docs: full audit — fix 4 wrong port/research labels + consolidated research-track

A full audit of `doc/` plus root-level markdown files (27 files) against
the actual ground truth in the C++ code and the local Java repository at
`/Users/tarikmoussa/Desktop/conformallab/` revealed four pre-existing
mis-labels and a stale test count.  All are corrected here.

Audit findings — corrected
─────────────────────────

1. **`InversiveDistanceFunctional` mis-labelled as Java port** (4 doc sites)
   Empirical verification:
       find /Users/tarikmoussa/Desktop/conformallab -iname "*nversive*"
       (zero matches)
   The class does NOT exist in `de.varylab.discreteconformal`.  The C++
   implementation is built from Luo 2004 + Glickenstein 2011 + Bowers-
   Stephenson 2004 — new research, not a port.
   Fixed in: java-parity.md, references.md, add-inversive-distance.md.

2. **HyperIdeal Hessian mis-labelled as "Java has analytic Hessian"**
   Empirical verification: `HyperIdealFunctional.java:295-298`:
       public boolean hasHessian() { return false; }
   Java has NO Hessian at all.  Both the FD (Phase 4a) and the block-FD
   (Phase 9b) Hessians in C++ are research beyond the Java port.  The
   chain rule (b,a) → ℓ → ζ → α/β is the *mathematical formulation*
   from Springborn 2020, not something Java implements.
   Fixed in: java-parity.md.

3. **Stale test count** README:87 said "28 suites, 170 tests" — current
   actual is 35 suites, 176 CGAL + 36 non-CGAL.  Fixed.

4. **Tutorial framing** — `add-inversive-distance.md` was framed as
   "porting an InversiveDistanceFunctional.java" that does not exist.
   Rewritten as "Implementing the Inversive-Distance functional from
   Luo 2004" with prominent verification block at top.

New document: `doc/roadmap/research-track.md`
─────────────────────────────────────────────

Consolidates everything in conformallab++ that goes beyond a Java port:

* Items already on `main`: HyperIdeal FD Hessian, period matrix τ
  partial-research components, Möbius holonomy storage.
* Items on open PRs: CP-Euclidean (PR #8, port), Inversive-Distance
  (PR #8, research), block-FD Hessian (PR #9, research).
* Planned research with full citations:
  - **Phase 9b-analytic** — full analytic HyperIdeal Hessian via
    Schläfli identity (Schläfli 1858/60) and chain rule through
    ζ₁₃/ζ₁₄/ζ₁₅, citing Springborn 2020 §4, Cho-Kim 1999,
    Glickenstein 2011 §4.  Includes acceptance-criteria checklist
    (per-case derivative cross-checks, gauge null space, PSD,
    measured ≥ 3× speed-up, LaTeX correctness note).
  - **Phase 9a.2-analytic** — analytic inversive-distance Hessian
    via Glickenstein 2011 eq. (4.6).
  - **Phase 10c** — full uniformization for genus g ≥ 2 (Fuchsian
    group representation) — fully new research, no Java reference.
  - **geometry-central** GC-1/2/3 exploratory track.

* Java backlog summary: 11 worth-porting Java classes identified by
  the parallel survey (FundamentalPolygonUtility, DiscreteHarmonicForm-
  Utility, DiscreteHolomorphicFormUtility, CanonicalBasisUtility,
  HyperbolicCyclicFunctional, QuasiisothermicUtility, KoebePolyhedron, …).
  ~6 500 Java lines, ~5 months of porting work, organised by phase.

Updated documents
─────────────────

* CLAUDE.md
  - New "Port-vs-research maintenance rule" with empirical verification
    command and the four corrected mis-labels.
  - Doc map: 23 → 24 documents (research-track.md added).

* README.md
  - Test count corrected (170 → 176+36).

* doc/math/references.md
  - Luo 2004 entry corrected ("new research" instead of "not yet ported").
  - New entries for Bowers-Stephenson 2004, Glickenstein 2011,
    Bobenko-Pinkall-Springborn 2010, Schläfli 1858/60.

* doc/roadmap/phases.md
  - Phase 9 reorganised: 9a split into 9a.1 (port) / 9a.2 (research),
    9b clarified as research (Java has no Hessian), 9c expanded with
    Java line counts and effort estimates.
  - Phase 10 reorganised: 10a/10b/10c with their Java prerequisites
    explicitly listed; 10c flagged as "fully new research".
  - Phase 10b' added: parallel research track (hyperbolic functional,
    quasi-isothermic, Möbius centering).
  - Phase 10c' added: optional Java-port additions (Koebe, circle
    patterns, electrostatic sphere).

* doc/roadmap/java-parity.md
  - Inversive-distance row: ❌ Java, ❌ C++ (Phase 9a.2) — new research.
  - CP-Euclidean row added: ✅ Java, ❌ C++ (Phase 9a.1) — port.
  - HyperIdeal Hessian row: ❌ Java, ⚠️ FD + block-FD in C++.
  - Worth-porting table replaced with the survey results (12 classes,
    Java line counts, suggested phases).
  - "HyperIdeal Hessian: FD vs analytic" section rewritten with the
    correction notice.

* doc/tutorials/add-inversive-distance.md
  - Rewritten end-to-end with prominent verification block at top.
  - Now correctly framed as "Implementing the Inversive-Distance
    functional from Luo 2004" — research, not port.
  - Includes the four required cross-validations:
    limit cases, Bowers-Stephenson round-trip, FD-vs-analytic,
    cross-validation against euclidean_functional at u=0.
  - New "How to know if it's a port or research" closing section
    with the empirical verification command.

Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>

doc/roadmap/research-track.md

@@ -0,0 +1,317 @@
+# Research Track — items beyond the Java port
+
+> **Purpose:** This document consolidates everything in conformallab++
+> that goes *beyond* a port of `de.varylab.discreteconformal`.  Items
+> listed here are **new research**, drawn from published mathematical
+> sources (not from Java code).  They are separated from the
+> port-tracking sheet `doc/roadmap/java-parity.md` so that the porting
+> work and the research work can be planned independently.
+>
+> **Created:** 2026-05-21, after a full doc audit that identified four
+> items previously mislabelled as "ports".  This document corrects the
+> record and extends it with the explicit research plan for Phase
+> 9b-analytic.
+
+---
+
+## How to read this document
+
+Every entry has the structure:
+
+```
+### <item>
+* Mathematical source(s):  <papers with year, journal, equation/section>
+* Java reference:          NONE  (or:  partial — <class>, with the note "<what>")
+* Status:                  🔲 planned · 🟡 PR open · ✅ landed · ❌ blocked
+* Acceptance criteria:     <what tests/proofs must pass>
+* Effort:                  small / medium / large
+* Phase:                   9b-analytic / 9c / 10a / 10b / 10c
+```
+
+The phase numbers match `doc/roadmap/phases.md`.
+
+---
+
+## Items already on `main` (research, not port)
+
+### Hyper-ideal Hessian — FD (Phase 4a, ✅ landed)
+* **Mathematical source:** symmetric central difference of the
+  analytic gradient `G = (β − Θ, α − θ)` (Springborn 2020 §4 for the
+  gradient itself).
+* **Java reference:** `HyperIdealFunctional.java:295-298` declares
+  `hasHessian() { return false; }` — **Java has no Hessian at all**.
+* **Status:** ✅ landed in `code/include/hyper_ideal_hessian.hpp` Phase 4a.
+* **Why a research item, not a port:** the existing Phase 4a label
+  describes only *when* it was added to the C++ project, not Java
+  parity.  The Hessian is a conformallab++ addition.
+* **Effort:** small (already done).
+
+### Period matrix τ for genus 1 (Phase 7, ✅ landed)
+* **Mathematical source:**
+  - Sechelmann (2016) *Variational Methods for Discrete Surface
+    Parameterization* §4 — SL(2,ℤ) reduction algorithm.
+  - Bobenko-Springborn (2004) §6 — period matrix definition.
+* **Java reference:** partial — `PeriodMatrixUtility.java` exists in Java
+  with similar functionality (this *is* a port).
+* **Status:** ✅ landed in `code/include/period_matrix.hpp`.
+* **Note:** listed here only because parts of `phase-9a-validation.md`
+  reference it as research; clarification — the genus-1 period matrix is
+  a Java port, the **genus g ≥ 2** extension (Phase 10b) is research.
+
+### Möbius holonomy in SU(1,1) (Phase 7, ✅ landed)
+* **Mathematical source:** Bobenko-Springborn (2004) §5; Sechelmann
+  (2016) §3 for the SU(1,1) representation.
+* **Java reference:** partial — Java has Möbius transformations but not
+  the holonomy-around-cut-graph computation in the same form.
+* **Status:** ✅ landed in `code/include/layout.hpp` (`MobiusMap` class).
+* **Why partially research:** the half-edge `uv` storage for proper
+  seam-aware texture atlasing is new in conformallab++.
+
+### Cross-API consistency tests (Phase 7 stubs, ✅ landed)
+* `EuclideanFunctional.GradientCheck_Hessian` and the spherical analog
+  were ported from Java `@Ignore` stubs and given real bodies.
+* See `doc/architecture/phase-9a-validation.md` for the full mapping.
+
+---
+
+## Items currently on open PRs
+
+### CP-Euclidean functional (Phase 9a.1, 🟡 PR #8)
+* **Mathematical source:** Bobenko, Pinkall, Springborn (2010).
+  *Discrete conformal maps and ideal hyperbolic polyhedra.*
+  Geometry & Topology 14, 379–426.
+* **Java reference:** ✅ `CPEuclideanFunctional.java` (260 lines + 88-line
+  `CPEuclideanFunctionalTest.java`).  **This one IS a port.**
+* **Status:** 🟡 PR #8 open, 10 tests including Java-test parity.
+* **Note:** listed here because the *face-based* DOF structure is new in
+  conformallab++ (existing functionals all have vertex/edge DOFs); the
+  trait API generalisation needed for it is research-flavoured but the
+  algorithm itself is a port.
+
+### Inversive-distance functional (Phase 9a.2, 🟡 PR #8)
+* **Mathematical sources:**
+  - **Luo, F.** (2004). *Combinatorial Yamabe Flow on Surfaces.*
+    Comm. Contemp. Math. 6(5), 765–780.  → edge-length formula §3,
+    gradient identity Lemma 3.1.
+  - **Bowers, P. L. & Stephenson, K.** (2004). *Uniformizing dessins
+    and Belyĭ maps via circle packing.* Memoirs of the AMS 170(805).
+    → inversive-distance identity `I_ij = (ℓ²−r_i²−r_j²)/(2 r_i r_j)`.
+  - **Glickenstein, D.** (2011). *Discrete conformal variations and
+    scalar curvature on piecewise flat manifolds.* J. Diff. Geom.
+    87(2), 201–238.  → §5 correspondence `I_ij = cos θ_e`, eq. 4.6
+    analytic Hessian (used later by Phase 9b-analytic mirror).
+* **Java reference:** ❌ **none.**  Verified empirically:
+  ```bash
+  $ find /Users/tarikmoussa/Desktop/conformallab -iname "*nversive*"
+  (zero matches)
+  ```
+* **Status:** 🟡 PR #8 open, 11 tests including limit-case verification
+  and cross-validation with `euclidean_functional.hpp` at `u = 0`.
+* **Acceptance criteria (all met):**
+  - Three limit-cases of Luo's `ℓ²` formula at machine precision
+    (tangent, orthogonal, inside-tangent).
+  - Bowers-Stephenson round-trip identity at machine precision.
+  - FD-vs-analytic gradient check ≤ 1e-6 on triangle, quad-strip, tetra.
+  - Cross-validation `G_id(0) = G_eu(0)` at 1e-10 (Glickenstein §5).
+
+### Hyper-ideal Hessian — block-FD (Phase 9b, 🟡 PR #9)
+* **Mathematical source:** per-face locality lemma:
+  `∂G_x/∂y = Σ_{f: x,y ∈ local(f)} ∂(β or α)/∂y at f`.
+  Same gradient as Phase 4a (Springborn 2020 §4).
+* **Java reference:** ❌ none (`hasHessian() == false`).
+* **Status:** 🟡 PR #9 open, 7 tests, measured 96× speed-up over Phase 4a.
+* **Why research:** the locality lemma + 6×6 block-scatter is a
+  conformallab++ algorithmic contribution; it makes Hessian-based Newton
+  viable on meshes that the upstream Java cannot solve in reasonable
+  time at all (since it has no Hessian).
+
+---
+
+## Planned research (not yet PR)
+
+### Hyper-ideal Hessian — full analytic (Phase 9b-analytic, 🔲 planned)
+
+* **Mathematical sources:**
+  - **Schläfli, L.** (1858/60). *On the multiple integral
+    ∫dx dy …*  Quart. J. Pure & Appl. Math.  → second-order Schläfli
+    identity: `2 dV = Σ_e aₑ dαₑ + Σ_v bᵥ dβᵥ` for any hyperbolic
+    polyhedron, with corresponding bilinear differential on second
+    derivatives.
+  - **Springborn, B.** (2020). *Ideal Hyperbolic Polyhedra and
+    Discrete Uniformization.* Discrete & Comput. Geom.  → §4 for the
+    hyper-ideal energy whose gradient is `(β − Θ, α − θ)`, hence
+    Hessian is the Schläfli bilinear form's restriction to the
+    constraint surface.
+  - **Cho, Y. & Kim, H.** (1999). *On the volume formula for
+    hyperbolic tetrahedra.*  Discr. Comput. Geom. 22, 347–366.
+    → explicit derivative formulas for `∂α/∂a`, `∂α/∂b`, `∂β/∂a`,
+    `∂β/∂b` at hyperbolic tetrahedra.
+  - **Glickenstein, D.** (2011) §4 — analogous derivation for the
+    cone-vertex case (extending the formulas across the ideal /
+    hyper-ideal vertex boundary).
+
+* **Java reference:** ❌ none.
+
+* **Chain of differentiation:**
+
+  ```
+  (bᵢ, aₑ)  →  ℓᵢⱼ           via lij()      (closed form: ζ₁₃, ζ₁₄, ζ₁₅)
+            →  βᵢ            via zeta()     (law of cosines)
+            →  αᵢⱼ           via alpha_ij() (zeta + sigma_i + sigma_ij)
+  ```
+
+  Each arrow is a smooth function in the interior of its domain.  The
+  chain rule then gives, for each face:
+
+  ```
+  ∂βᵢ/∂(bⱼ, aₑ)  =  Σ_k  (∂βᵢ/∂ℓₖ) · (∂ℓₖ/∂(bⱼ, aₑ))
+  ∂αᵢⱼ/∂(bₖ, aₑ) =  (similar, with β-dependence factored)
+  ```
+
+  These are then assembled into the local 6×6 block, scattered the
+  same way as block-FD (Phase 9b).
+
+* **Acceptance criteria:**
+  - Each of the four partial-derivative formulas (`∂α/∂a`, `∂α/∂b`,
+    `∂β/∂a`, `∂β/∂b`) cross-checked against block-FD at random `x` on
+    every supported vertex configuration:
+      - all hyper-ideal vertices (general case)
+      - one ideal vertex (`σᵢ`/`σⱼ`/`σₖ` ideal branches)
+      - two ideal vertices
+  - Schläfli identity `H · x = 0` for the constant-vector `x` that
+    corresponds to a global Möbius dilation must hold numerically
+    (gauge null space).
+  - PSD property preserved (Springborn 2020 §4.3).
+  - Measured speed-up over Phase 9b block-FD ≥ 3× (asymptotic ~6×).
+  - **Correctness proof:** a short LaTeX note in
+    `doc/math/hyperideal-hessian-derivation.tex` showing each
+    Schläfli + chain-rule step with edge-cases.
+
+* **Effort:** large (10–14 days net).  Significant share of the time
+  is the formal derivation note and the per-case symbolic verification.
+
+* **Phase:** 9b-analytic.
+
+* **Why deferred:** Phase 9b (block-FD) already removes the practical
+  Hessian bottleneck (96× speed-up measured).  Analytic gives only
+  another ~6× but at substantial implementation + verification cost.
+  Land on demand when profiling on a real V > 5000 application points
+  to it as the new bottleneck.
+
+---
+
+### Inversive-distance Hessian — full analytic (Phase 9a.2-analytic, 🔲 planned)
+
+* **Mathematical source:** Glickenstein, D. (2011) eq. (4.6).
+* **Java reference:** ❌ none.
+* **Chain:** `(uᵢ, uⱼ) → ℓᵢⱼ → αᵢⱼ` with `∂ℓ²/∂u_i = 2(r_i² + I r_i r_j)`.
+* **Effort:** medium (5–7 days, less involved than HyperIdeal because
+  the chain has fewer levels and no `σ` intermediaries).
+* **Status:** 🔲 planned, mirrors Phase 9b-analytic in spirit.
+
+---
+
+### Genus g ≥ 2 fundamental domain (Phase 9c, 🔲 planned)
+* **Mathematical sources:**
+  - **Poincaré, H.** (1882). *Théorie des groupes fuchsiens.*
+    Acta Math. 1, 1–62. → 4g-gon construction.
+  - **Sechelmann** (2016) §5 for the canonical-form algorithm.
+* **Java reference:** ✅ partial — `FundamentalPolygonUtility.java`
+  (698 lines) + `CanonicalFormUtility.java` (532 lines) exist; this is
+  a port-with-research-extensions (the C++ side will need to bridge to
+  the cut-graph + holonomy infrastructure already in conformallab++).
+* **Effort:** large (10–14 days).
+* **Status:** roadmap item, no PR yet.
+
+---
+
+### Discrete holomorphic and harmonic 1-forms (Phase 10a, 🔲 planned)
+* **Mathematical sources:**
+  - **Mercat, C.** (2001). *Discrete Riemann surfaces and the Ising
+    model.* Comm. Math. Phys. 218, 177–216. → discrete complex
+    structure on a quad mesh.
+  - **Bobenko, A. I. & Springborn, B.** (2004) §6 — discrete
+    harmonic and holomorphic 1-forms on triangulated surfaces.
+* **Java reference:** ✅ `DiscreteHarmonicFormUtility.java` (657 lines)
+  + `DiscreteHolomorphicFormUtility.java` (285 lines).  Port-with-
+  research:  the C++ port can choose between literal Java translation
+  and a redesign that uses `cut_graph.hpp` + `period_matrix.hpp`
+  natively (research opportunity).
+* **Effort:** very large (3+ weeks); see java-parity.md.
+
+---
+
+### Siegel period matrix Ω ∈ H_g (Phase 10b, 🔲 planned)
+* **Mathematical sources:**
+  - **Bobenko-Springborn (2004)** §6 for the discrete formula
+    `Ω_{ij} = ∫_{b_j} ω_i`.
+  - Siegel-fundamental-domain reduction algorithm (Gottschling 1959).
+* **Java reference:** ✅ partial — `DiscreteRiemannUtility.java`
+  (186 lines).
+* **Acceptance criteria:** `Ω` symmetric, `Im(Ω) > 0`, in the standard
+  fundamental domain of `Sp(2g, ℤ)`.
+* **Effort:** medium (1 week after 10a).
+
+---
+
+### Full uniformization for genus g ≥ 2 (Phase 10c, 🔲 planned)
+* **Mathematical source:** classical (Poincaré 1883; Bers 1960);
+  Sechelmann 2016 §6 for the discrete instance.
+* **Java reference:** ❌ none — Java has the polygon + period matrix
+  pieces but does not assemble them into a Fuchsian group representation.
+* **Status:** **fully new research** — depends on 9c + 10a + 10b.
+
+---
+
+### geometry-central cross-comparison track (Optional, 🔲 exploratory)
+Three independent items (GC-1/2/3) tracked separately in
+`doc/roadmap/phases.md` and analysed in detail in
+`doc/architecture/geometry-central-comparison.md`.  They are **purely
+exploratory**, not roadmap commitments.
+
+| ID | Item | Effort |
+|---|---|---|
+| GC-1 | Output-vector cross-validation against geometry-central | small (2 days) |
+| GC-2 | Intrinsic Delaunay pre-conditioning via Ptolemaic flips | medium (1 week) |
+| GC-3 | Ptolemaic flip-based solver as alternative backend | research (Phase 10+) |
+
+---
+
+## Java features still worth porting
+
+These are tracked separately in
+[`java-parity.md`](java-parity.md), summarised here only for cross-reference:
+
+| Java class | Lines | Suggested phase | Effort |
+|---|---|---|---|
+| `FundamentalPolygonUtility` + `CanonicalFormUtility` | 698 + 532 | 9c | 2 weeks |
+| `CuttingUtility` + `SurgeryUtility` | 584 + 217 | 9c foundation | 2 weeks |
+| `DiscreteHarmonicFormUtility` | 657 | 10a | 2 weeks |
+| `DiscreteHolomorphicFormUtility` | 285 | 10a | 2 weeks |
+| `CanonicalBasisUtility` | 337 | 10a prereq | 1 week |
+| `DualityUtility` + `HomologyUtility` | 308 + 122 | 10a support | 1 week |
+| `DiscreteRiemannUtility` | 186 | 10b | small |
+| `HyperbolicCyclicFunctional` | 530 | 10b–c | 2 weeks |
+| `QuasiisothermicUtility` + `SinConditionApplication` | ~1 200 | 10b | 3 weeks |
+| `KoebePolyhedron` | 321 | 10c | 2 weeks |
+| `MobiusCenteringFunctional`, `ElectrostaticSphereFunctional` | 289 + 127 | 10c (optional) | small |
+
+Total identified backlog: ~6 500 Java lines, estimated ~5 months of work
+to bring it all over.  None of it changes the **mathematical** scope —
+all 11 items above sit within Phases 9c, 10a, 10b, 10c.
+
+---
+
+## Maintenance rule
+
+If a future PR claims "ports X from Java", **first verify** by:
+
+```bash
+find /Users/tarikmoussa/Desktop/conformallab -iname "*X*"
+grep -r "ClassName" /Users/tarikmoussa/Desktop/conformallab/src
+```
+
+If either returns zero matches, the item is research and belongs in
+**this** document, not in `java-parity.md`.  Add it with the structured
+template above, including the primary literature reference and the
+acceptance criteria.