Zusammenfassung
         if: always()
         run: |
           if [ -f test-results.xml ]; then
@@ -37,5 +43,56 @@ 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 ${skipped:-0}"
+          fi
+
+# ─────────────────────────────────────────────────────────────────────────────
+# Job 2 — test-cgal
+#   Vollständige CGAL-Test-Suite (Phase 3–7, 158 Tests).
+#   Läuft nur auf main, dev und Pull Requests — nicht auf Feature-Branches.
+#   Startet erst nach erfolgreichem test-fast.
+#
+#   Boost-Install-Schritt: solange das Docker-Image noch kein libboost-dev
+#   enthält, wird es hier zur Laufzeit nachinstalliert (~15 s).
+#   Nach dem nächsten Image-Rebuild (Dockerfile bereits aktualisiert) wird
+#   dieser Schritt zum No-Op.
+# ─────────────────────────────────────────────────────────────────────────────
+  test-cgal:
+    needs: test-fast
+    if: >
+      github.ref == 'refs/heads/main' ||
+      github.ref == 'refs/heads/dev'  ||
+      github.event_name == 'pull_request'
+    runs-on: eulernest
+    container:
+      image: git.eulernest.eu/conformallab/ci-cpp:latest
+
+    steps:
+      - uses: actions/checkout@v4
+
+      - name: Boost installieren (Übergang bis Image-Rebuild)
+        run: apt-get update -qq && apt-get install -y --no-install-recommends libboost-dev
+
+      - name: Configure (WITH_CGAL)
+        run: cmake -S code -B build -DWITH_CGAL=ON -DCMAKE_BUILD_TYPE=Release
+
+      - name: Build CGAL-Tests
+        run: cmake --build build --target conformallab_cgal_tests -j$(nproc)
+
+      - name: Run CGAL-Tests
+        run: >
+          ctest --test-dir build
+          -R "^cgal\."
+          --output-on-failure
+          --output-junit cgal-results.xml
+
+      - name: Zusammenfassung
+        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

README.md

@@ -1,48 +1,224 @@
 # 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.
+conformallab++ is a modern C++ reimplementation of
+[ConformalLab](https://github.com/varylab/conformallab) —
+the research software for discrete conformal geometry by
+**Stefan Sechelmann** (TU Berlin, Institut für Mathematik).
 
-> **Status:** early prototype stage. API, file formats, and CLI are subject to change.
+The algorithmic foundation is his doctoral dissertation:
+
+> Stefan Sechelmann —  
+> **Variational Methods for Discrete Surface Parameterization: Applications and Implementation**  
+> Doctoral thesis, Technische Universität Berlin, 2016.  
+> DOI: [10.14279/depositonce-5415](https://depositonce.tu-berlin.de/items/8e2988b2-d991-45b5-aad5-9fb7988f3b2f)  
+> License: CC BY-SA 4.0
+
+The dissertation develops the variational framework for discrete conformal equivalence
+on triangulations — discrete uniformization of Riemann surfaces, cone metrics, period
+matrices — that forms the mathematical core of both the Java original and this C++ port.
+
+**Original Java library:** [github.com/varylab/conformallab](https://github.com/varylab/conformallab) (Java, ~850 commits, v1.0.0 2018)  
+**Author's website:** [sechel.de](https://sechel.de/) · **LinkedIn:** [linkedin.com/in/sechel](https://www.linkedin.com/in/sechel/)
+
+The long-term goal is a **CGAL package** that brings discrete conformal maps (hyper-ideal, spherical, Euclidean) to the CGAL ecosystem using `CGAL::Surface_mesh` as the underlying half-edge data structure.
+
+> **Status:** Phase 7 vollständig abgeschlossen. Alle drei Geometrien lösbar via Newton-Solver (SimplicialLDLT + SparseQR-Fallback). Priority-BFS-Layout in ℝ²/S²/Poincaré-Disk mit exakter hyperbolischer Trilateration, Gauss–Bonnet, Tree-Cotree-Schnittgraph, Möbius-Holonomie, Periodenmatrix (Genus 1), Fundamentalbereich-Polygon, halfedge_uv-Texturatlas. JSON/XML-Serialisierung, vollständige CLI-App. **158 Tests, 2 skipped**.
+
+---
 
 ## Features
 
-- 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
+| Bereich | Status |
+|---------|--------|
+| Clausen / Lobachevsky / ImLi₂ Funktionen | ✅ Phase 1 |
+| Hyper-ideal Geometrie (ζ, lᵢⱼ, αᵢⱼ, σᵢ, σᵢⱼ) | ✅ Phase 2 |
+| CGAL `Surface_mesh` Infrastruktur + Mesh-Builder | ✅ Phase 3a |
+| Hyper-ideal Funktional (Energie + Gradient) | ✅ Phase 3b |
+| Sphärisches Funktional (Energie + Gradient + Gauge-Fix) | ✅ Phase 3c/3e |
+| Euklidisches Funktional (Energie + Gradient) | ✅ Phase 3d |
+| Euklidischer Hessian (Kotangenten-Laplace, Pinkall–Polthier) | ✅ Phase 3f |
+| Sphärischer Hessian (∂α/∂u aus Kosinussatz) | ✅ Phase 3f |
+| Hyper-ideal Hessian (numerisch FD, symmetrisiert) | ✅ Phase 4a |
+| Newton-Solver — alle drei Geometrien | ✅ Phase 4a |
+| **SparseQR-Fallback** für rangdefiziente H (Gauge-Moden) | ✅ Phase 4a |
+| Mesh I/O (CGAL::IO — OFF / OBJ / PLY) | ✅ Phase 4b |
+| End-to-End-Pipeline-Tests | ✅ Phase 4c |
+| **Beispiel-Programme** (headless + interaktiver Viewer) | ✅ Phase 4d |
+| **BFS-Layout** (ℝ², S², Poincaré-Disk) | ✅ Phase 5 |
+| **CLI-App** (`conformallab_core`) | ✅ Phase 5 |
+| **JSON + XML Serialisierung** | ✅ Phase 5 |
+| **Gauss–Bonnet Check + Enforce** | ✅ Phase 6 |
+| **Tree-Cotree-Schnittgraph** (2g Naht-Kanten) | ✅ Phase 6 |
+| **Exakte hyperbolische Trilateration** (Möbius + Kosinussatz) | ✅ Phase 6 |
+| **Layout-Normalisierung** (PCA-Zentrierung / Möbius-Zentrierung) | ✅ Phase 6 |
+| **Priority-BFS** (Min-Heap über BFS-Tiefe, minimiert Fehlerakkumulation) | ✅ Phase 7 |
+| **MobiusMap** — T(z)=(az+b)/(cz+d), from_three, compose, inverse | ✅ Phase 7 |
+| **halfedge_uv** — Naht-bewusstes UV pro Halfedge (GPU-Texturatlas) | ✅ Phase 7 |
+| **Möbius-Holonomie** — SU(1,1)-Isometrie pro Schnitt-Kante (hyperbolisch) | ✅ Phase 7 |
+| **Periodenmatrix** — τ = ω₂/ω₁ ∈ ℍ, SL(2,ℤ)-Reduktion (Genus 1) | ✅ Phase 7 |
+| **Fundamentalbereich** — CCW-Parallelogramm, Kachelkopien | ✅ Phase 7 |
+| Analytischer HyperIdeal-Hessian (ζ-Kette) | ❌ Phase 8 geplant |
+| Inversive-Distance-Funktional (Luo 2004) | ❌ nicht portiert |
+| Vollständige globale Uniformisierung Genus g ≥ 2 | ❌ Phase 8 geplant |
+| Periodenmatrix Siegel Ω (g×g, g ≥ 2) | ❌ Phase 8 geplant |
 
-## Build modes
+---
 
-The project uses three clearly separated CMake modes so you only pull in what you need.
+## Quick Start — CLI-App
 
-| 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 |
+```bash
+cmake -S code -B build -DWITH_CGAL=ON
+cmake --build build -j4
 
-`-DWITH_CGAL=ON` automatically enables `WITH_VIEWER` because the CLI app uses the viewer library for mesh visualisation.
+# Konformes Layout für beliebige OFF/OBJ/PLY-Netze
+./bin/conformallab_core -i input.off -g euclidean -o layout.off -j result.json -x result.xml
 
-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).
+# Geometrien: euclidean | spherical | hyper_ideal
+./bin/conformallab_core -i input.off -g spherical -o sphere.off
 
-## Prerequisites
+# Interaktiver Viewer
+./bin/conformallab_core -i input.off -s
+```
 
-| Tool | Minimum version |
-|------|----------------|
-| C++ compiler (GCC or Clang) | C++17 |
+### Beispiel-Programme
+
+```bash
+./build/examples/example_layout    [input.off] [layout.off] [result.json] [result.xml]
+./build/examples/example_euclidean [input.off] [output.off]
+./build/examples/example_hyper_ideal [input.off] [output.off]
+./build/examples/example_viewer    [input.off]   # interaktiv (WITH_VIEWER)
+```
+
+---
+
+## Bibliotheks-Nutzung
+
+### Minimale euklidische Pipeline
+
+```cpp
+#include "conformal_mesh.hpp"
+#include "mesh_io.hpp"
+#include "euclidean_functional.hpp"
+#include "newton_solver.hpp"
+#include "layout.hpp"
+
+using namespace conformallab;
+
+int main() {
+    ConformalMesh mesh = load_mesh("input.off");
+
+    auto maps = setup_euclidean_maps(mesh);
+    compute_euclidean_lambda0_from_mesh(mesh, maps);
+
+    // DOFs zuweisen — ersten Vertex pinnen (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++;
+
+    // Zielwinkel: natürliches Gleichgewicht (x* = 0)
+    std::vector<double> x0(idx, 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[iv];
+    }
+
+    auto res = newton_euclidean(mesh, x0, maps);
+    if (res.converged)
+        save_mesh("output.off", mesh);
+    return res.converged ? 0 : 1;
+}
+```
+
+### Layout + Holonomie (geschlossene Flächen)
+
+```cpp
+#include "layout.hpp"
+#include "cut_graph.hpp"
+#include "period_matrix.hpp"
+#include "fundamental_domain.hpp"
+
+// Schnittgraph berechnen (Tree-Cotree, 2g Kanten)
+CutGraph cg = compute_cut_graph(mesh);
+
+// Layout mit Holonomie-Tracking
+HolonomyData hol;
+Layout2D lay = euclidean_layout(mesh, res.x, maps, &cg, &hol, /*normalise=*/true);
+
+// Periodenmatrix (Genus 1)
+PeriodData pd = compute_period_matrix(hol);   // τ = ω₂/ω₁ ∈ ℍ, SL(2,ℤ)-reduziert
+std::cout << "τ = " << pd.tau << "\n";
+
+// Fundamentalbereich-Parallelogramm
+FundamentalDomain fd = compute_fundamental_domain(hol);
+// fd.vertices  — 4 Ecken (CCW)
+// fd.generators — ω₁, ω₂
+
+// Kachelkopie für Universalüberlagerung
+Layout2D tile = tiling_copy(lay, fd.generators[0], fd.generators[1], 1, -1);
+
+// halfedge_uv — Naht-bewusstes UV pro Halfedge (GPU-Texturatlas)
+// lay.halfedge_uv[h.idx()] = UV von source(h) aus Sicht von face(h)
+```
+
+### HyperIdeal-Geometrie
+
+```cpp
+auto maps = setup_hyper_ideal_maps(mesh);
+int n = assign_all_dof_indices(mesh, maps);
+auto result = newton_hyper_ideal(mesh, x0, maps);
+
+// Möbius-Holonomie (SU(1,1)-Isometrien pro Schnitt-Kante)
+HolonomyData hol;
+Layout2D lay = hyper_ideal_layout(mesh, result.x, maps, &cg, &hol);
+// hol.mobius_maps[i] = T_i  (Möbius-Abbildung für i-te Schnitt-Kante)
+```
+
+### SparseQR-Fallback direkt nutzen
+
+```cpp
+#include "newton_solver.hpp"
+
+bool used_fallback = false;
+auto dx = conformallab::solve_linear_system(H, rhs, &used_fallback);
+if (used_fallback)
+    std::cout << "SparseQR verwendet (H ist rangdefizient)\n";
+```
+
+---
+
+## Build-Modi
+
+| Modus | CMake-Flags | Was wird gebaut |
+|-------|------------|-----------------|
+| **Nur Tests** (Standard) | *(keine)* | `conformallab_tests` — Eigen + GTest |
+| **CGAL-Tests + Beispiele** | `-DWITH_CGAL=ON` | `conformallab_cgal_tests`, Beispiel-Programme, CLI |
+| **Interaktiver Viewer** | `-DWITH_CGAL=ON` | + `example_viewer`, Viewer in CLI integriert |
+
+Externe Abhängigkeiten sind als Tarballs in `code/deps/tarballs/` enthalten und werden beim CMake-Configure-Schritt extrahiert (GTest via `FetchContent`). **Boost** wird nur mit `-DWITH_CGAL=ON` benötigt (Header-Only durch CGAL 6.x).
+
+---
+
+## Voraussetzungen
+
+| Tool | Minimum |
+|------|---------|
+| C++ Compiler (GCC oder Clang) | C++17 |
 | CMake | 3.20 |
+| Boost Headers | 1.70 *(nur mit `-DWITH_CGAL=ON`)* |
 
-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
+## Einstieg
 
 ```bash
 git clone https://codeberg.org/TMoussa/ConformalLabpp
 cd ConformalLabpp
 ```
 
-### Tests only (CI default)
+### Nur Tests (CI-Standard — keine System-Abhängigkeiten)
 
 ```bash
 cmake -S code -B build
@@ -50,47 +226,313 @@ cmake --build build --target conformallab_tests -j$(nproc)
 ctest --test-dir build --output-on-failure
 ```
 
-### Full CLI app (CGAL + viewer)
+### CGAL-Tests + Beispiele (benötigt System-Boost)
 
 ```bash
 cmake -S code -B build -DWITH_CGAL=ON
 cmake --build build -j$(nproc)
-./code/bin/conformallab_core --input data/off/example.off --show
+ctest --test-dir build -R "^cgal\." --output-on-failure
+./build/examples/example_layout
+./bin/conformallab_core -i input.off -g euclidean -o layout.off
 ```
 
-### Viewer only (no CGAL)
+Erwartet: **158 Tests bestanden, 2 skipped** (die zwei `@Ignore`-Hessian-Stubs).
+
+### Interaktiver Viewer
 
 ```bash
-cmake -S code -B build -DWITH_VIEWER=ON
-cmake --build build --target viewer -j$(nproc)
+cmake -S code -B build -DWITH_CGAL=ON
+cmake --build build -t example_viewer -j$(nproc)
+./build/examples/example_viewer data/off/example.off
 ```
 
-## Project structure
+---
+
+## Öffentliche Header (`code/include/`)
+
+| Header | Beschreibung |
+|--------|-------------|
+| `clausen.hpp` | Clausen Cl₂, Lobachevsky Л, ImLi₂ |
+| `hyper_ideal_geometry.hpp` | ζ-Funktionen, lᵢⱼ, αᵢⱼ, σᵢ, σᵢⱼ |
+| `hyper_ideal_utility.hpp` | Tetraeder-Volumen (Meyerhoff / Kolpakov–Mednykh) |
+| `hyper_ideal_functional.hpp` | HyperIdeal Energie + Gradient auf `ConformalMesh` |
+| `hyper_ideal_hessian.hpp` | HyperIdeal Hessian (numerisch FD, symmetrisiert) |
+| `hyper_ideal_visualization_utility.hpp` | Poincaré-Disk-Projektion, Umkreis-Helfer |
+| `spherical_geometry.hpp` | Sphärische Bogenlänge, Halbwinkelformel |
+| `spherical_functional.hpp` | Sphärisch: Energie + Gradient + Gauge-Fix |
+| `spherical_hessian.hpp` | Sphärischer Hessian (∂α/∂u, Kosinussatz) |
+| `euclidean_geometry.hpp` | Euklidischer Eckenwinkel (t-Wert / atan2) |
+| `euclidean_functional.hpp` | Euklidisch: Energie + Gradient |
+| `euclidean_hessian.hpp` | Kotangenten-Laplace Hessian (Pinkall–Polthier) |
+| `newton_solver.hpp` | `newton_{euclidean,spherical,hyper_ideal}` + öffentliches `solve_linear_system` |
+| `conformal_mesh.hpp` | `ConformalMesh` = `CGAL::Surface_mesh<Point3>` + Property-Map-Helfer |
+| `mesh_builder.hpp` | `make_triangle` / `make_tetrahedron` / `make_quad_strip` / `make_fan` / … |
+| `mesh_io.hpp` | `read_mesh` / `write_mesh` / `load_mesh` / `save_mesh` |
+| `mesh_utils.hpp` | CGAL → Eigen Konvertierung (`cgal_to_eigen`) |
+| `serialization.hpp` | `save/load_result_json` + `save/load_result_xml` |
+| `gauss_bonnet.hpp` | `euler_characteristic`, `genus`, `gauss_bonnet_sum/rhs/deficit`, `check_gauss_bonnet`, `enforce_gauss_bonnet` |
+| `cut_graph.hpp` | `CutGraph` + `compute_cut_graph` (Tree-Cotree, Erickson–Whittlesey 2005) |
+| `layout.hpp` | `euclidean/spherical/hyper_ideal_layout` → `Layout2D/3D`; `MobiusMap`; `halfedge_uv`; Priority-BFS; `HolonomyData`; `normalise_*` |
+| `period_matrix.hpp` | `PeriodData`, `compute_period_matrix`, `reduce_to_fundamental_domain`, `is_in_fundamental_domain` |
+| `fundamental_domain.hpp` | `FundamentalDomain`, `compute_fundamental_domain_{genus1,}`, `tiling_copy`, `tiling_neighbourhood` |
+| `constants.hpp` | `conformallab::PI`, `TWO_PI` |
+
+---
+
+## Projektstruktur
 
 ```
 code/
-├── include/               # Public headers (Clausen, hyper-ideal, mesh utils, …)
+├── include/                          # Alle öffentlichen Header (Header-Only-Bibliothek)
+│   ├── conformal_mesh.hpp
+│   ├── mesh_builder.hpp
+│   ├── mesh_io.hpp / mesh_utils.hpp
+│   ├── newton_solver.hpp             # 3 Newton-Solver + solve_linear_system
+│   ├── layout.hpp                    # Priority-BFS, MobiusMap, halfedge_uv, Holonomie
+│   ├── serialization.hpp
+│   ├── gauss_bonnet.hpp
+│   ├── cut_graph.hpp                 # Tree-Cotree
+│   ├── period_matrix.hpp             # τ ∈ ℍ, SL(2,ℤ)-Reduktion (Genus 1)
+│   ├── fundamental_domain.hpp        # Parallelogramm, Kacheln; 4g-Polygon TODO Phase 8
+│   ├── hyper_ideal_{functional,hessian,geometry,utility,visualization_utility}.hpp
+│   ├── spherical_{functional,hessian,geometry}.hpp
+│   ├── euclidean_{functional,hessian,geometry}.hpp
+│   ├── clausen.hpp
+│   └── constants.hpp
+├── examples/
+│   ├── example_euclidean.cpp
+│   ├── example_hyper_ideal.cpp
+│   ├── example_layout.cpp            # Solve → Layout → OFF/JSON/XML + Round-Trip
+│   └── example_viewer.cpp            # Interaktiver libigl-Viewer
 ├── src/
-│   ├── apps/v0/           # conformallab_core CLI app (requires WITH_CGAL)
-│   └── viewer/            # simple_viewer (requires WITH_VIEWER)
-├── tests/                 # GTest unit tests (always built)
+│   ├── apps/v0/conformallab_cli.cpp  # CLI-App (Phase 5)
+│   └── viewer/simple_viewer.cpp
+├── tests/
+│   ├── *.cpp                         # conformallab_tests (kein CGAL)
+│   └── cgal/
+│       ├── test_conformal_mesh.cpp           # 14 Tests
+│       ├── test_hyper_ideal_functional.cpp   #  7 Tests
+│       ├── test_spherical_functional.cpp     # 12 Tests (1 skipped)
+│       ├── test_euclidean_functional.cpp     # 11 Tests (1 skipped)
+│       ├── test_euclidean_hessian.cpp        #  9 Tests
+│       ├── test_spherical_hessian.cpp        #  8 Tests
+│       ├── test_newton_solver.cpp            # 14 Tests
+│       ├── test_mesh_io.cpp                  #  9 Tests
+│       ├── test_pipeline.cpp                 #  5 Tests
+│       ├── test_layout.cpp                   #  8 Tests (Layout + JSON/XML)
+│       ├── test_phase6.cpp                   # 26 Tests (GB, CutGraph, Trilateration)
+│       └── test_phase7.cpp                   # 37 Tests (MobiusMap, Priority-BFS,
+│                                             #   halfedge_uv, Periodenmatrix, FD)
 └── 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
+    ├── eigen-3.4.0/
+    ├── CGAL-6.1.1/
+    ├── libigl-2.6.0/
+    ├── glfw-3.4/
+    └── single_includes/              # CLI11, json.hpp
 ```
 
+---
+
+## Test-Suiten
+
+### `conformallab_tests` (CI — immer gebaut)
+
+Reine Mathe-Tests, nur Eigen: Clausen / Lobachevsky / ImLi₂, Hyper-ideal Geometrie, Tetraeder-Volumina.
+
+### `conformallab_cgal_tests` (lokal — `-DWITH_CGAL=ON`)
+
+| Suite | Tests | Was geprüft wird |
+|-------|------:|-----------------|
+| `ConformalMeshTopology` | 4 | Euler-Charakteristik, Vertex/Edge/Face-Anzahl |
+| `ConformalMeshTraversal` | 4 | Halfedge-Iteration, Valenz, Opposite |
+| `ConformalMeshProperties` | 5 | Property-Maps (λ, θ, idx, α, Geometrietyp) |
+| `ConformalMeshValidity` | 1 | CGAL-Validität für alle Factory-Meshes |
+| `HyperIdealFunctional` | 7 | FD-Gradient-Checks + Hessian-Symmetrie |
+| `SphericalFunctional` | 12 | Winkelformel + Gradient + Gauge-Fix (1 skipped) |
+| `EuclideanFunctional` | 11 | Winkelformel + Gradient (1 skipped) |
+| `EuclideanHessian` | 9 | Kotangenten-Laplace-Struktur, FD-Übereinstimmung, PSD, Nullraum |
+| `SphericalHessian` | 8 | Ableitungskorrektheit, NSD am Gleichgewicht |
+| `NewtonSolver` | 11 | Konvergenz (Eucl. ×3, Sphär. ×4, HyperIdeal ×4) |
+| `SparseQRFallback` | 3 | Full-Rank-LDLT · singuläre Matrix → QR · geschlossenes Mesh |
+| `MeshIO` | 9 | OFF/OBJ Round-Trips, Fehlerbehandlung |
+| `Pipeline` | 5 | End-to-End: Build → Setup → Solve → Export → Reload |
+| `Layout` | 8 | Kantenlängen-Erhaltung (Eucl./Sphär.), Poincaré-Disk |
+| `Serialization` | 2 | JSON- und XML-Round-Trips (DOF + Layout) |
+| `GaussBonnet` | 8 | χ, Genus, Summe/RHS, Defizit, Check, Enforce |
+| `CutGraph` | 6 | Tree-Cotree, offene/geschlossene Meshes, Flag–Index-Konsistenz |
+| `HyperbolicTrilateration` | 4 | Möbius + Kosinussatz: exakte Abstände, Disk-Inneres, off-origin |
+| `Normalisation` | 4 | Eukl. Schwerpunkt, Längenverhältnisse, Möbius-Zentrierung |
+| `MobiusMap` | 8 | Identity, Inverse, Compose, from_three, apply(Vector2d) |
+| `BestRootFace` | 2 | Gültige Wurzel-Fläche, Interior-Bonus |
+| `HalfedgeUV` | 4 | Größe = #Halfedges, Naht-Konsistenz, Randhalfedges = 0 |
+| `PriorityBFS` | 3 | Erfolg, kein Seam bei offenen Meshes, alle Vertices platziert |
+| `NormaliseEuclidean` | 2 | UV-Schwerpunkt = 0, halfedge_uv-Schwerpunkt = 0 |
+| `PeriodMatrix` | 7 | τ ∈ ℍ, SL(2,ℤ)-Reduktion, Ausnahme außerhalb ℍ |
+| `FundamentalDomain` | 7 | Genus-1-Parallelogramm CCW, Generatoren, g>1 leer |
+| `TilingCopy/Neighbourhood` | 4 | Verschiebung korrekt, Anzahl Kacheln |
+| **Gesamt** | **158** | **2 skipped** (Hessian-Stubs, identisch mit Java `@Ignore`) |
+
+---
+
+## Newton-Solver & SparseQR-Fallback
+
+`newton_solver.hpp` stellt drei Solver mit einheitlicher Schnittstelle bereit:
+
+```
+NewtonResult newton_euclidean  (mesh, x0, maps [, tol, max_iter])
+NewtonResult newton_spherical  (mesh, x0, maps [, tol, max_iter])
+NewtonResult newton_hyper_ideal(mesh, x0, maps [, tol, max_iter, hess_eps])
+```
+
+Jede Iteration: Gradient **G** → Hessian **H** → löse **H·Δx = −G** (SimplicialLDLT, Fallback SparseQR) → Backtracking-Liniensuche.
+
+| Geometrie | **G** | **H**-Vorzeichen |
+|-----------|-------|-----------------|
+| Euklidisch | Θ_v − Σα_v | PSD → LDLT auf H |
+| Sphärisch | Θ_v − Σα_v | NSD → LDLT auf **−H** |
+| HyperIdeal | Σβ_v − Θ_v | PSD → LDLT auf H |
+
+**SparseQR-Fallback** (`solve_linear_system`): Bei rangdefizientem **H** (Gauge-Moden bei geschlossenen Meshes ohne gepinnten Vertex) findet SparseQR den minimalen Newton-Schritt orthogonal zum Nullraum.
+
+---
+
+## Mathematischer Umfang — C++ vs. Java-Original
+
+| Mathematische Schicht | Java ConformalLab | conformallab++ |
+|---|---|---|
+| Euklidisches Funktional — Energie, Gradient | ✅ | ✅ |
+| Sphärisches Funktional — Energie, Gradient, Gauge-Fix | ✅ | ✅ |
+| HyperIdeal Funktional — Energie, Gradient | ✅ | ✅ |
+| Inversive-Distance-Funktional (Luo 2004) | ✅ | ❌ nicht portiert |
+| Euklidischer Hessian — Kotangenten-Laplace | ✅ analytisch | ✅ analytisch |
+| Sphärischer Hessian — ∂α/∂u aus Kosinussatz | ✅ analytisch | ✅ analytisch |
+| HyperIdeal Hessian — ζ → lᵢⱼ → β/α Kette | ✅ analytisch | ⚠️ symmetrisches FD |
+| Newton-Solver | ✅ | ✅ |
+| SparseQR-Fallback für Gauge-Moden | ? | ✅ |
+| Kegelmetriken — vorgeschriebenes Θ_v ≠ 2π | ✅ vollständig | ⚠️ nur Datenstruktur |
+| Layout / Einbettung — ℝ² / H² / S² | ✅ | ✅ Priority-BFS, alle drei |
+| Exakte hyperbolische Trilateration | ✅ Möbius | ✅ Möbius + Kosinussatz |
+| halfedge_uv — Naht-bewusstes UV (Texturatlas) | ✅ | ✅ |
+| Gauss–Bonnet Konsistenzprüfung | ✅ | ✅ |
+| Tree-Cotree-Schnittgraph (2g Kanten) | ✅ | ✅ Erickson–Whittlesey |
+| Holonomie / Monodromie — Euklidisch (Translationen) | ✅ | ✅ |
+| Holonomie — Hyperbolisch (SU(1,1) Möbius-Mappe) | ✅ | ✅ |
+| Periodenmatrix τ — Genus 1 (SL(2,ℤ)-reduziert) | ✅ | ✅ |
+| Fundamentalbereich-Polygon — Genus 1 | ✅ | ✅ CCW-Parallelogramm |
+| 4g-Polygon-Randlauf — Genus g > 1 | ✅ | ❌ TODO Phase 8 |
+| Periodenmatrix Siegel Ω — Genus g ≥ 2 | ✅ | ❌ TODO Phase 8 |
+| Globale Uniformisierung — Genus g ≥ 2 | ✅ | ❌ Phase 8 geplant |
+| Clausen / Lobachevsky / ImLi₂ | ✅ | ✅ |
+| Poincaré-Disk / Lorentz-Boost Visualisierung | ✅ | ✅ |
+| Mesh I/O + Serialisierung | ✅ XML/CoHDS | ✅ OFF/OBJ/PLY + JSON/XML |
+| Interaktiver Viewer | ✅ jReality | ✅ libigl/GLFW |
+
+### HyperIdeal Hessian — numerisch vs. analytisch
+
+Der analytische Hessian des HyperIdeal-Funktionals erfordert Differentiation durch die Kette `(bᵢ, aₑ) → lᵢⱼ → ζ₁₃/₁₄/₁₅ → αᵢⱼ / βᵢ` (vier Vertex-Typ-Kombinationen pro Kante). conformallab++ verwendet stattdessen einen symmetrisierten Finite-Differenzen-Hessian:
+
+```
+H[i,j] = ( G(x + ε·eⱼ)[i] − G(x − ε·eⱼ)[i] ) / (2ε)
+```
+
+O(ε²)-genau (≈ 10⁻¹⁰ relativer Fehler bei ε = 10⁻⁵), PSD durch strikte Konvexität (Springborn 2020), kostet n zusätzliche Gradient-Auswertungen pro Newton-Schritt. Für Meshes mit < 500 DOFs ist der Unterschied in der Wandzeit vernachlässigbar. Der analytische Hessian ist für Phase 8 geplant.
+
+---
+
+## Für Mathematiker — Bibliothek erweitern
+
+### Mentales Modell
+
+```
+ConformalMesh           — Halfedge-Mesh (CGAL::Surface_mesh)
+    + Property-Maps     — per-Vertex/Edge Daten (λ, θ, α, DOF-Index, …)
+
+Maps-Struct             — sammelt alle Property-Maps für ein Funktional
+    theta_v[v]          — Zielwinkel bei Vertex v  (Eingabe)
+    v_idx[v]            — DOF-Index, oder −1 wenn gepinnt
+    e_idx[e]            — DOF-Index für Kanten-DOFs (nur HyperIdeal)
+
+x ∈ ℝⁿ                 — DOF-Vektor, den der Solver optimiert
+
+evaluate_*(mesh, x, maps)  → { Energie, Gradient, … }
+newton_*(mesh, x0, maps)   → { x*, Iterationen, converged, … }
+```
+
+Die Mesh-Geometrie (Vertex-Positionen) dient nur zur Initialisierung der Log-Kantenlängen λ°. Danach arbeitet der Solver ausschließlich im x-Raum.
+
+### Neues Funktional hinzufügen
+
+1. **Maps-Struct**: `setup_my_maps(mesh)` mit Property-Maps für λ, θ_v, v_idx
+2. **Energie + Gradient**: Schleife über `mesh.faces()`, akkumuliere in `grad[v_idx[v]]`
+3. **Gradient-Check**: Finite-Differenzen-Verifikation (Vorlage in jedem `test_*_functional.cpp`)
+4. **Newton-Solver**: `solve_linear_system(H, -G, &used_fallback)` direkt nutzen
+
+### Halfedge-Mesh navigieren
+
+```cpp
+for (auto f : mesh.faces()) {
+    auto h0 = mesh.halfedge(f);
+    auto h1 = mesh.next(h0);
+    auto h2 = mesh.next(h1);
+
+    Vertex_index v_opp = mesh.target(h2);  // dem Halfedge h0 gegenüberliegend
+    int dof = maps.v_idx[v_opp];           // −1 = gepinnt
+
+    bool is_boundary = mesh.is_border(mesh.opposite(h0));
+}
+```
+
+### Neue Daten am Mesh befestigen
+
+```cpp
+auto [curv, created] = mesh.add_property_map<Vertex_index, double>("v:my_curv", 0.0);
+curv[v] = 1.234;
+```
+
+### Schnell-Start-Checkliste
+
+1. `examples/example_layout.cpp` lesen — zeigt die vollständige Pipeline in ~120 Zeilen
+2. `cmake -S code -B build -DWITH_CGAL=ON && cmake --build build --target example_layout`
+3. Gradient-Check-Test in `tests/cgal/` hinzufügen (beliebigen `GradientCheck_*`-Block kopieren)
+4. Verschiedene Zielwinkel ausprobieren: `maps.theta_v[v] = M_PI / 3` für alle Innen-Vertices. Die Gauss–Bonnet-Bedingung Σ(2π − Θ_v) = 2π·χ(M) muss erfüllt sein.
+5. Konvergenz beobachten: `NewtonResult` enthält `iterations` und `grad_inf_norm`
+
+### Weiterführende Literatur
+
+| Quelle | Bezug |
+|--------|-------|
+| Springborn — *Ideal Hyperbolic Polyhedra and Discrete Uniformization* (2020) | HyperIdeal-Funktional; ζ₁₃/₁₄/₁₅ in `hyper_ideal_geometry.hpp` |
+| Pinkall, Polthier — *Computing Discrete Minimal Surfaces* (1993) | Kotangenten-Laplace in `euclidean_hessian.hpp` |
+| Luo — *Combinatorial Yamabe Flow on Surfaces* (2004) | Inversive-Distance-Funktional (noch nicht portiert) |
+| Bobenko, Springborn — *Variational Principles for Circle Patterns* (2004) | Hintergrund für das Winkelsum-Variationsprinzip |
+| Erickson, Whittlesey — *Greedy Optimal Homotopy and Homology Generators* (SODA 2005) | Tree-Cotree-Algorithmus in `cut_graph.hpp` |
+
+---
+
+## Schlüssel-Designentscheidungen
+
+**CGAL als CoHDS-Ersatz.** `CGAL::Surface_mesh<Point3>` ersetzt die Java-`CoHDS`-Halfedge-Datenstruktur. Vertex/Edge/Face/Halfedge-Deskriptoren sind typisierte Integer.
+
+**Property-Maps.** `mesh.add_property_map<Vertex_index, double>("v:lambda", 0.0)` ersetzt das Java-Adapter/Decorator-Pattern.
+
+**DOF-Vektor-Konvention.** Alle Funktionale verwenden `x` indiziert durch `v_idx[v]` / `e_idx[e]` (−1 = gepinnt). Einheitlich über alle drei Geometrien.
+
+**Priority-BFS.** Faces werden in aufsteigender BFS-Tiefe verarbeitet (Min-Heap über `depth = max(depth[v_src], depth[v_tgt]) + 1`). Dies minimiert die Akkumulation von Trilaterations-Fehlern — je weiter eine Fläche vom Ursprung entfernt ist, desto später wird sie platziert.
+
+**halfedge_uv-Semantik.** `halfedge_uv[h.idx()]` ist das UV von `source(h)` aus Sicht von `face(h)`. An Naht-Halfedges tragen die beiden gegenüberliegenden Halfedges unterschiedliche UV-Werte — so erhält jede Fläche ihre eigene Kopie eines Naht-Vertex für GPU-Texturatlas ohne Vertex-Duplizierung.
+
+**HyperIdeal Hessian via FD.** Der analytische Hessian durch `ζ13/14/15 → lij → β/α` ist auf Phase 8 verschoben. Ein symmetrischer FD-Hessian ist O(ε²)-genau, PSD durch strikte Konvexität und für < 500 DOFs ausreichend.
+
+**Sphärischer Hessian Vorzeichen.** Die sphärische Energie ist **konkav** (Hessian NSD). Newton löst `(−H)·Δx = G`, das Vorzeichen wird transparent in `newton_spherical` behandelt.
+
+---
+
 ## CI
 
-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:
+Tests laufen automatisch bei Push auf `main`, `dev` und `claude/**`-Branches via selbst-gehostetem Gitea-Actions-Runner (`eulernest`, ARM64). **Nur `conformallab_tests` läuft in CI** (kein Boost/CGAL dort).
 
 ```bash
+# CI-Image neu bauen und pushen
 docker buildx build \
   --platform linux/arm64 \
   -f .gitea/docker/Dockerfile.ci-cpp \
@@ -99,6 +541,157 @@ docker buildx build \
   .gitea/docker/
 ```
 
-## License
+---
 
-conformallab++ is released under the MIT License (see [LICENSE](LICENSE)).
+## Roadmap
+
+> **Legende:**  ✅ abgeschlossen · 🔲 geplant
+>
+> **Grenze Portierung / neue Forschung:**  
+> Phase 1–7 sind direkte Portierungen aus dem Java-Original bzw. seiner Dissertation.  
+> Ab Phase 8 geht die Arbeit über den Umfang der Java-Bibliothek hinaus.  
+> — Phase 8 (CGAL-Paket) ist **Infrastruktur**, kein neuer Algorithmus.  
+> — Phase 9 (Inversive-Distance, Analytischer Hessian) ist **Portierung** ausstehender Java-Features.  
+> — Phase 10+ ist **eigenständige Forschung**, die über das Java-Original hinausgeht.
+
+---
+
+### ◼ Portierungsphase abgeschlossen
+
+```
+Phase 1   Clausen / Lobachevsky / ImLi₂                     ✅
+Phase 2   Hyper-ideal Geometrie (ζ, lᵢⱼ, αᵢⱼ, σᵢ)         ✅
+Phase 3   CGAL-Infrastruktur + alle drei Funktionale
+          + analytische Hessians (Eucl. + Sphär.)            ✅
+Phase 4   Newton-Solver (SimplicialLDLT + SparseQR-Fallback)
+          + Mesh-I/O + Beispielprogramme                     ✅
+Phase 5   Priority-BFS-Layout + CLI + JSON/XML               ✅  95 Tests
+
+Phase 6   Layout-Parität I                                   ✅  121 Tests
+          → gauss_bonnet.hpp  — χ, Genus, Σ(2π-Θ_v) Check + Enforce
+          → cut_graph.hpp     — Tree-Cotree (Erickson–Whittlesey 2005), 2g Schnitt-Kanten
+          → Exakte hyperbolische Trilateration (Möbius + Kosinussatz)
+          → normalise_{euclidean,hyperbolic,spherical}
+
+Phase 7   Layout-Parität II                                  ✅  158 Tests
+          → MobiusMap  — T(z)=(az+b)/(cz+d), from_three, compose, inverse
+          → halfedge_uv — naht-bewusstes UV pro Halfedge (GPU-Texturatlas)
+          → Möbius-Holonomie als SU(1,1)-Isometrie (hyperbolisch)
+          → period_matrix.hpp — τ = ω₂/ω₁ ∈ ℍ, SL(2,ℤ)-Reduktion
+          → fundamental_domain.hpp — CCW-Parallelogramm, Kachelung
+```
+
+---
+
+### ◼ Infrastruktur (über Java-Bibliothek hinaus)
+
+```
+Phase 8   CGAL-Paket-Struktur                               🔲  (nächste Phase)
+
+  Ziel: conformallab++ als eigenständiges CGAL-Paket, das in CGAL integriert
+  werden kann und dessen Konventionen vollständig erfüllt.
+
+  8a — Traits-Klasse & Konzepte
+       → include/CGAL/Conformal_map_traits.h
+         Trennt MeshType, KernelType, ScalarType vom Algorithmus.
+         Ermöglicht Nutzung mit beliebigem CGAL-kompatiblem Mesh.
+       → Konzept-Checks (static_assert / CGAL_concept_check)
+
+  8b — Öffentliche CGAL-Header-Hierarchie
+       → include/CGAL/Discrete_conformal_map.h  (zentraler Nutzer-Header)
+       → include/CGAL/Conformal_newton_solver.h
+       → include/CGAL/Conformal_layout.h
+       → include/CGAL/Conformal_cut_graph.h
+       → include/CGAL/conformal_map_package.h  (Package-Description)
+       Alle bestehenden include/conformallab/*.hpp bleiben als Impl.-Detail.
+
+  8c — Dokumentation im CGAL-Stil
+       → doc/Conformal_map/PackageDescription.txt
+       → doc/Conformal_map/fig/  (Pipeline-Diagramme)
+       → Doxygen-Kommentare für alle öffentlichen Konzepte + Funktionen
+       → User_manual.md + Reference_manual.md
+
+  8d — CGAL-Testformat
+       → test/Conformal_map/  (CMakeLists.txt im CGAL-Format)
+       Bestehende GTest-Tests bleiben; CGAL-Tests kommen als zweites Format.
+
+  8e — Declarative YAML-Pipeline
+       → Leichtgewichtiges YAML-Format für reproduzierbare Experimente
+         (Spezifikation bereits in doc/architecture/overall_pipeline.md)
+       → Validator: prüft require/provide-Tokens vor der Ausführung
+       → Einbindung in CLI-App: conformallab_core --pipeline experiment.yml
+```
+
+---
+
+### ◼ Ausstehende Portierung (Java-Features noch nicht übertragen)
+
+```
+Phase 9   Verbleibende Java-Parität                         🔲
+
+  9a — Inversive-Distance-Funktional (Luo 2004 / Bowers–Stephenson)
+       → inversive_distance_functional.hpp  (folgt exakt dem Muster der
+         bestehenden drei Funktionale — niedrigstes Risiko)
+       → newton_inversive_distance()
+       → Neue Test-Suite: test_inversive_distance.cpp
+
+  9b — Analytischer HyperIdeal-Hessian
+       → Direkte Ableitung durch die Kette
+         (b_i, a_e) → l_ij → ζ₁₃/ζ₁₄/ζ₁₅ → α_ij / β_i
+       → Ersetzt den symmetrischen FD-Hessian in hyper_ideal_hessian.hpp
+       → Relevant für Meshes > 500 DOFs (aktueller FD-Hessian ist dort langsam)
+       → Aufwand: ~2 Wochen (viele verschachtelte Fallunterscheidungen)
+
+  9c — 4g-Polygon-Randlauf (Genus g > 1)
+       → Boundary-Walk auf dem aufgeschnittenen Mesh
+       → Befüllt fundamental_domain.hpp für g > 1 (aktuell: leeres Objekt)
+       → Algorithmus-Skizze bereits als TODO(Phase 8) in fundamental_domain.hpp
+```
+
+---
+
+### ◼ Neue Forschung (über das Java-Original hinaus)
+
+> Ab hier gibt es keine direkte Java-Referenzimplementierung mehr.
+> Jedes Item ist eigenständige mathematische Arbeit.
+
+```
+Phase 10  Globale Uniformisierung Genus g ≥ 2               🔲  (Forschung)
+
+  10a — Holomorphe Differentiale auf diskreten Flächen
+        Integration ω_i längs der b-Zyklen des Schnittgraphen.
+        Mathematische Grundlage: Bobenko–Springborn (2004), §6.
+
+  10b — Siegel-Periodenmatrix Ω ∈ H_g (g×g, g ≥ 2)
+        Ω_ij = ∫_{b_j} ω_i  — komplexe symmetrische Matrix,
+        Im(Ω) positiv definit (Siegel-Oberhalbebene H_g).
+        Reduktion auf den Siegel-Fundamentalbereich via Sp(2g,ℤ).
+
+  10c — Vollständige Uniformisierung
+        Für g ≥ 2: Einbettung als H²/Γ mit Γ ⊂ PSL(2,ℝ) Fuchssche Gruppe.
+        Erfordert 10a + 10b + stabilen Cut-Graph für g ≥ 2 (Phase 9c).
+```
+
+---
+
+## Ursprung & Danksagung
+
+conformallab++ wäre ohne die Grundlagenarbeit von **Stefan Sechelmann** nicht möglich.
+Die Algorithmen, die Variationsformulierung und die Idee, diskrete konforme Geometrie
+als Newton-Problem auf Winkel-Summen-Energie-Funktionalen zu behandeln, stammen aus:
+
+| | |
+|---|---|
+| **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) |
+| **Java-Originalbibliothek** | [github.com/varylab/conformallab](https://github.com/varylab/conformallab) |
+| **Website** | [sechel.de](https://sechel.de/) |
+| **LinkedIn** | [linkedin.com/in/sechel](https://www.linkedin.com/in/sechel/) |
+
+Die Dissertation steht unter Creative Commons Attribution ShareAlike 4.0 (CC BY-SA 4.0).
+
+---
+
+## Lizenz
+
+conformallab++ steht unter der MIT-Lizenz (siehe [LICENSE](LICENSE)).

code/CMakeLists.txt

@@ -106,5 +106,10 @@ if(WITH_CGAL)
         RUNTIME_OUTPUT_DIRECTORY ${CMAKE_CURRENT_SOURCE_DIR}/bin)
 endif()
 
+# ── Example programs (require WITH_CGAL; viewer example also needs WITH_VIEWER) ─
+if(WITH_CGAL)
+    add_subdirectory(examples)
+endif()
+
 # ── Tests (always) ────────────────────────────────────────────────────────────
 add_subdirectory(tests)

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/conformal_mesh.hpp

@@ -0,0 +1,94 @@
+#pragma once
+// 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.
+using Kernel = CGAL::Simple_cartesian<double>;
+using Point3 = Kernel::Point_3;
+using Point2 = Kernel::Point_2;
+
+// ── Mesh type ────────────────────────────────────────────────────────────────
+using ConformalMesh = CGAL::Surface_mesh<Point3>;
+
+// ── Index/descriptor aliases (CGAL 6.x naming) ───────────────────────────────
+using Vertex_index   = ConformalMesh::Vertex_index;
+using Halfedge_index = ConformalMesh::Halfedge_index;
+using Edge_index     = ConformalMesh::Edge_index;
+using Face_index     = ConformalMesh::Face_index;
+
+// ── Geometry type constant (replaces Java CoFace.type enum) ─────────────────
+enum class GeometryType : int {
+    Euclidean  = 0,
+    Hyperbolic = 1,
+    Spherical  = 2
+};
+
+// ── Standard property-map bundles ────────────────────────────────────────────
+
+// Add the vertex properties used by all conformal-map functionals.
+// Returns {lambda, theta, idx}.
+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);
+}
+
+// Add the edge intersection-angle property used by the hyperbolic 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;
+}
+
+// Add the face geometry-type property.
+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 +1,17 @@
+#pragma once
+// 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/cut_graph.hpp

@@ -0,0 +1,152 @@
+#pragma once
+// 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
+// ─────────────────────────────────────────────────────────────────────────────
+
+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;
+
+    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_cut_graph
+// ─────────────────────────────────────────────────────────────────────────────
+//
+// Implements the standard tree-cotree algorithm (Erickson–Whittlesey 2005):
+//
+//   Step 1: BFS primal spanning tree T  (V−1 primal tree edges).
+//   Step 2: BFS dual spanning tree T*   (F−1 dual/primal edges, avoiding
+//           edges whose primal crosses T).
+//   Step 3: cut edges = primal edges neither in T nor "used" by 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 +1,43 @@
+#pragma once
+
+// 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
+//
+// Corresponds to 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 +1,323 @@
+#pragma once
+// 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 ─────────────────────────────────────────────────
+
+using EuclVMapD = ConformalMesh::Property_map<Vertex_index, double>;
+using EuclVMapI = ConformalMesh::Property_map<Vertex_index, int>;
+using EuclEMapD = ConformalMesh::Property_map<Edge_index,   double>;
+using EuclEMapI = ConformalMesh::Property_map<Edge_index,   int>;
+
+// ── Persistent map bundle ─────────────────────────────────────────────────────
+
+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)
+};
+
+// Create and attach property maps with sensible defaults.
+// theta_v = 2π (flat vertex), phi_e = π (interior edge, flat surface).
+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 DOF indices 0..n-1 for all vertices only (no edge DOFs).
+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.
+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 variable DOFs (vertices + edges).
+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 (Euclidean):
+//   λ°_e = 2·log(|p_i − p_j|)   (natural log of Euclidean edge length squared)
+//
+// 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 ──────────────────────────────────────────────────────────
+
+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;
+}
+
+static inline std::size_t eucl_hidx(Halfedge_index h)
+{
+    return static_cast<std::size_t>(static_cast<std::uint32_t>(h));
+}
+
+// ── Gradient ──────────────────────────────────────────────────────────────────
+//
+// G_v = Θ_v − Σ_{faces adj. v} α_v(face)
+// G_e = α_opp(face⁺) + α_opp(face⁻) − φ_e
+//
+// Corner-angle storage (h_alpha):
+//   h_alpha[h] = corner angle OPPOSITE to the edge of halfedge h in its face.
+//   h_alpha[h0] = α3,  h_alpha[h1] = α1,  h_alpha[h2] = α2
+// (same convention as SphericalFunctional)
+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;
+}
+
+// ── Energy via Gauss-Legendre path integral ───────────────────────────────────
+//
+// E(x) = ∫₀¹ ⟨G(tx), x⟩ dt   (10-point GL quadrature, same as SphericalFunctional)
+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) ──────────────────────────────────────
+
+struct EuclideanResult {
+    double              energy   = 0.0;
+    std::vector<double> 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 ─────────────────────────────────────────
+//
+// Tests |G[i] − (E(x+εeᵢ) − E(x−εeᵢ))/(2ε)| / max(1,|G[i]|) < tol
+// for all variable DOFs.
+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 +1,91 @@
+#pragma once
+// 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 {
+
+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;
+};
+
+// ── From side lengths ─────────────────────────────────────────────────────────
+//
+// Given three Euclidean side lengths l12, l23, l31 > 0 satisfying the triangle
+// inequality, compute the corner angles.
+//
+// Returns valid=false if the triangle inequality is violated (any t-value ≤ 0).
+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
+    };
+}
+
+// ── From effective log-lengths Λ̃ ─────────────────────────────────────────────
+//
+// Converts to side lengths l_ij = exp(Λ̃_ij / 2), applying the centering
+// trick for numerical safety, then delegates to euclidean_angles_from_lengths.
+//
+// The centering constant μ = (Λ̃12 + Λ̃23 + Λ̃31) / 6 ensures
+//   l12 · l23 · l31 = 1  (geometric mean = 1)
+// which keeps all l values near 1 and prevents 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 +1,211 @@
+#pragma once
+// 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).
+struct EuclCotWeights { double cot1, cot2, cot3; bool valid; };
+
+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 Hessian (cotangent Laplacian) ──────────────────────────────────
+//
+// Returns the n×n sparse Hessian matrix H where n = euclidean_dimension(mesh, m).
+//
+// Only vertex DOFs are supported.  Edge DOFs (m.e_idx[e] >= 0) produce
+// additional mixed-derivative entries that are not yet implemented; this
+// function asserts they are absent.
+//
+// x   – current DOF vector (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 ──────────────────────────────────────────
+//
+// Compares the analytical Hessian column-by-column against
+//   H_fd[:, j] = (G(x + ε·eⱼ) − G(x − ε·eⱼ)) / (2ε).
+//
+// Returns true if max relative error < tol for every entry.
+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 +1,204 @@
+#pragma once
+// 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
+// ─────────────────────────────────────────────────────────────────────────────
+
+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;
+
+    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)
+// ─────────────────────────────────────────────────────────────────────────────
+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).
+// ─────────────────────────────────────────────────────────────────────────────
+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.
+// ─────────────────────────────────────────────────────────────────────────────
+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.
+// ─────────────────────────────────────────────────────────────────────────────
+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 +1,145 @@
+#pragma once
+// 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) ─────────────────────────────────────────────
+
+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;
+}
+
+inline double gauss_bonnet_sum(const ConformalMesh& m, const EuclideanMaps&   mp)
+    { return gauss_bonnet_sum(m, mp.theta_v); }
+inline double gauss_bonnet_sum(const ConformalMesh& m, const SphericalMaps&   mp)
+    { return gauss_bonnet_sum(m, mp.theta_v); }
+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) ───────────────────────────────────────────────
+
+inline double gauss_bonnet_rhs(const ConformalMesh& mesh)
+{
+    return TWO_PI * static_cast<double>(euler_characteristic(mesh));
+}
+
+// ── Deficit: lhs − rhs  (0 = Gauss–Bonnet 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);
+}
+
+// ── check_gauss_bonnet — throws std::runtime_error if |deficit| > tol ─────────
+
+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());
+    }
+}
+
+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).
+
+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;
+}
+
+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 +1,344 @@
+#pragma once
+// 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 ─────────────────────────────────────────────────
+
+using VMapD = ConformalMesh::Property_map<Vertex_index,   double>;
+using VMapI = ConformalMesh::Property_map<Vertex_index,   int>;
+using EMapD = ConformalMesh::Property_map<Edge_index,     double>;
+using EMapI = ConformalMesh::Property_map<Edge_index,     int>;
+
+// ── Persistent map bundle ─────────────────────────────────────────────────────
+
+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 Θ_v   (parameter, not variable)
+    EMapD theta_e;  // target intersection angle θ_e
+};
+
+// Add all needed persistent property maps and return handles.
+// Defaults: theta_v = 2π (regular cone), theta_e = π (orthogonal circles).
+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 variable 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;
+}
+
+// Assign DOF indices 0..n-1: vertices first, then edges.
+// All vertices and edges become variable.  Returns total DOF count.
+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 ─────────────────────────────────────────────────────────
+
+struct HyperIdealResult {
+    double              energy   = 0.0;
+    std::vector<double> gradient;     // empty when gradient was not requested
+};
+
+// ── Internal helpers ──────────────────────────────────────────────────────────
+
+// Get the DOF value from x, or 0.0 if pinned.
+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 halfedge 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));
+}
+
+// ── Per-face angle kernel ─────────────────────────────────────────────────────
+
+struct FaceAngles {
+    double alpha12, alpha23, alpha31;  // dihedral angles at each edge
+    double beta1,   beta2,   beta3;    // interior angles at each vertex
+    double a12, a23, a31;              // edge DOF values (used in energy)
+    double b1,  b2,  b3;              // vertex DOF values
+    bool   v1b, v2b, v3b;             // whether each vertex is variable
+};
+
+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 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 ───────────────────────────────────────────────────────────
+
+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 ─────────────────────────────────────────
+//
+// Returns true if  |G[i] − fd[i]| / max(1, |G[i]|)  <  tol  for all DOFs.
+// eps = step size, tol = tolerance (same defaults as 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 +1,137 @@
+#pragma once
+// 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 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.
+//   b_i, b_j – vertex log scale factors  (used only if vertex is hyper-ideal)
+//   a_ij     – edge intersection-angle variable
+//   vi_var   – true if vertex i is hyper-ideal (has a DOF b_i)
+//   vj_var   – true if vertex j 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
+//   βi, βj, βk     – 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 +1,87 @@
+#pragma once
+// hyper_ideal_hessian.hpp
+//
+// Phase 4a — Hessian of the hyper-ideal discrete conformal functional.
+//
+// ┌──────────────────────────────────────────────────────────────────────────┐
+// │  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; an analytical Hessian is left for a    │
+// │  future phase.                                                           │
+// │                                                                          │
+// │  Here we compute the Hessian by symmetric finite differences of the     │
+// │  gradient, which is exact to O(ε²) and sufficient for Newton's method   │
+// │  at the meshes typical in Phase 4 (< 500 DOFs):                         │
+// │                                                                          │
+// │    H[i,j] = (G(x + ε·eⱼ)[i] − G(x − ε·eⱼ)[i]) / (2ε)                │
+// │                                                                          │
+// │  The hyper-ideal energy is strictly convex (Springborn 2020), so H is   │
+// │  positive semi-definite everywhere and Eigen::SimplicialLDLT applies     │
+// │  directly.                                                               │
+// └──────────────────────────────────────────────────────────────────────────┘
+
+#include "hyper_ideal_functional.hpp"
+#include <Eigen/Sparse>
+#include <vector>
+#include <cmath>
+
+namespace conformallab {
+
+// ── Numerical Hessian via symmetric finite differences ────────────────────────
+//
+// Returns the n×n sparse Hessian, where n = hyper_ideal_dimension(mesh, m).
+// eps: finite-difference step size (default 1e-5 gives ~1e-10 relative error).
+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));   // dense upper bound
+
+    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 Hessian ───────────────────────────────────────────────────────
+//
+// The FD Hessian is symmetric in exact arithmetic; floating-point rounding
+// can introduce tiny asymmetries.  This helper returns (H + Hᵀ)/2.
+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;
+}
+
+} // namespace conformallab

code/include/hyper_ideal_utility.hpp

@@ -4,6 +4,7 @@
 // Ported from de.varylab.discreteconformal.functional.HyperIdealUtility (Java).
 
 #include "clausen.hpp"
+#include "constants.hpp"
 
 #include <Eigen/Dense>
 #include <cmath>
@@ -16,10 +17,10 @@ namespace conformallab {
 // Corresponds to 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);
@@ -81,26 +82,26 @@ 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 +1,129 @@
+#pragma once
+// 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/Dense>
+#include <array>
+#include <cmath>
+
+namespace conformallab {
+
+// ---------------------------------------------------------------------------
+// Circumcenter of three 2-D points
+// ---------------------------------------------------------------------------
+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`.
+// `center` must lie on the hyperboloid: center[3]² - ‖center.head<3>()‖² = 1.
+// ---------------------------------------------------------------------------
+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 to the Poincaré disk (jReality convention:
+// add 1 to the w-coordinate, then dehomogenize the 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()
+// ---------------------------------------------------------------------------
+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/layout.hpp

@@ -0,0 +1,808 @@
+#pragma once
+// 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).
+// ─────────────────────────────────────────────────────────────────────────────
+struct MobiusMap {
+    using C = std::complex<double>;
+    C a{1.0, 0.0};
+    C b{0.0, 0.0};
+    C c{0.0, 0.0};
+    C d{1.0, 0.0};
+
+    C apply(C z) const { return (a * z + b) / (c * z + d); }
+
+    Eigen::Vector2d apply(const Eigen::Vector2d& p) const {
+        C w = apply(C(p.x(), p.y()));
+        return Eigen::Vector2d(w.real(), w.imag());
+    }
+
+    static MobiusMap identity() { return {C(1), C(0), C(0), C(1)}; }
+
+    MobiusMap inverse()               const { return {d, -b, -c, a}; }
+    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 };
+    }
+
+    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 ──────────────────────────────────────────────────────────────
+
+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;
+    bool has_seam = false;   ///< true when a vertex was reached via two paths
+};
+
+struct Layout3D {
+    std::vector<Eigen::Vector3d> pos;
+    bool success  = false;
+    bool has_seam = false;
+};
+
+/// 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
+    std::vector<MobiusMap>       mobius_maps;     ///< hyperbolic (Phase 7)
+    std::vector<std::size_t>     cut_edge_indices;
+};
+
+// ── 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
+
+// ── Vertex Voronoi area weights ───────────────────────────────────────────────
+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 ──────────────────────────────────────────────────────
+
+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);
+}
+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);
+}
+
+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 ──────────────────────────────────────────────────────────
+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 ──────────────────────────────────────────────────────────
+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()];
+            holonomy->translations.push_back(Eigen::Vector2d(diff.x(), diff.y()));
+        }
+    }
+
+    if (normalise) normalise_spherical(result);
+    return result;
+}
+
+// ── HyperIdeal layout (Poincaré disk) — exact trilateration ──────────────────
+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>;
+            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 ──────────────────────────────────────────
+
+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";
+    }
+}
+
+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/mesh_builder.hpp

@@ -0,0 +1,150 @@
+#pragma once
+// 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
+//
+// Returns a mesh with 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 ──────────────────────────────────────────────────────
+//
+// 4 vertices, 4 faces, 6 edges.
+// 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
+//
+// 4 vertices, 2 faces, 5 edges (1 interior edge v1–v2 shared by both faces).
+// Useful for testing edge-interior vs edge-boundary 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 ─────────────────────────────────────────────────
+//
+// n triangles sharing a central vertex; forms a disk topology (boundary).
+// Used to verify valence-n vertex traversal.
+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) ───────────────────────
+//
+// The four vertices of a regular tetrahedron projected onto the unit sphere.
+// Starting from (±1,±1,±1), dividing by √3 gives unit-length positions.
+// All edge lengths equal arccos(−1/3) ≈ 1.9106 radians.
+// Used for SphericalFunctional tests (all four faces are valid spherical triangles).
+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) ────────────────────
+//
+// One face of a regular octahedron: the triangle (1,0,0)→(0,1,0)→(0,0,1).
+// All edge lengths equal arccos(0) = π/2.
+// The corner angles are all π/2 (right-angled spherical triangle).
+// base log-length: λ° = 2·log(sin(π/4)) = 2·log(1/√2) = −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 +1,65 @@
+#pragma once
+// 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 ──────────────────────────────────────────────────────────────────────
+//
+// Reads a polygon mesh from file into `mesh` (clears any 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 ─────────────────────────────────────────────────────────────────────
+//
+// Writes `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);
+}
+
+// ── Convenience: throwing wrappers ────────────────────────────────────────────
+
+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;
+}
+
+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/newton_solver.hpp

@@ -0,0 +1,326 @@
+#pragma once
+// 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 <Eigen/SparseCholesky>
+#include <Eigen/SparseQR>
+#include <Eigen/OrderingMethods>
+#include <Eigen/Dense>
+#include <algorithm>
+#include <cmath>
+
+namespace conformallab {
+
+// ── Result ────────────────────────────────────────────────────────────────────
+
+struct NewtonResult {
+    std::vector<double> x;            ///< DOF vector at termination
+    int                 iterations;   ///< Newton steps taken
+    double              grad_inf_norm;///< max |G_i| 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.
+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 ────────────────────────────────────────────────────
+//
+// Minimises the Euclidean discrete conformal energy by solving G(x) = 0.
+// The Hessian H is PSD; Eigen::SimplicialLDLT is used directly.
+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 ───────────────────────────────────────────────────
+//
+// Solves G(x) = 0 for the spherical discrete conformal functional.
+// The Hessian H is NSD at the solution; −H is PSD, so we factorise −H and
+// solve (−H)·Δx = G  ⟺  H·Δx = −G.
+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 ──────────────────────────────────────────────────
+//
+// Solves G(x) = 0 for the hyper-ideal discrete conformal functional.
+//
+// Gradient sign convention (opposite to Euclidean/Spherical):
+//   G_v = Σ β_v − Θ_v,   G_e = Σ α_e − θ_e   (actual − target)
+//
+// The hyper-ideal energy is strictly convex (Springborn 2020), so H is PSD
+// and SimplicialLDLT (with SparseQR fallback) applies directly.
+//
+// The Hessian is computed by symmetric finite differences of G (see
+// hyper_ideal_hessian.hpp); replace with an analytical Hessian in Phase 5.
+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;
+}
+
+} // namespace conformallab

code/include/p2_utility.hpp

@@ -0,0 +1,102 @@
+#pragma once
+
+// 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 ──────────────────────────────────────────────────────
+
+// Intersection of two lines l1, l2  (or  line through two points p1, p2)
+// via the cross product.  Works for any P2 element.
+// Corresponds to Java P2.pointFromLines / P2.lineFromPoints.
+inline Eigen::Vector3d pointFromLines(const Eigen::Vector3d& l1,
+                                      const Eigen::Vector3d& l2) {
+    return l1.cross(l2);
+}
+
+// ── Euclidean perpendicular bisector ─────────────────────────────────────────
+
+// Returns the homogeneous line coordinates (a, b, c) of the perpendicular
+// bisector of the segment [p, q] in the Euclidean plane.
+// Coordinates: ax + by + c = 0  (after dehomogenizing p and q).
+//
+// Corresponds to 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 ─────────────────────
+
+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 matrix that represents the coordinate frame
+// anchored at p0 with p1 defining the positive x-direction.
+// Euclidean case: columns are [dehom(p0), unit_dir(p0→p1), perp_dir].
+//
+// Template parameter S allows float / double / long double.
+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;
+}
+
+// Find the 3×3 Euclidean isometry (as a projective matrix) that maps
+// the frame (s1, s2) to the frame (t1, t2).
+//
+// Corresponds to Java P2.makeDirectIsometryFromFrames(s1, s2, t1, t2, Pn.EUCLIDEAN)
+// and P2Big.makeDirectIsometryFromFrames(...) (the BigDecimal / high-precision variant).
+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 +1,152 @@
+#pragma once
+// 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
+// ─────────────────────────────────────────────────────────────────────────────
+
+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;
+
+    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).
+// ─────────────────────────────────────────────────────────────────────────────
+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.
+// ─────────────────────────────────────────────────────────────────────────────
+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.
+// ─────────────────────────────────────────────────────────────────────────────
+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/serialization.hpp

@@ -0,0 +1,289 @@
+#pragma once
+// 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 solver result (+ optional 2D 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 DOF vector from a JSON result file.
+// Returns the DOF vector; fills out res fields if non-null.
+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 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 solver result from an XML file written by save_result_xml.
+// Returns the DOF vector; fills res/geom/layout2d if non-null.
+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 +1,475 @@
+#pragma once
+// 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 ─────────────────────────────────────────────────
+
+using SpherVMapD = ConformalMesh::Property_map<Vertex_index, double>;
+using SpherVMapI = ConformalMesh::Property_map<Vertex_index, int>;
+using SpherEMapD = ConformalMesh::Property_map<Edge_index,   double>;
+using SpherEMapI = ConformalMesh::Property_map<Edge_index,   int>;
+
+// ── Persistent map bundle ─────────────────────────────────────────────────────
+
+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 Θ_v  (default 2π)
+    SpherEMapD theta_e;  // target edge angle θ_e  (default π)
+    SpherEMapD lambda0;  // base log-length λ°_e   (default 0.0)
+};
+
+// 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.
+// For real meshes, set lambda0 from mesh geometry via
+// compute_lambda0_from_mesh() below.
+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 DOF indices 0..n-1 for all vertices (only vertex DOFs).
+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 edges.
+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 variable DOFs.
+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;
+}
+
+// Set lambda0 from mesh vertex positions (unit-sphere assumed):
+//   λ°_e = 2·log(sin(l_e / 2))   where l_e = arccos(p_i · p_j).
+// Requires vertices to lie on the unit sphere.
+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 ─────────────────────────────────────────────────────────
+
+struct SphericalResult {
+    double              energy   = 0.0;
+    std::vector<double> gradient;
+};
+
+// ── Internal helpers ──────────────────────────────────────────────────────────
+
+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;
+}
+
+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 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 halfedges using the convention:
+//   h_alpha[h] = corner angle at source(prev(h)) = corner angle at the vertex
+//                ACROSS FROM the edge of halfedge h in its face.
+// This convention makes both the vertex and edge 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
+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) ──────────────────────────────────────
+
+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 ─────────────────────────────────────────
+//
+// Tests |G[i] − fd[i]| / max(1, |G[i]|) < tol for all DOFs.
+// Same defaults as the hyper-ideal gradient check (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).
+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 gauge shift in-place: x_v ← x_v + t* for all variable vertices.
+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 +1,84 @@
+#pragma once
+// 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 ────────────────────────────────────────────
+
+// l(λ) = 2·asin(min(exp(λ/2), 1)).
+// Clamps exp(λ/2) to [0, 1] so the arcsin 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 ──────────────────────────────────
+
+struct SphericalFaceAngles {
+    double alpha1, alpha2, alpha3;  // corner angles at v1, v2, v3
+    bool   valid;                   // false when the three lengths fail the
+                                    // spherical triangle inequality
+};
+
+// Compute corner angles from spherical arc lengths using the half-angle formula.
+//
+// Convention (matching the halfedge cycle h0→v1→v2, h1→v2→v3, h2→v3→v1):
+//   l12 – arc length of edge opposite v3 (edge e12)
+//   l23 – arc length of edge opposite v1 (edge e23)
+//   l31 – arc length of edge opposite v2 (edge e31)
+//
+// Half-angle formula (spherical law of cosines):
+//   α_k = 2·atan2(sqrt(sin(s-a)·sin(s-b)), sqrt(sin(s)·sin(s-c)))
+// where a,b are the two edges ADJACENT to vertex k, c is the opposite edge.
+//
+// Equivalently (in terms of s-deficiencies):
+//   α1 = 2·atan2( sqrt(sin(s12)·sin(s31)), sqrt(sin(s)·sin(s23)) )
+//   α2 = 2·atan2( sqrt(sin(s12)·sin(s23)), sqrt(sin(s)·sin(s31)) )
+//   α3 = 2·atan2( sqrt(sin(s23)·sin(s31)), sqrt(sin(s)·sin(s12)) )
+//
+// where s = (l12+l23+l31)/2 and s_ij = s - l_ij.
+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 +1,256 @@
+#pragma once
+// 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).
+struct SpherCotWeights { double w12, w23, w31; bool valid; };
+
+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 Hessian ────────────────────────────────────────────────────────
+//
+// Returns the n×n sparse Hessian matrix H where n = spherical_dimension(mesh, m).
+// x – current DOF vector.
+//
+// Derivation: G_v = θ_v − Σ_f α_v^f  →  H[i,j] = −Σ_f ∂α_i^f/∂u_j
+//
+// For a face (v1,v2,v3) with arc-lengths l12,l23,l31 and angles α1,α2,α3,
+// differentiating the spherical law of cosines
+//   cos(l_opp) = cos(l_a)cos(l_b) + sin(l_a)sin(l_b)cos(α)
+// gives:
+//   ∂α1/∂l12 = [cot(l12)cos(α1) − cot(l31)] / sin(α1)     (adjacent side)
+//   ∂α1/∂l31 = [cot(l31)cos(α1) − cot(l12)] / sin(α1)     (adjacent side)
+//   ∂α1/∂l23 = sin(l23) / [sin(l12)sin(l31)sin(α1)]        (opposite side)
+//
+// Chain rule with ∂l_ij/∂u_k = tan(l_ij/2)  (from l = 2·asin(exp(λ/2))):
+//   ∂α1/∂u1 = ∂α1/∂l12·t12 + ∂α1/∂l31·t31
+//   ∂α1/∂u2 = ∂α1/∂l12·t12 + ∂α1/∂l23·t23
+//   ∂α1/∂u3 = ∂α1/∂l23·t23 + ∂α1/∂l31·t31
+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;
+}
+
+// ── Finite-difference Hessian check ──────────────────────────────────────────
+//
+// Compares the analytical Hessian column-by-column against
+//   H_fd[:, j] = (G(x + ε·eⱼ) − G(x − ε·eⱼ)) / (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/src/apps/v0/conformallab_cli.cpp

@@ -1,107 +1,314 @@
-#include <CGAL/Simple_cartesian.h>
-#include <CGAL/Surface_mesh.h>
-#include <CGAL/IO/polygon_mesh_io.h>
-
-
-#include "viewer_utils.h"
-#include "mesh_utils.hpp"
+// 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 "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);
+
+        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);
     }
 
+    // ── 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);
 
-
-    
-    // #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;
-    }
-
-    return EXIT_SUCCESS;
+    std::cerr << "Unknown geometry: " << geometry << "\n";
+    return EXIT_FAILURE;
 }

code/tests/CMakeLists.txt

@@ -1,8 +1,18 @@
 add_executable(conformallab_tests
+    # ── Fully ported (pure math, no HDS) ────────────────────────────────────
     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
+
+    # ── Stubs: blocked until HDS port (Phase 4) ──────────────────────────────
+    # All tests call GTEST_SKIP() with a clear explanation.
+    test_hyper_ideal_functional.cpp
+    test_hyper_ideal_hyperelliptic_utility.cpp
+    test_spherical_functional.cpp
 )
 
 target_include_directories(conformallab_tests SYSTEM PRIVATE
@@ -17,3 +27,8 @@ target_link_libraries(conformallab_tests PRIVATE GTest::gtest_main)
 
 include(GoogleTest)
 gtest_discover_tests(conformallab_tests DISCOVERY_TIMEOUT 60)
+
+# ── CGAL test suite (requires -DWITH_CGAL=ON) ────────────────────────────────
+if(WITH_CGAL)
+    add_subdirectory(cgal)
+endif()

code/tests/cgal/CMakeLists.txt

@@ -0,0 +1,81 @@
+# 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 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-Parität: Geometrie-Utility-Tests ─────────────────────────────────
+    #    Portiert aus CuttinUtilityTest, UnwrapUtilityTest,
+    #    ConvergenceUtilityTests, HomologyTest (Tests 1–6).
+    #    Test 7 (Genus-2-Homologie) als GTEST_SKIP-Stub bis Phase 8.
+    test_geometry_utils.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
+)
+
+# Suppress warnings from CGAL/Boost headers
+target_compile_options(conformallab_cgal_tests PRIVATE
+    $<$<CXX_COMPILER_ID:GNU,Clang,AppleClang>:-Wno-unused-parameter>
+)
+
+target_link_libraries(conformallab_cgal_tests PRIVATE GTest::gtest_main)
+
+include(GoogleTest)
+gtest_discover_tests(conformallab_cgal_tests
+    TEST_PREFIX "cgal."
+    DISCOVERY_TIMEOUT 60
+)

code/tests/cgal/test_conformal_mesh.cpp

@@ -0,0 +1,240 @@
+// 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_euclidean_functional.cpp

@@ -0,0 +1,269 @@
+// 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           (SKIPPED)
+//   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 <gtest/gtest.h>
+#include <cmath>
+#include <vector>
+
+using namespace conformallab;
+
+// ════════════════════════════════════════════════════════════════════════════
+// @Ignore in Java: no Hessian implemented yet
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(EuclideanFunctional, GradientCheck_Hessian)
+{
+    GTEST_SKIP() << "@Ignore in Java – Hessian not yet implemented";
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// 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 +1,213 @@
+// 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 +1,342 @@
+// test_geometry_utils.cpp
+//
+// Portierung der Java ConformalLab Geometrie-Utility-Tests.
+//
+// Java-Quelle                           Java-Testmethode                         Status
+// ─────────────────────────────────────────────────────────────────────────────────────
+// CuttinUtilityTest.java                testIsInConvexTextureFace_False          PORTIERT
+// CuttinUtilityTest.java                testIsInConvexTextureFace_True           PORTIERT
+// UnwrapUtilityTest.java                testGetAngleReturnsPI                    PORTIERT
+// ConvergenceUtilityTests.java          testGetTextureCircumRadius               PORTIERT
+// ConvergenceUtilityTests.java          testGetTextureTriangleArea               PORTIERT
+// ConvergenceUtilityTests.java          testScaleInvariantCircumCircleRadius     PORTIERT
+// HomologyTest.java                     testHomology                             GEBLOCKT
+//
+// ─── Geometrische Grundlage ──────────────────────────────────────────────────────────
+//
+//  Tests 1–2  Punkt-in-konvexem-Dreieck (2D UV-Raum, baryzentrische Vorzeichen-Methode)
+//             Java: CuttingUtility.isInConvexTextureFace(pp, face, adapters)
+//             Hinweis: Java-Test 2 hat ein 5-elementiges T-Array mit w=0 (Punkt im
+//             Unendlichen), was ein Tippfehler im Original ist. Hier werden
+//             äquivalente, wohlgeformte Koordinaten verwendet.
+//
+//  Test 3     Eckenwinkel für kollineare Vertices über den Kosinussatz.
+//             Java: UnwrapUtility.getAngle(edge, adapters) — gibt den Winkel am
+//             Zielknoten zurück. Für v0=(-1,0,0), v1=(0,0,0), v2=(1,0,0) ist
+//             der Winkel bei v1 genau π (Dreiecksungleichung entartet).
+//
+//  Tests 4–5  2D Umkreisradius und Dreiecksfläche.
+//             Java: ConvergenceUtility.getTextureCircumCircleRadius(face)
+//                   ConvergenceUtility.getTextureTriangleArea(face)
+//             Formeln: Area = |det([B-A, C-A])| / 2
+//                      R    = (a·b·c) / (4·Area)
+//
+//  Test 6     Skaleninvarianter Umkreisradius über ein Mesh.
+//             Java: ConvergenceUtility.getMaxMeanSumScaleInvariantCircumRadius(hds)
+//             Gibt [max, mean, sum] von  R_f / sqrt(total_texture_area)  zurück.
+//             Invariant unter uniformer Skalierung der Texturkoordinaten (Test mit
+//             homogenem Gewicht w: Position = (T[0]/w, T[1]/w)).
+//
+// ─── GEBLOCKT (Test 7) ───────────────────────────────────────────────────────────────
+//
+//  Test 7     Genus-2 Homologie-Generatoren
+//             Java: HomologyTest.testHomology
+//             Erwartet: getGeneratorPaths(root, ...).size() == 4  (2g = 4 für g = 2)
+//             C++-Äquivalent: compute_cut_graph(mesh).cut_edge_indices.size() == 4
+//             BLOCKED: Kein Genus-2-Testmesh in mesh_builder.hpp vorhanden.
+//             TODO(Phase 8): make_genus2_surface() in mesh_builder.hpp implementieren
+//             oder brezel2.obj via load_mesh importieren, dann GTEST_SKIP entfernen.
+//
+// ─────────────────────────────────────────────────────────────────────────────────────
+
+#include "cut_graph.hpp"        // für Test 7 (Genus-2 TODO)
+#include "conformal_mesh.hpp"
+#include "mesh_builder.hpp"
+#include <gtest/gtest.h>
+#include <Eigen/Dense>
+#include <array>
+#include <cmath>
+#include <vector>
+
+using namespace conformallab;
+
+// ─────────────────────────────────────────────────────────────────────────────
+// Lokale Geometrie-Hilfsfunktionen
+// (portiert aus Java CuttingUtility / ConvergenceUtility)
+// ─────────────────────────────────────────────────────────────────────────────
+
+/// Punkt-in-Dreieck Test (2D, baryzentrische Vorzeichenmethode).
+/// Gibt true zurück wenn p strikt innerhalb oder auf dem Rand von v0-v1-v2 liegt.
+/// 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 Dreiecksfläche (halbes Kreuzprodukt).
+/// 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 Umkreisradius: 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);
+}
+
+/// Skaleninvarianter Umkreisradius für ein Mesh:
+///   scale_R_f = R_f / sqrt(total_area)
+/// Gibt {max, mean, sum} über alle Flächen zurück.
+/// Java: ConvergenceUtility.getMaxMeanSumScaleInvariantCircumRadius
+///
+/// Homogene Koordinaten: 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)
+{
+    // Gesamtfläche
+    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: Punkt-in-konvexem-Dreieck (2D UV-Raum)
+//             Java: CuttinUtilityTest.testIsInConvexTextureFace_False / _True
+// ════════════════════════════════════════════════════════════════════════════
+
+// Test 1: Punkt liegt weit außerhalb — exakte Java-Koordinaten
+TEST(CuttingUtility, IsInConvexTextureFace_False)
+{
+    // Winziges Dreieck um (0.7488, 0.0629) — Java-Testkoordinaten (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);
+    // Testpunkt weit entfernt bei (0.447, 0.000228)
+    Eigen::Vector2d pp(0.44661534423161037, 2.2808373704822393e-4);
+
+    EXPECT_FALSE(point_in_triangle_2d(pp, v0, v1, v2));
+}
+
+// Test 2: Punkt liegt innerhalb
+// Hinweis: Das originale Java-Array p2 hat 5 Elemente mit w=0 (Tippfehler im
+// Java-Original). Hier werden äquivalente, wohlgeformte Koordinaten verwendet,
+// die dasselbe geometrische Szenario abbilden.
+TEST(CuttingUtility, IsInConvexTextureFace_True)
+{
+    // Dreieck: (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);
+    // Schwerpunkt des Dreiecks — liegt immer innen
+    Eigen::Vector2d pp(1e-8 / 3.0, 1e-8 / 3.0);
+
+    EXPECT_TRUE(point_in_triangle_2d(pp, v0, v1, v2));
+}
+
+// Zusätzlich: einfaches Einheitsdreieck für Klarheit
+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));  // jenseits Hypotenuse
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// Test 3 — UnwrapUtility: Eckenwinkel = π für kollineare Vertices
+//          Java: UnwrapUtilityTest.testGetAngleReturnsPI
+// ════════════════════════════════════════════════════════════════════════════
+
+// Java: v0=(-1,0,0), v1=(0,0,0), v2=(1,0,0) kollinear.
+// Kante e von v2 nach v1. getAngle(e) = Winkel bei v1 = π.
+//
+// C++: Kosinussatz mit Kantenlängen 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, entartet)
+    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));   // numerisches Clamp
+    double angle = std::acos(cos_angle);
+    EXPECT_NEAR(M_PI, angle, 1e-15);
+}
+
+// Gegenkontrolle: gleichseitiges Dreieck → Winkel = 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 Umkreisradius
+//          Java: ConvergenceUtilityTests.testGetTextureCircumRadius
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(ConvergenceUtility, TextureCircumRadius_RightTriangle)
+{
+    // A=(0,0), B=(1,0), C=(0,1): rechtwinkliges gleichschenkliges Dreieck
+    // Seiten: 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-Variante mit B.T={0.5,0.5,0,1}
+    // Seiten: √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 Dreiecksfläche
+//          Java: ConvergenceUtilityTests.testGetTextureTriangleArea
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(ConvergenceUtility, TextureTriangleArea_RightTriangle)
+{
+    // A=(0,0), B=(1,0), C=(0,1) → Fläche = 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) → Fläche = 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: Skaleninvarianter Umkreisradius
+//          Java: ConvergenceUtilityTests.testScaleInvariantCircumCircleRadius
+//
+// Mesh: 4 Vertices (v1..v4), 2 Flächen (f1: v1-v2-v3, f2: v1-v3-v4).
+// Skaleninvariante Größe: R_f / sqrt(total_area) — invariant unter
+// uniformer Skalierung (homogeneous weight w: pos = (x/w, y/w)).
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(ConvergenceUtility, ScaleInvariantCircumRadius_BaseScale)
+{
+    // Positionen bei 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} };
+
+    // Einzelflächen-Prüfung (Java testGetTextureTriangleArea-Anforderung)
+    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);
+
+    // Erwartet: sin(π/4) = √2/2 für max und mean (beide Dreiecke identisch)
+    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)
+{
+    // Skalierung durch w=2: alle Positionen halbiert (homogene Koordinaten)
+    // 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} };
+
+    // Flächen sind ein Viertel der ursprünglichen (Längen halbiert → 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);
+
+    // Skaleninvariante Größe muss identisch zu w=1 sein
+    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 Homologie-Generatoren
+//          Java: HomologyTest.testHomology
+//
+// GEBLOCKT — kein Genus-2-Testmesh vorhanden.
+//
+// Java-Test:
+//   CoHDS hds = TestUtility.readOBJ("brezel2.obj");  // Genus-2-Brezel-Fläche
+//   List<Set<CoEdge>> paths = getGeneratorPaths(hds.getVertex(0), weightAdapter);
+//   Assert.assertEquals(4, paths.size());             // 2g = 4 für g = 2
+//
+// C++-Äquivalent (sobald entsprechendes Mesh verfügbar):
+//   ConformalMesh mesh = load_mesh("brezel2.obj");    // oder make_genus2_surface()
+//   CutGraph cg = compute_cut_graph(mesh);
+//   EXPECT_EQ(4u, cg.cut_edge_indices.size());        // 2g = 4
+//   EXPECT_EQ(2,  cg.genus);
+//
+// TODO(Phase 8): Eine der folgenden Optionen implementieren und GTEST_SKIP entfernen:
+//   Option A — Programmatisch: mesh_builder.hpp um make_genus2_surface() erweitern.
+//              Ein Genus-2-Mesh lässt sich als zwei miteinander verbundene Tori
+//              konstruieren (handle attachment).
+//   Option B — Dateibasiert: brezel2.obj aus dem Java-Projekt (Pfad:
+//              conformallab/src-test/.../brezel2.obj) via load_mesh importieren.
+//              Erfordert den Dateipfad zur Laufzeit als CMake-Variable.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(HomologyGenerators, Genus2_FourGeneratorPaths_BLOCKED)
+{
+    GTEST_SKIP()
+        << "TODO(Phase 8): Genus-2-Testmesh fehlt.\n"
+           "  Sobald mesh_builder.hpp make_genus2_surface() bereitstellt\n"
+           "  oder brezel2.obj via load_mesh importiert wird, hier prüfen:\n"
+           "    CutGraph cg = compute_cut_graph(mesh);\n"
+           "    EXPECT_EQ(4u, cg.cut_edge_indices.size());  // 2g = 4 fuer g = 2\n"
+           "    EXPECT_EQ(2,  cg.genus);\n"
+           "  Java-Quelle: HomologyTest.testHomology (brezel2.obj, 4 Generatoren).";
+}

code/tests/cgal/test_hyper_ideal_functional.cpp

@@ -0,0 +1,196 @@
+// 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_layout.cpp

@@ -0,0 +1,369 @@
+// 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 +1,170 @@
+// 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_solver.cpp

@@ -0,0 +1,500 @@
+// 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 +1,406 @@
+// 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 +1,485 @@
+// 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 +1,292 @@
+// 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_spherical_functional.cpp

@@ -0,0 +1,340 @@
+// 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     (SKIPPED)
+//   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 <gtest/gtest.h>
+#include <cmath>
+#include <vector>
+
+using namespace conformallab;
+
+// ════════════════════════════════════════════════════════════════════════════
+// @Ignore in Java: no Hessian implemented
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(SphericalFunctional, GradientCheck_Hessian)
+{
+    GTEST_SKIP() << "@Ignore in Java – Hessian not implemented";
+}
+
+// ════════════════════════════════════════════════════════════════════════════
+// 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 +1,205 @@
+// 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_discrete_elliptic_utility.cpp

@@ -0,0 +1,39 @@
+// Port of de.varylab.discreteconformal.util.DiscreteEllipticUtilityTest (Java/JUnit).
+// Tests the normalizeModulus function that moves a complex number tau into the
+// fundamental domain of the modular group SL(2,Z).
+
+#include "discrete_elliptic_utility.hpp"
+#include <gtest/gtest.h>
+#include <complex>
+#include <cmath>
+
+using namespace conformallab;
+
+// Corresponds to Java testNormalizeModulus()
+TEST(DiscreteEllipticUtilityTest, NormalizeModulus) {
+    // tau already in fundamental domain → should be returned unchanged
+    std::complex<double> tau(0.45, 1.1);
+    auto tauNorm = normalizeModulus(tau);
+    EXPECT_NEAR(0.45, tauNorm.real(), 1E-12);
+    EXPECT_NEAR(1.1,  tauNorm.imag(), 1E-12);
+
+    // tau = i/3  (|tau| < 1) → inversion gives 3i
+    tau = std::complex<double>(0.0, 1.0 / 3.0);
+    tauNorm = normalizeModulus(tau);
+    EXPECT_NEAR(3.0, tauNorm.imag(), 1E-12);
+    EXPECT_NEAR(0.0, tauNorm.real(), 1E-12);
+}
+
+// Corresponds to Java testNormalizeModulusPeriodShift()
+// Two tau values that differ by a T-shift (integer shift of Re) must normalize
+// to the same point in the fundamental domain.
+TEST(DiscreteEllipticUtilityTest, NormalizeModulusPeriodShift) {
+    std::complex<double> tau1(0.3,  1.0);
+    std::complex<double> tau2(-0.7, 1.0);   // tau2 = tau1 - 1
+
+    auto n1 = normalizeModulus(tau1);
+    auto n2 = normalizeModulus(tau2);
+
+    EXPECT_NEAR(n1.real(), n2.real(), 1E-12) << "real parts should be equal";
+    EXPECT_NEAR(n1.imag(), n2.imag(), 1E-12) << "imag parts should be equal";
+}

code/tests/test_hyper_ideal_functional.cpp

@@ -0,0 +1,37 @@
+// Stub for de.varylab.discreteconformal.functional.HyperIdealFunctionalTest (Java/JUnit).
+//
+// STATUS: BLOCKED – requires HDS port (Phase 4).
+//
+// These tests evaluate gradient and Hessian of the HyperIdealFunctional on
+// actual mesh data (CoHDS + HyperIdealGenerator).  They cannot be ported
+// until the HalfEdge data structure (CoHDS), the functional evaluation
+// framework, and the mesh generators are available in C++.
+//
+// Java tests and their status:
+//   testHessian()                           – @Ignore in Java (skipped here too)
+//   testGradientWithHyperIdealAndIdealPoints – blocked: needs HDS
+//   testGradientInTheExtendedDomain          – blocked: needs HDS
+//   testGradientWithHyperellipticCurve       – blocked: needs HDS
+//   testFunctionalAtNaNValue                 – blocked: needs HDS
+
+#include <gtest/gtest.h>
+
+TEST(HyperIdealFunctionalTest, TestHessian_IgnoredInJava) {
+    GTEST_SKIP() << "@Ignore in Java – skipped here too";
+}
+
+TEST(HyperIdealFunctionalTest, GradientWithHyperIdealAndIdealPoints) {
+    GTEST_SKIP() << "Blocked: requires HDS port (CoHDS + HyperIdealFunctional)";
+}
+
+TEST(HyperIdealFunctionalTest, GradientInTheExtendedDomain) {
+    GTEST_SKIP() << "Blocked: requires HDS port (CoHDS + HyperIdealFunctional)";
+}
+
+TEST(HyperIdealFunctionalTest, GradientWithHyperellipticCurve) {
+    GTEST_SKIP() << "Blocked: requires HDS port (CoHDS + HyperIdealFunctional)";
+}
+
+TEST(HyperIdealFunctionalTest, FunctionalAtNaNValue) {
+    GTEST_SKIP() << "Blocked: requires HDS port (CoHDS + HyperIdealFunctional)";
+}

code/tests/test_hyper_ideal_hyperelliptic_utility.cpp

@@ -0,0 +1,26 @@
+// Stub for de.varylab.discreteconformal.functional.HyperIdealHyperellipticUtilityTest.
+//
+// STATUS: BLOCKED – requires HDS port (Phase 4).
+//
+// Tests compute intersection angles of circles associated with hyper-ideal
+// vertices using CoHDS + HalfEdgeUtils.  All three tests operate on mesh
+// data structures that are not yet available in C++.
+//
+// Java tests and their status:
+//   testCalculateCircleIntersections         – blocked: needs CoHDS + HalfEdgeUtils
+//   testCalculateCircleIntersectionsInfinite – blocked: needs CoHDS + HalfEdgeUtils
+//   testLawsonHyperellipticAngles            – blocked: needs CoHDS + HyperIdealGenerator
+
+#include <gtest/gtest.h>
+
+TEST(HyperIdealHyperellipticUtilityTest, CalculateCircleIntersections) {
+    GTEST_SKIP() << "Blocked: requires HDS port (CoHDS + HalfEdgeUtils)";
+}
+
+TEST(HyperIdealHyperellipticUtilityTest, CalculateCircleIntersectionsInfinite) {
+    GTEST_SKIP() << "Blocked: requires HDS port (CoHDS + HalfEdgeUtils)";
+}
+
+TEST(HyperIdealHyperellipticUtilityTest, LawsonHyperellipticAngles) {
+    GTEST_SKIP() << "Blocked: requires HDS port (CoHDS + HyperIdealGenerator)";
+}

code/tests/test_hyper_ideal_visualization_utility.cpp

@@ -0,0 +1,50 @@
+// Port of de.varylab.discreteconformal.plugin.HyperIdealVisualizationPluginTest (Java/JUnit).
+//
+// Tests the conversion from a hyperbolic circle (hyperboloid model)
+// to its Euclidean representation (Poincaré disk model).
+//
+// Java test: static method HyperIdealVisualizationPlugin
+//              .getEuclideanCircleFromHyperbolic(double[] center, double radius)
+// C++ port:  conformallab::getEuclideanCircleFromHyperbolic(Vector4d, double)
+//              in hyper_ideal_visualization_utility.hpp
+
+#include "hyper_ideal_visualization_utility.hpp"
+#include <gtest/gtest.h>
+#include <cmath>
+
+using namespace conformallab;
+
+// Corresponds to Java testGetEuclideanCircleFromHyperbolic_Centered()
+//
+// A hyperbolic circle centered at the origin (0,0,0,1) with radius 1.
+// In the Poincaré disk this maps to a Euclidean circle centered at (0,0)
+// with radius  sinh(1) / (cosh(1) + 1).
+TEST(HyperIdealVisualizationUtilityTest, EuclideanCircleFromHyperbolic_Centered)
+{
+    Eigen::Vector4d center(0.0, 0.0, 0.0, 1.0);
+    const double radius = 1.0;
+
+    auto result = getEuclideanCircleFromHyperbolic(center, radius);
+
+    const double expected_r = std::sinh(1.0) / (std::cosh(1.0) + 1.0);
+    EXPECT_NEAR(0.0,        result[0], 1E-12) << "Euclidean cx should be 0";
+    EXPECT_NEAR(0.0,        result[1], 1E-12) << "Euclidean cy should be 0";
+    EXPECT_NEAR(expected_r, result[2], 1E-12) << "Euclidean radius mismatch";
+}
+
+// Corresponds to Java testGetEuclideanCircleFromHyperbolic_OffCenter()
+//
+// A hyperbolic circle centered at (sinh(1),0,0,cosh(1)) with radius 1.
+// By symmetry (center is on the x-axis, circle is symmetric about it):
+//   • the Euclidean center lies on the x-axis  → cy = 0
+//   • the Euclidean center equals the radius   → cx = r  (the circle passes through the Poincaré origin)
+TEST(HyperIdealVisualizationUtilityTest, EuclideanCircleFromHyperbolic_OffCenter)
+{
+    Eigen::Vector4d center(std::sinh(1.0), 0.0, 0.0, std::cosh(1.0));
+    const double radius = 1.0;
+
+    auto result = getEuclideanCircleFromHyperbolic(center, radius);
+
+    EXPECT_NEAR(result[0], result[2], 1E-12) << "cx should equal Euclidean radius";
+    EXPECT_NEAR(0.0, result[1],       1E-12) << "cy should be 0 (x-axis symmetry)";
+}

code/tests/test_p2_utility.cpp

@@ -0,0 +1,87 @@
+// Port of de.varylab.discreteconformal.math.P2BigTest (Java/JUnit).
+// Tests 2-D projective geometry utilities: perpendicular bisectors,
+// point-from-lines, and direct isometries in the Euclidean plane.
+//
+// The Java test compared double precision (P2) against BigDecimal precision
+// (P2Big) to 1E-10.  Here we compare double against long double to the
+// same tolerance.
+
+#include "p2_utility.hpp"
+#include <gtest/gtest.h>
+#include <Eigen/Dense>
+#include <cmath>
+
+using namespace conformallab;
+
+// Corresponds to Java P2BigTest.testMakeDirectIsometryFromFramesEuclidean()
+//
+// Computes the Euclidean isometry mapping frame (s1,s2) to frame (t1,t2)
+// with both double and long-double precision, and checks:
+//   1. The two precisions agree to 1E-10 (precision stability).
+//   2. The matrix actually maps s1→t1 and s2→t2.
+TEST(P2UtilityTest, MakeDirectIsometryFromFramesEuclidean) {
+    using V3d  = Eigen::Vector3d;
+    using V3ld = Eigen::Matrix<long double, 3, 1>;
+
+    V3d s1(-1.4142135623730963, 0.0, 1.0);
+    V3d s2( 1.4142135623730951, 0.0, 1.0);
+    V3d t1(-2.828427124746189,  2.4494897427831805, 1.0);
+    V3d t2( 0.0,               2.4494897427831783, 1.0);
+
+    // double precision
+    auto T = makeDirectIsometryFromFramesEuclidean<double>(s1, s2, t1, t2);
+
+    // long double precision (analogous to Java's BigDecimal P2Big)
+    V3ld s1l = s1.cast<long double>();
+    V3ld s2l = s2.cast<long double>();
+    V3ld t1l = t1.cast<long double>();
+    V3ld t2l = t2.cast<long double>();
+    auto Tl = makeDirectIsometryFromFramesEuclidean<long double>(s1l, s2l, t1l, t2l);
+
+    // 1. double vs long double must agree to 1E-10
+    for (int i = 0; i < 3; ++i)
+        for (int j = 0; j < 3; ++j)
+            EXPECT_NEAR((double)Tl(i,j), T(i,j), 1E-10)
+                << "element (" << i << "," << j << ") differs between precisions";
+
+    // 2. T must map s1 → t1 and s2 → t2  (verify isometry correctness)
+    auto map_s1 = T * s1;
+    auto map_s2 = T * s2;
+    EXPECT_NEAR(euclideanDistanceP2(map_s1, t1), 0.0, 1E-9) << "T*s1 should equal t1";
+    EXPECT_NEAR(euclideanDistanceP2(map_s2, t2), 0.0, 1E-9) << "T*s2 should equal t2";
+}
+
+// Corresponds to Java P2BigTest.testPerpendicularBisector()
+TEST(P2UtilityTest, PerpendicularBisector) {
+    Eigen::Vector3d p1(0.5, 0.0, 1.0);
+    Eigen::Vector3d q1(0.0, 0.5, 1.0);
+
+    auto bisector = perpendicularBisectorEuclidean(p1, q1);
+
+    EXPECT_NEAR( 0.5, bisector(0), 1E-10);
+    EXPECT_NEAR(-0.5, bisector(1), 1E-10);
+    EXPECT_NEAR( 0.0, bisector(2), 1E-10);
+}
+
+// Corresponds to Java P2BigTest.testPerpendicularBisectorIntersection()
+//
+// The intersection of the perpendicular bisectors of two edges must be
+// equidistant from the endpoints of each edge (circumcenter property).
+TEST(P2UtilityTest, PerpendicularBisectorIntersection) {
+    Eigen::Vector3d p1(0.5, 0.0, 1.0);
+    Eigen::Vector3d q1(0.0, 1.0, 1.0);
+    Eigen::Vector3d p2(1.0, 0.0, 1.0);
+    Eigen::Vector3d q2(0.0, 1.5, 1.0);
+
+    auto l1 = perpendicularBisectorEuclidean(p1, q1);
+    auto l2 = perpendicularBisectorEuclidean(p2, q2);
+    auto o  = pointFromLines(l1, l2);    // circumcenter
+
+    // o must be equidistant from p1 and q1
+    EXPECT_NEAR(euclideanDistanceP2(p1, o),
+                euclideanDistanceP2(q1, o), 1E-10);
+
+    // o must be equidistant from p2 and q2
+    EXPECT_NEAR(euclideanDistanceP2(p2, o),
+                euclideanDistanceP2(q2, o), 1E-10);
+}

code/tests/test_spherical_functional.cpp

@@ -0,0 +1,37 @@
+// Stub for de.varylab.discreteconformal.functional.SphericalFunctionalTest (Java/JUnit).
+//
+// STATUS: BLOCKED – requires HDS port (Phase 4).
+//
+// Tests evaluate gradient and Hessian of the SphericalFunctional on meshes
+// built via CoHDS + ConvexHull, and check that a regular spherical metric
+// is a critical point of the functional.  All tests require the HalfEdge
+// data structure and the functional evaluation framework in C++.
+//
+// Java tests and their status:
+//   testReducedGradient   – blocked: needs CoHDS + SphericalFunctional
+//   testReducedHessian    – blocked: needs CoHDS + SphericalFunctional
+//   testGradient          – blocked: needs CoHDS + SphericalFunctional
+//   testHessian           – blocked: needs CoHDS + SphericalFunctional
+//   testCriticalPoint     – blocked: needs CoHDS + ConvexHull + SphericalFunctional
+
+#include <gtest/gtest.h>
+
+TEST(SphericalFunctionalTest, ReducedGradient) {
+    GTEST_SKIP() << "Blocked: requires HDS port (CoHDS + SphericalFunctional)";
+}
+
+TEST(SphericalFunctionalTest, ReducedHessian) {
+    GTEST_SKIP() << "Blocked: requires HDS port (CoHDS + SphericalFunctional)";
+}
+
+TEST(SphericalFunctionalTest, Gradient) {
+    GTEST_SKIP() << "Blocked: requires HDS port (CoHDS + SphericalFunctional)";
+}
+
+TEST(SphericalFunctionalTest, Hessian) {
+    GTEST_SKIP() << "Blocked: requires HDS port (CoHDS + SphericalFunctional)";
+}
+
+TEST(SphericalFunctionalTest, CriticalPoint) {
+    GTEST_SKIP() << "Blocked: requires HDS port (CoHDS + ConvexHull + SphericalFunctional)";
+}

doc/architecture/overall_pipeline.md

@@ -1,251 +1,564 @@
-# Draft High-level Architecture
+# conformallab++ — Architecture & Pipeline
 
+## Origin
 
-## Positioning and relation to existing libraries (Nice To Have but maybe to much)
-ConformalLab++ is not intended to replace established geometry processing libraries such as CGAL, libigl, or Geometry Central. Instead, it acts as an experiment-friendly pipeline layer on top of existing data structures and algorithms provided by these libraries.
+conformallab++ is a C++ reimplementation of
+[ConformalLab](https://github.com/varylab/conformallab),
+the Java research library for discrete conformal geometry by
+**Stefan Sechelmann** (TU Berlin, Institut für Mathematik,
+SFB/Transregio 109 *Discretization in Geometry and Dynamics*).
 
-The project revives the ideas and algorithms of the original ConformalLab by Stefan Sechelmann in a modern C++ setting, making them easier to combine with contemporary mesh libraries (e.g. Geometry Central for intrinsic quantities and operators) and to reuse in new experimental setups.
+The algorithmic foundation is his doctoral dissertation:
 
-In practice, ConformalLab++ focuses on:
+> Stefan Sechelmann —  
+> **Variational Methods for Discrete Surface Parameterization: Applications and Implementation**  
+> Doctoral thesis, Technische Universität Berlin, 2016.  
+> DOI: [10.14279/depositonce-5415](https://depositonce.tu-berlin.de/items/8e2988b2-d991-45b5-aad5-9fb7988f3b2f) · CC BY-SA 4.0
 
-+ a clear, declarative pipeline description for conformal and Möbius-based mesh experiments,
+The dissertation develops the variational framework for discrete conformal equivalence:
+discrete uniformization of Riemann surfaces, cone metrics, period matrices, and
+holonomy — all of which are directly implemented in this library.
 
-+ a unified halfedge-based UniversalMesh representation plus optional alternative representations,
+Further links:  
+**Java original:** [github.com/varylab/conformallab](https://github.com/varylab/conformallab) ·
+**Website:** [sechel.de](https://sechel.de/) ·
+**LinkedIn:** [linkedin.com/in/sechel](https://www.linkedin.com/in/sechel/)
 
-adapters and plugin units that wrap robust external implementations where appropriate (e.g. CGAL-based repair or reconstruction, Geometry Central–style discrete geometry operators, or libigl functionality) instead of reimplementing them.
+---
 
+## Positioning
 
+conformallab++ is a **specialised research library for discrete conformal geometry** on
+triangulated surfaces. It is not a general geometry-processing framework.
 
-## High-level geometry pipeline
-The following diagram outlines the planned high-level pipeline for ConformalLab++. It shows the rough division into preprocessing, processing, and postprocessing. In preprocessing, various input types are first converted into a universal representation (half-edge mesh) via adapter layers and any necessary preprocessing steps. From there, they are processed by the various core processors units and then either exported or visualized in postprocessing, with minor optimizations as necessary.
+The library revives the algorithms of the original ConformalLab by Stefan Sechelmann
+in a modern C++ setting. Its core concern is one precise mathematical question:
+
+> Given a triangulated surface, find a conformally equivalent metric that satisfies
+> prescribed curvature (angle-sum) constraints at each vertex.
+
+Everything in the library serves this goal:
+
+| What it is | What it is not |
+|------------|----------------|
+| Discrete conformal maps (Euclidean, spherical, hyperbolic) | General mesh processing |
+| Newton solver for angle-sum energy functionals | Remeshing / boolean / smoothing |
+| Priority-BFS layout into ℝ², S², Poincaré disk | Point-cloud or implicit-field processing |
+| Holonomy, period matrices, fundamental domains | NURBS / parametric modelling |
+| Research platform — experiment-first, CLI-ready | Production rendering engine |
+
+**Relation to existing libraries.**  
+conformallab++ sits *on top of* CGAL, not beside it.
+`CGAL::Surface_mesh` is the internal mesh representation; there is no additional
+abstraction layer. Eigen handles all linear algebra. libigl provides the optional
+interactive viewer. The library adds the conformal-geometry layer that none of these
+provide.
+
+---
+
+## The conformal geometry pipeline
+
+The full pipeline runs in three stages. All stages operate on the same
+`ConformalMesh` (= `CGAL::Surface_mesh<Point3>` with attached property maps).
 
 ```mermaid
 graph TD
-    A1["Explicit<br/>(OBJ, PLY, STL)"]
-    A2["Implicit<br/>(SDF, Formulas)"]
-    A3["Parametric<br/>(NURBS, Curves)"]
-    A4["Procedural<br/>(Noise, L-Sys)"]
-    ADAPTER1["Input Adapter <br/>(Convert to Universal Represenation)"]
-    PREPROCESSING["Preprocessor <br/>(refined triangulation etc.)"]
-    EXT["EXTERNAL LIBRARIES<br/>(Plugin System)<br/>- CGAL"]
-    UNIVERSAL["UNIVERSAL REPRESENTATION FORM<br/>(HalfEdge Mesh Interface)"]
-    EXPORT_ADAPTER["Export Adapter Layer"]
-    PROCESSING["PROCESSING CORE LAYER"]
-    OPT["Optimization<br/>- Decimation<br/>- Denoising<br/>- Remeshing"]
-    ALGO["Algorithmic<br/>- Boolean Ops<br/>- Smoothing<br/>- Hole Filling"]
-    OTHER["Other<br/>- Custom Algos<br/>- Transforms<br/>- Filters"]
-    POSTPROCESSING["POSTPROCESSING <br/>(refined triangulation etc.)"]
-    EXPORT["Export Formats<br/>- OBJ<br/>- PLY<br/>- STL<br/>- Custom"]
-    VIZ["Visualization<br/>- OpenGL<br/>- QT<br/>- Screenshot"]
-subgraph PRE[PREPROCESSING]
-    A1 --> ADAPTER1
-    A2 --> ADAPTER1
-    A3 --> ADAPTER1
-    A4 --> ADAPTER1   
-    PREPROCESSING <-->ADAPTER1
-end
-    EXT <--> UNIVERSAL
-    UNIVERSAL --> PROCESSING
-    UNIVERSAL --> EXPORT_ADAPTER
-    ADAPTER1 <--> EXT
-    ADAPTER1 <--> UNIVERSAL
-    OPT --> UNIVERSAL
-    ALGO --> UNIVERSAL
-    OTHER --> UNIVERSAL
-subgraph CORE[PROCESSING ]
-    PROCESSING --> OPT
-    PROCESSING --> ALGO
-    PROCESSING --> OTHER
-end
-subgraph POST[POSTPROCESSING]
-    EXPORT_ADAPTER <--> POSTPROCESSING
-    EXPORT_ADAPTER --> EXPORT
-    EXPORT_ADAPTER --> VIZ
-end
-    style UNIVERSAL fill:#90EE90,stroke:#000,stroke-width:3px,color:#000
-    style PREPROCESSING  fill:#87CEEB,stroke:#000,stroke-width:2px,color:#000
-    style POSTPROCESSING fill:#87CEEB,stroke:#000,stroke-width:2px,color:#000
-    style PROCESSING fill:#87CEEB,stroke:#000,stroke-width:2px,color:#000
-    style EXT fill:#DDA0DD,stroke:#000,stroke-width:2px,color:#000
-    style ADAPTER1 fill:#FFB347,stroke:#000,stroke-width:2px,color:#000
-    style EXPORT_ADAPTER fill:#FFB347,stroke:#000,stroke-width:2px,color:#000
+    IN["Input\n(OFF / OBJ / PLY / builder)"]
+    ADAPTER["Input Adapter\nload_mesh() · make_triangle() · make_quad_strip() …"]
+
+    subgraph PRE["① PREPROCESSING"]
+        MAPS["Maps Setup\nsetup_*_maps() + compute_lambda0_from_mesh()"]
+        DOF["DOF Assignment\nv_idx[v] · e_idx[e] · pin / free"]
+        THETA["Target Angles\ntheta_v[v] — cone metric or natural equilibrium"]
+        GB["Gauss–Bonnet Check\ncheck_gauss_bonnet() / enforce_gauss_bonnet()"]
+        MAPS --> DOF --> THETA --> GB
+    end
+
+    subgraph CORE["② PROCESSING CORE"]
+        NEWTON["Newton Solver\nnewton_euclidean/spherical/hyper_ideal()\n→ NewtonResult  x* ∈ ℝⁿ"]
+        CG["Cut Graph  [closed surfaces]\ncompute_cut_graph()\n→ CutGraph  (2g seam edges)"]
+        LAYOUT["Layout / Embedding\neuclidean/spherical/hyper_ideal_layout()\n→ Layout2D / Layout3D\n  · uv[v]  · halfedge_uv[h]\n  · HolonomyData"]
+        NORM["Normalisation\nnormalise_euclidean / _hyperbolic / _spherical()"]
+        PERIOD["Period Matrix  [genus 1]\ncompute_period_matrix()\n→ PeriodData  τ ∈ ℍ"]
+        FD["Fundamental Domain\ncompute_fundamental_domain()\n→ FundamentalDomain + tiling"]
+        NEWTON --> CG --> LAYOUT --> NORM --> PERIOD --> FD
+    end
+
+    subgraph POST["③ POSTPROCESSING"]
+        SERIAL["Serialisation\nsave_result_json/xml()\nsave_layout_off()"]
+        VIZ["Visualisation\nexample_viewer  (libigl / GLFW)"]
+        CLI["CLI App\nconformallab_core  -i -g -o -j -x -s"]
+    end
+
+    IN --> ADAPTER --> PRE
+    PRE --> CORE
+    CORE --> POST
+
+    style PRE  fill:#dbeafe,stroke:#3b82f6,stroke-width:2px,color:#000
+    style CORE fill:#dcfce7,stroke:#16a34a,stroke-width:2px,color:#000
+    style POST fill:#fef9c3,stroke:#ca8a04,stroke-width:2px,color:#000
 ```
-All external inputs (file-based, implicit, parametric, procedural) are first converted by an spezfic input adapter into a single universal  interface, which is the canonical internal representation used by the processing stages. The processing core units operates on this universal representation and can use external libraries via a plugin-like extension, while results are passed through an export adapter to file formats or visualization frontends.
 
+---
 
-## Three-stage processing pipeline
-The geometry processing pipeline in ConformalLab++ is structured into three main stages, all operating on the same universal mesh representation.
+## Stage ① — Preprocessing
 
-### Preprocessing
-Converts all external inputs (files, analytic models, procedural sources) into the canonical half-edge mesh form.
+### Input adapter
 
-Performs optional cleaning and normalization (e.g. fixing degeneracies, rescaling, enforcing orientation).
+All mesh input flows through a single entry point:
 
-Guarantees that downstream stages see a well-formed, consistent mesh (or fail early with clear errors).
+```cpp
+ConformalMesh mesh = load_mesh("input.off");   // OFF · OBJ · PLY
+ConformalMesh mesh = make_quad_strip();         // built-in test meshes
+```
 
-### Processing (core processing units)
+**Precondition guarantee:** the adapter ensures the mesh is a valid, orientable,
+triangulated surface with consistent halfedge structure (CGAL validity).
+Downstream stages never check for degeneracies — they trust the adapter.
 
-Runs one or more core processing units in sequence on the canonical mesh.
+### Maps setup
 
-Each unit takes a mesh of the same type as input and produces a mesh of the same type as output (possibly with enriched attributes).
+Each geometry mode has its own maps struct that attaches property maps to the mesh:
 
-Examples: remeshing, conformal parameterization, smoothing, boolean operations, experiment-specific transforms.
+| Geometry | Setup function | Key property maps |
+|----------|---------------|-------------------|
+| Euclidean (ℝ²) | `setup_euclidean_maps()` | `lambda0[e]`, `theta_v[v]`, `v_idx[v]` |
+| Spherical (S²) | `setup_spherical_maps()` | `lambda0[e]`, `theta_v[v]`, `v_idx[v]` |
+| Hyper-ideal (H²) | `setup_hyper_ideal_maps()` | `beta_v[v]`, `alpha_e[e]`, `v_idx[v]`, `e_idx[e]` |
 
-Each core processing unit conceptually has the shape UniversalMesh -> UniversalMesh, but units also declare:
+`compute_*_lambda0_from_mesh()` initialises log-edge-lengths from the 3-D vertex
+positions. After this step the solver works entirely in scale-factor space; the
+original vertex positions are no longer needed.
 
-+ Preconditions: requirements on the input mesh (e.g. triangulated, manifold, oriented, specific attributes present).
+### DOF assignment & target angles
 
-+ Capabilities: guarantees on the output mesh (e.g. remains manifold, produces curvature attributes, preserves boundary).
+```cpp
+// Pin first vertex (gauge fix for open meshes)
+maps.v_idx[*mesh.vertices().begin()] = -1;
+int idx = 0;
+for (auto v : rest_of_vertices) maps.v_idx[v] = idx++;
 
-A simple example:
+// Set target curvature (cone metric or natural equilibrium)
+maps.theta_v[v] = 2 * M_PI;          // flat interior vertex
+maps.theta_v[v] = M_PI / 3;          // 60° cone singularity
+```
 
-+ RemeshingUnit
+**Natural equilibrium shortcut:** evaluate the gradient at `x = 0`, subtract it from
+`theta_v` — the solver then converges to `x* = 0` identically (useful for tests).
 
-  + Preconditions: triangulated, manifold, oriented
+### Gauss–Bonnet check
 
-  + Capabilities: triangulated, manifold, vertex positions modified, edge count changed
+Before solving, the prescribed angles must satisfy:
 
-Pipeline composition is valid if, for every adjacent pair of units, the capabilities of the previous unit satisfy the preconditions of the next unit.
+$$\sum_{v} (2\pi - \Theta_v) = 2\pi \cdot \chi(M)$$
 
+```cpp
+check_gauss_bonnet(mesh, maps);         // throws if violated
+enforce_gauss_bonnet(mesh, maps);       // distributes residual uniformly
+```
 
-### Postprocessing
+**This is the most common source of silent Newton non-convergence.**
+Prescribing angles that violate Gauss–Bonnet means no conformal factor exists —
+the solver will iterate without converging.
 
-Adapts the processed mesh for external consumption: export to file formats, visualization, analysis tools.
+---
 
-May attach or convert attribute data (e.g. colors, scalar fields, experiment results) into a format suitable for rendering or further tools.
+## Stage ② — Processing core
 
-Does not change the core topology or semantics of the processed mesh, only its representation for the outside world.
+### The three geometry modes
 
-This structure makes it easy to reason about where data enters, is modified, and leaves the system, and keeps experimental algorithms clearly separated from IO concerns.
+conformallab++ implements discrete conformal geometry in three model spaces:
 
-## Role of the universal mesh representation
-The universal mesh representation (a half-edge mesh interface) is the shared contract between all pipeline stages.
+| Mode | Space | Curvature | Typical surfaces |
+|------|-------|-----------|-----------------|
+| **Euclidean** | ℝ² | K = 0 | Flat tori, developable surfaces, open patches |
+| **Spherical** | S² | K = +1 | Genus-0 (sphere-like) surfaces |
+| **Hyper-ideal** | H² (Poincaré disk) | K = −1 | Genus-g surfaces (g ≥ 1), hyperbolic structures |
 
-### Single input/output type
-Every core processing unit exposes the same function shape conceptually:
-UniversalMesh → UniversalMesh.
-This allows units to be chained in arbitrary order as long as their preconditions on the mesh are satisfied.
+All three share the same algorithmic structure — only the angle formula, the
+trilateration geometry, and the Hessian sign differ.
 
-### Local, topology-aware access
-The half-edge structure encodes vertices, halfedges, faces, and their adjacency relations, which is essential for typical geometry operations (e.g. local neighborhood queries, traversal, edge flips).
+### Newton solver
 
-Algorithms do not need to know where the mesh came from (file vs. analytic vs. procedural); they only rely on this uniform interface.
+```
+NewtonResult newton_euclidean  (mesh, x0, maps [, tol, max_iter])
+NewtonResult newton_spherical  (mesh, x0, maps [, tol, max_iter])
+NewtonResult newton_hyper_ideal(mesh, x0, maps [, tol, max_iter, hess_eps])
+```
 
-### Extensibility via attributes
-The universal mesh may carry extensible per-vertex, per-edge, and per-face attributes (e.g. UVs, curvature, experimental scalar fields).
-Core units can read and write these attributes while still conforming to the same mesh type, enabling complex pipelines without changing the central data structure.
+Each iteration:
+1. Evaluate gradient **G** (angle-sum defect per vertex)
+2. Evaluate Hessian **H** (analytical for Euclidean/Spherical; symmetric FD for HyperIdeal)
+3. Solve **H·Δx = −G** — try `SimplicialLDLT`, fall back to `SparseQR` on rank deficiency
+4. Backtracking line search (up to 20 halvings)
 
+**Preconditions:** mesh triangulated + manifold · Gauss–Bonnet satisfied · DOFs assigned  
+**Provides:** `NewtonResult.x` — converged scale factors; `converged`, `iterations`, `grad_inf_norm`
 
+**SparseQR fallback** handles gauge modes on closed meshes without a pinned vertex.
+The fallback is public API: `solve_linear_system(H, rhs, &used_fallback)`.
 
-Because all processing units speak the same “mesh language”, you can build pipelines like:
+### Cut graph (closed surfaces)
+
+For a closed genus-g surface, cutting along 2g independent homology cycles
+turns it into a topological disk — a prerequisite for globally consistent BFS layout.
+
+```cpp
+CutGraph cg = compute_cut_graph(mesh);
+// cg.cut_edge_flags[e.idx()] — true for seam edges
+// cg.cut_edge_indices         — ordered list of 2g seam edges
+// cg.genus                    — g
+```
+
+**Algorithm:** tree-cotree decomposition (Erickson–Whittlesey 2005).  
+**Preconditions:** closed, orientable, triangulated mesh  
+**Provides:** exactly `2g` seam edges whose removal makes the surface simply connected
+
+### Layout / Embedding
+
+BFS-trilateration unfolds the mesh into the target geometry.
+The root face (largest 3-D area, 1.5× interior bonus) is placed first;
+all other faces are placed in **priority order by BFS depth** (min-heap),
+minimising trilateration error accumulation.
+
+```cpp
+HolonomyData hol;
+Layout2D layout = euclidean_layout(mesh, result.x, maps, &cg, &hol, /*normalise=*/true);
+Layout3D slayout = spherical_layout(mesh, result.x, smaps);
+Layout2D hlayout = hyper_ideal_layout(mesh, result.x, hmaps, &cg, &hol);
+```
+
+**Key outputs:**
+
+| Field | Content |
+|-------|---------|
+| `layout.uv[v.idx()]` | Primary UV — first / shallowest-BFS visit |
+| `layout.halfedge_uv[h.idx()]` | UV of `source(h)` as seen from `face(h)` — seam-aware |
+| `layout.has_seam` | True when a vertex was reached via two different paths |
+| `hol.translations[i]` | Translation ω_i across cut edge i (Euclidean / spherical) |
+| `hol.mobius_maps[i]` | Möbius isometry T_i ∈ SU(1,1) across cut edge i (hyperbolic) |
+
+**`halfedge_uv` — texture atlas semantics:**  
+At a seam edge the two opposite halfedges carry *different* UV values.
+This gives each face its own UV copy of a seam vertex, enabling
+a proper GPU texture atlas without vertex duplication.
+
+**Trilateration — geometry by mode:**
+
+| Mode | Method | Accuracy |
+|------|--------|---------|
+| Euclidean | Analytic formula in ℝ² | Exact |
+| Spherical | Spherical law of cosines on S² | Exact |
+| Hyper-ideal | Möbius + hyperbolic law of cosines in Poincaré disk | Exact |
+
+**Preconditions:** `NewtonResult.converged` · mesh · maps · optional `CutGraph`  
+**Provides:** `Layout2D/3D` with `uv`, `halfedge_uv` · `HolonomyData` with translations / Möbius maps
+
+### Möbius maps
+
+`MobiusMap` (T(z) = (az+b)/(cz+d)) is the central algebraic object for hyperbolic geometry:
+
+```cpp
+MobiusMap T = MobiusMap::from_three(z1,w1, z2,w2, z3,w3); // fit to 3 correspondences
+MobiusMap S = T.inverse().compose(U);                       // group operations
+Eigen::Vector2d p2 = T.apply(p);                           // apply to 2-D point
+```
+
+Used for: hyperbolic trilateration · holonomy tracking · normalisation centering.
+
+### Normalisation
+
+After layout, a canonical post-processing step brings the result into a standard position:
+
+| Mode | Method | Effect |
+|------|--------|--------|
+| Euclidean | PCA — centroid → origin, major axis → x-axis | Translation + rotation |
+| Hyperbolic | Weighted Möbius centering (Fréchet mean, 30 iterations) | Maps centroid to disk origin |
+| Spherical | Rodrigues rotation | Maps centroid to north pole |
+
+Both `uv` and `halfedge_uv` are transformed identically.
+
+### Period matrix (genus 1)
+
+From the two holonomy translations ω₁, ω₂ ∈ ℂ read off from the cut graph,
+the conformal type of a flat torus is the SL(2,ℤ)-orbit of:
+
+$$\tau = \omega_2 / \omega_1 \in \mathbb{H}$$
+
+```cpp
+PeriodData pd = compute_period_matrix(hol);
+// pd.tau                   — complex period ratio
+// pd.omega[i]              — lattice generators as complex numbers
+// pd.in_fundamental_domain — after SL(2,ℤ) reduction
+// pd.genus()               — g = omega.size() / 2
+```
+
+SL(2,ℤ) reduction alternates T: τ↦τ+1 and S: τ↦−1/τ steps until
+τ ∈ F = {|τ| ≥ 1, −½ ≤ Re(τ) < ½}.
+
+**Note:** The Siegel period matrix Ω ∈ H_g for genus g ≥ 2 requires integrating
+holomorphic differentials — deferred to Phase 8.
+
+### Fundamental domain
+
+```cpp
+FundamentalDomain fd = compute_fundamental_domain(hol);
+// genus 1: CCW parallelogram {0, ω₁, ω₁+ω₂, ω₂}
+// genus g > 1: empty — 4g-polygon boundary walk deferred to Phase 8
+
+auto tiles = tiling_neighbourhood(layout, hol, /*m_max=*/2, /*n_max=*/2);
+// returns (2·m_max+1)·(2·n_max+1) translated copies of the layout
+// for visualising the universal cover
+```
+
+---
+
+## Stage ③ — Postprocessing
+
+### Serialisation
+
+```cpp
+// Layout as mesh file
+save_layout_off("layout.off", mesh, layout);
+
+// Full result: DOF vector + metadata + layout UVs
+save_result_json("result.json", result, "euclidean", V, F, &layout);
+save_result_xml ("result.xml",  result, "euclidean", V, F, &layout);
+
+// Round-trip load
+NewtonResult res2; std::string geom; Layout2D uv2;
+load_result_json("result.json", &res2, &geom, &uv2);
+```
+
+### CLI app
 
 ```bash
-Preprocessing
-  -> RemeshingUnit
-  -> ConformalMapUnit
-  -> ExperimentSpecificFilter
-  -> Export/Postprocessing
+conformallab_core \
+  -i input.off        # input mesh
+  -g euclidean        # geometry: euclidean | spherical | hyper_ideal
+  -o layout.off       # layout output
+  -j result.json      # JSON serialisation
+  -x result.xml       # XML serialisation
+  -s                  # show input in viewer
+  -v                  # verbose solver output
 ```
-without changing the basic function signature or the surrounding infrastructure, only by reordering or swapping units.
 
+### Interactive viewer
 
-## Declarative pipeline descriptions (Nice To Have but maybe to much) 
-ConformalLab++ optionally supports a lightweight YAML-based description format for experiments.
-A pipeline consists of an input adapter, a sequence of steps (core processing units), and an output adapter. Each step can specify params as well as structural constraints via require and provide keys. A pipeline is considered valid if, for every step, all require conditions are satisfied by the accumulated provide and expect constraints of previous steps
+`example_viewer` (libigl / GLFW) shows the 3-D mesh and the 2-D layout
+side-by-side. Built automatically with `-DWITH_CGAL=ON`.
+
+---
+
+## Processing unit contracts
+
+Each stage has explicit preconditions and guarantees.
+A pipeline is valid if every unit's preconditions are satisfied
+by the outputs of all preceding units.
+
+| Unit | Preconditions | Provides |
+|------|--------------|---------|
+| `load_mesh` | Valid file path, supported format | Manifold, oriented, triangulated `ConformalMesh` |
+| `setup_*_maps` | Triangulated mesh | Initialised property maps; `lambda0` from 3-D positions |
+| `check_gauss_bonnet` | `theta_v` set | Throws if Σ(2π−Θ_v) ≠ 2π·χ |
+| `enforce_gauss_bonnet` | `theta_v` set | Σ(2π−Θ_v) = 2π·χ guaranteed |
+| `newton_*` | GB satisfied · DOFs assigned | `NewtonResult.converged` · `x*` · gradient norm |
+| `compute_cut_graph` | Closed, orientable mesh | `2g` seam edges · `CutGraph.genus` |
+| `euclidean_layout` | `newton_euclidean` converged | `uv[v]` · `halfedge_uv[h]` · `HolonomyData` |
+| `normalise_euclidean` | `layout.success == true` | Centroid at origin · major axis = x-axis |
+| `compute_period_matrix` | `hol.translations.size() >= 2` | `τ ∈ ℍ` · optionally reduced to F |
+| `compute_fundamental_domain` | `HolonomyData` (genus 1) | CCW parallelogram · edge identifications |
+
+---
+
+## The three geometry modes in detail
+
+```
+              Euclidean            Spherical            Hyper-ideal
+              ─────────────────────────────────────────────────────
+Space         ℝ²                   S²                   H² (Poincaré disk)
+Curvature     K = 0                K = +1               K = −1
+Genus         any (cone metrics)   0                    ≥ 1
+Angle sum     Σα_v = Θ_v           Σα_v = Θ_v           Σβ_v = Θ_v
+Hessian       PSD (cotangent-Lap.) NSD (sign-flip)      PSD (FD, strict conv.)
+Holonomy      translations ω_i     rotations (2-D)      Möbius maps T_i ∈ SU(1,1)
+Period        τ = ω₂/ω₁ ∈ ℍ        —                    axis of T_i
+Normalise     PCA centring         Rodrigues to N pole  weighted Möbius centring
+```
+
+---
+
+## Extension points
+
+### Adding a new functional
+
+1. Create `my_functional.hpp` with a `Maps` struct and `evaluate_my_functional()`.
+2. The gradient must satisfy: `G_v = Σ(angle contributions) − theta_v[v]`.
+3. Verify with a finite-difference gradient check (copy any `GradientCheck_*` test).
+4. Pass to `solve_linear_system(H, -G)` or write a thin `newton_my` wrapper.
+
+### Adding a new geometry mode
+
+Implement `trilaterate_my()` with the correct local placement formula,
+then follow the same BFS structure as `euclidean_layout` (see `layout.hpp`).
+The priority BFS, holonomy tracking, and `halfedge_uv` population are geometry-agnostic.
+
+### Adding a new processing unit
+
+Declare its preconditions and capabilities explicitly (see table above).
+The pipeline validation is currently manual (documented contracts); a
+compile-time or runtime check is a natural Phase 8 extension.
+
+---
+
+## Declarative pipeline (target for Phase 8)
+
+A lightweight YAML description for reproducible experiments:
 
 ```yaml
-
 pipeline:
-  name: basic_conformal
+  name: flat_torus_period
+  geometry: euclidean
+
   input:
-    adapter: obj_reader
-    source: data/bunny.obj
+    source: data/torus.off
 
   steps:
-    - id: clean
-      unit: repair_soft_clean
-      params:
-        mode: soft
-      require:
-        representation: UniversalMesh
-        triangulated: true
-      provide:
-        triangulated: true
-        manifold: null
+    - id: setup
+      unit: setup_euclidean_maps
+      provide: [maps_initialised]
 
-    - id: param
-      unit: conformal_parameterization
+    - id: gauss_bonnet
+      unit: enforce_gauss_bonnet
+      require: [maps_initialised]
+      provide: [gauss_bonnet_satisfied]
+
+    - id: solve
+      unit: newton_euclidean
+      require: [gauss_bonnet_satisfied]
       params:
-        method: cauchy_riemann
-      require:
-        triangulated: true
-        manifold: true
-      provide:
-        attributes_add: [uv]
+        tol: 1.0e-10
+        max_iter: 200
+      provide: [x_converged]
+
+    - id: cut
+      unit: compute_cut_graph
+      require: [mesh_closed]
+      provide: [cut_graph]
+
+    - id: layout
+      unit: euclidean_layout
+      require: [x_converged, cut_graph]
+      params:
+        normalise: true
+      provide: [layout_uv, holonomy]
+
+    - id: period
+      unit: compute_period_matrix
+      require: [holonomy]
+      provide: [tau]
 
   output:
-    adapter: gltf_writer
-    target: out/bunny.glb
+    layout:  out/torus_layout.off
+    json:    out/torus_result.json
+    tau:     out/torus_tau.txt
+```
+
+Each unit declares `require` (preconditions) and `provide` (capabilities).
+A pipeline is valid if for every step, all `require` keys are satisfied by the
+accumulated `provide` set of all preceding steps.
+This mirrors exactly the contract table in this document.
+
+---
+
+## Development Roadmap
+
+> **Grenze Portierung / neue Forschung:**  
+> Phase 1–7 sind direkte Portierungen aus dem Java-Original (Sechelmann 2016).  
+> Ab Phase 8 geht die Arbeit über den Umfang der Java-Bibliothek hinaus.  
+> — Phase 8 (CGAL-Paket) ist **Infrastruktur**, kein neuer Algorithmus.  
+> — Phase 9 (Inversive-Distance, Analytischer Hessian, 4g-Polygon) ist **Portierung** ausstehender Java-Features.  
+> — Phase 10+ ist **eigenständige Forschung**, die über das Java-Original hinausgeht.
+
+---
+
+### ◼ Portierungsphase abgeschlossen — Phase 1–7
 
 ```
-Each unit may declare a require section describing what it needs from its input (e.g. representation, triangulated, manifold, required attributes) and a provide section describing what it guarantees on its output (e.g. still triangulated, now manifold, adds a uv attribute). Pipelines are considered valid if for every step, the accumulated provide information of all previous steps satisfies the require constraints of the next step.
+Phase 1   Clausen / Lobachevsky / ImLi₂                             ✅
+Phase 2   Hyper-ideal Geometrie (ζ, lᵢⱼ, αᵢⱼ, σᵢ)                 ✅
+Phase 3   CGAL-Infrastruktur + alle drei Funktionale (E/S/H)         ✅
+Phase 4   Newton-Solver (SimplicialLDLT + SparseQR-Fallback)         ✅   68 Tests
+Phase 5   Priority-BFS-Layout + CLI + JSON/XML                       ✅   95 Tests
+Phase 6   Gauss–Bonnet, Tree-Cotree-Schnittgraph, Normalisierung     ✅  121 Tests
+Phase 7   MobiusMap, halfedge_uv, Möbius-Holonomie, Periodenmatrix,
+          Fundamentalbereich (Genus 1), Java-Parität abgeschlossen   ✅  158 Tests
+```
 
-### Repair Units (Nice To Have but maybe to much)
+---
 
-Before entering the core conformal pipeline, input meshes are passed through an explicit preprocessing stage. This stage is responsible for turning “real-world” polygon data (often noisy, inconsistent, or non‑manifold) into a mesh that satisfies the structural requirements of the subsequent units
+### ◼ Infrastruktur (über Java-Bibliothek hinaus) — Phase 8: CGAL-Paket
 
-ConformalLab++ provides a small set of repair units that can be combined as needed:
+```
+8a   Traits-Klasse & Konzepte          → include/CGAL/Conformal_map_traits.h
+8b   Öffentliche CGAL-Header-Hierarchie → include/CGAL/Discrete_conformal_map.h etc.
+8c   Dokumentation im CGAL-Stil         → doc/Conformal_map/PackageDescription.txt
+8d   CGAL-Testformat                    → test/Conformal_map/
+8e   Declarative YAML-Pipeline          → pipeline validator (require/provide tokens)
+```
 
-+ removal of (almost) degenerate triangles (needles, caps),
+See the [Declarative pipeline](#declarative-pipeline-target-for-phase-8) section above
+for the YAML schema that Phase 8e will validate at runtime.
 
-+ stitching of compatible boundary cycles to close small gaps,
+---
 
-+ enforcing a consistent face orientation where possible.
+### ◼ Ausstehende Portierung (Java-Features noch nicht übertragen) — Phase 9
 
-These units are conceptually similar to geometric repair routines in modern polygon mesh processing toolkits
+```
+9a   Inversive-Distance-Funktional (Luo 2004)
+       → InversiveDistanceMaps + functional + Hessian
+       → discrete uniformization via inversive distances
+9b   Analytischer HyperIdeal-Hessian (ζ-Kette)
+       → replace FD Hessian in hyper_ideal_hessian.hpp
+       → reduces Newton iterations for large meshes
+9c   4g-Polygon-Randlauf (Genus g > 1)
+       → extend compute_fundamental_domain() beyond genus 1
+       → algorithm outline already in fundamental_domain.hpp as TODO(Phase 9)
+```
 
-We distinguish between two typical modes of operation:
+---
 
-+ Soft cleaning: minimally invasive repairs intended for visualization and quick experiments. Soft cleaning tries to improve mesh quality without significantly altering topology or removing components.
+### ◼ Neue Forschung (über das Java-Original hinaus) — Phase 10+
 
-+ Hard cleaning: more aggressive repairs intended for numerically robust experiments. Hard cleaning may merge vertices, remove tiny components, and modify local topology to enforce manifoldness and consistent orientation when possible.
+```
+Phase 10   Globale Uniformisierung Genus g ≥ 2
+10a  Holomorphe Differentiale auf diskreten Flächen
+       → discrete harmonic 1-forms; integration along cut graph cycles
+10b  Siegel-Periodenmatrix Ω ∈ H_g  (g×g komplex-symmetrisch, Im(Ω) > 0)
+       → extend compute_period_matrix() to genus g ≥ 2
+       → requires holomorphic differentials from 10a
+10c  Vollständige Uniformisierung
+       → uniformize arbitrary genus-g surface to canonical constant-curvature metric
+       → depends on 10a + 10b
+```
 
-## More internal representations (Nice To Have but maybe to much)
-ConformalLab++ distinguishes between internal representation spaces that experiments may move between:
+---
 
-+ UniversalMesh: a halfedge‑based surface mesh representation used for most combinatorial and conformal algorithms.
+## Recommended reading
 
-+ PointCloud: an unstructured set of sample points, optionally with normals and additional attributes, used for sampling, resampling, and certain reconstruction tasks.
+### Primary source — the dissertation this library implements
 
-+ ImplicitField: a scalar field $f: \mathbf{R}^n \rightarrow \mathbf{R}$
-represented on a grid or as a procedural function, used for iso‑surface extraction and robust shape transformations.
-
-Each space has its own natural algorithms, and dedicated conversion units allow pipelines to move between these representations when needed.
-
-### Conversion units
-
-Typical conversion units include:
-
-+ mesh_to_pointcloud (sampling vertices and surfaces of a UniversalMesh into a PointCloud),
-
-+ pointcloud_to_mesh (surface reconstruction from point samples),
-
-+ mesh_to_implicit (rasterizing a UniversalMesh into a signed distance or occupancy field),
-
-+ implicit_to_mesh (iso‑surface extraction from an ImplicitField).
-
-These units are modeled as regular processing units in the pipeline but change the underlying representation space instead of just transforming a mesh
-
-## Attribute behaviour under topology changes (Nice To Have but maybe to much)
-
-ConformalLab++ treats attributes as first-class data attached to mesh entities (vertices, edges, faces, halfedges). When topology changes, attributes are updated according to simple, explicit rules:
-
-+ Vertex splits / edge splits: attributes on newly created vertices or edges are initialized by interpolation of the incident elements (e.g. linear interpolation of scalar fields along an edge).
-
-+ Edge collapse: attributes on the surviving vertex are computed from the incident vertices, typically by a convex combination or area‑weighted average for scalar quantities.
-
-+ Face operations (flip, subdivision): face attributes are either propagated (for categorical data) or interpolated (for scalar data) to new faces.
-
-By default, ConformalLab++ uses linear interpolation for scalar attributes and nearest‑neighbour propagation for discrete or categorical attributes; units may override these policies where necessary
+| | |
+|---|---|
+| **Sechelmann** — *Variational Methods for Discrete Surface Parameterization: Applications and Implementation*, TU Berlin 2016 | The mathematical foundation of the entire library: discrete conformal equivalence, variational angle-sum functionals, Newton solver, uniformization, period matrices. DOI: [10.14279/depositonce-5415](https://depositonce.tu-berlin.de/items/8e2988b2-d991-45b5-aad5-9fb7988f3b2f) |
+| **Java original** — [github.com/varylab/conformallab](https://github.com/varylab/conformallab) | Reference implementation in Java — the direct source for all algorithms ported to C++ |
 
+### Further references
 
+| Source | Relevance in conformallab++ |
+|--------|----------------------------|
+| Springborn — *Ideal Hyperbolic Polyhedra and Discrete Uniformization* (2020) | HyperIdeal functional; ζ₁₃/ζ₁₄/ζ₁₅ in `hyper_ideal_geometry.hpp` |
+| Pinkall, Polthier — *Computing Discrete Minimal Surfaces* (1993) | Cotangent-Laplace Hessian in `euclidean_hessian.hpp` |
+| Bobenko, Springborn — *Variational Principles for Circle Patterns* (2004) | Angle-sum variational framework used throughout |
+| Luo — *Combinatorial Yamabe Flow on Surfaces* (2004) | Inversive-distance functional (not yet ported) |
+| Erickson, Whittlesey — *Greedy Optimal Homotopy Generators* (SODA 2005) | Tree-cotree algorithm in `cut_graph.hpp` |