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/java-parity.md

@@ -15,10 +15,11 @@ as the reference implementation for expected behaviour, edge cases, and test cas
 | Euclidean functional — energy, gradient | ✅ | ✅ | |
 | Spherical functional — energy, gradient, gauge-fix | ✅ | ✅ | |
 | HyperIdeal functional — energy, gradient | ✅ | ✅ | |
-| Inversive-distance functional (Luo 2004) | ✅ | ❌ Phase 9a | `InversiveDistanceFunctional.java` |
+| Inversive-distance functional (Luo 2004) | ❌ *(not in Java)* | ❌ Phase 9a.2 | **No Java source.** Implemented in C++ from Luo 2004 + Glickenstein 2011 + Bowers-Stephenson 2004 — **new research, not a port**. Verified: `find /Users/tarikmoussa/Desktop/conformallab -iname "*nversive*"` returns zero. |
+| CP-Euclidean functional (BPS 2010) | ✅ | ❌ Phase 9a.1 | `CPEuclideanFunctional.java` (260 lines) — face-based circle packing |
 | Euclidean Hessian — cotangent Laplacian | ✅ analytic | ✅ analytic | Pinkall–Polthier (1993) |
 | Spherical Hessian — ∂α/∂u via law of cosines | ✅ analytic | ✅ analytic | |
-| HyperIdeal Hessian — ζ → lᵢⱼ → β/α chain | ✅ analytic | ⚠️ symmetric FD | Phase 9b |
+| HyperIdeal Hessian — analytic via ζ → l → β/α | ❌ *(`hasHessian()==false`)* | ⚠️ FD (Phase 4a) → block-FD (Phase 9b) | **Java has NO Hessian for HyperIdeal** (verified: `HyperIdealFunctional.java:295-298` declares `hasHessian() { return false; }`).  Both C++ Hessian variants are **new research beyond Java**; analytic Schläfli-based variant is Phase 9b-analytic. |
 | Newton solver | ✅ | ✅ | |
 | SparseQR fallback for gauge modes | unknown | ✅ | New in C++ |
 | Cone metrics — prescribed Θᵥ ≠ 2π | ✅ fully | ⚠️ data structure only | |
@@ -48,12 +49,22 @@ They are candidates for Phase 9 or Phase 10.
 
 | Java class | Description | Phase |
 |---|---|---|
-| `InversiveDistanceFunctional` | Inversive-distance conformal energy | 9a |
-| `DiscreteHarmonicFormUtility` | Discrete harmonic 1-forms | 10a prerequisite |
-| `DiscreteHolomorphicFormUtility` | Holomorphic differentials on discrete surfaces | 10a |
-| `DiscreteRiemannUtility` | Discrete Riemann surfaces | 10 |
-| `CanonicalBasisUtility` | Canonical homology basis for genus g | 9c / 10 |
-| `HomologyUtility` | Homology computation | 9c |
+| `CPEuclideanFunctional` | Face-based circle-packing energy (BPS 2010) | 9a.1 |
+| `FundamentalPolygonUtility` (698 lines) | Construction + canonicalisation of 4g-gons for genus-g | 9c |
+| `CanonicalFormUtility` (532 lines) | High-level wrapper for 9c — drives canonicalisation pipeline | 9c |
+| `CuttingUtility` + `SurgeryUtility` (~800 lines) | Mesh cuts and gluing operations needed for fundamental domains | 9c (foundation) |
+| `DiscreteHarmonicFormUtility` (657 lines) | Discrete harmonic 1-forms via cotangent Laplacian (Hodge theory) | 10a prerequisite |
+| `DiscreteHolomorphicFormUtility` (285 lines) | Holomorphic differentials via Mercat complex structure | 10a (Bobenko-Springborn 2004 §6) |
+| `CanonicalBasisUtility` (337 lines) | Symplectic homology basis with intersection-form normalisation | 10a prerequisite |
+| `DiscreteRiemannUtility` (186 lines) | Period matrix τ, Siegel reduction (genus g) | 10b |
+| `DualityUtility` (308 lines), `HomologyUtility` (122 lines) | Primal/dual cohomology, cycle generators | 10a support |
+| `HyperbolicCyclicFunctional` (530 lines) | Discrete hyperbolic conformal energy (analogue of Euclidean) — completes the geometry suite | 10b–c |
+| `QuasiisothermicUtility` + `SinConditionApplication` (~1200 lines) | Lawson-correspondence parametrisation, sin-condition functional | 10b |
+| `KoebePolyhedron` (321 lines) | Koebe–Andreev–Thurston circle-packing construction | 10c |
+| `ElectrostaticSphereFunctional`, `MobiusCenteringFunctional` | Sphere-domain pre-processing functionals | 10c (optional) |
+
+Note: items marked as *new research* (e.g. Inversive Distance, HyperIdeal Hessian variants)
+are tracked separately in `doc/roadmap/research-track.md`.
 | `HomotopyUtility` | Homotopy generators | 9c |
 | `SpanningTreeUtility` | Spanning tree algorithms | 8 / infrastructure |
 | `SurgeryUtility` | Mesh surgery (cut/glue) | — |
@@ -65,22 +76,30 @@ They are candidates for Phase 9 or Phase 10.
 
 ---
 
-## HyperIdeal Hessian: FD vs. analytic
+## HyperIdeal Hessian — correction of an earlier mis-claim
 
-The Java library computes the HyperIdeal Hessian analytically through the chain:
+> **2026-05-21 audit:** A previous version of this document claimed
+> "the Java library computes the HyperIdeal Hessian analytically through the
+> chain (bᵢ, aₑ) → lᵢⱼ → ζ₁₃/ζ₁₄/ζ₁₅ → αᵢⱼ/βᵢ".  **This is incorrect.**
+> The Java source file `HyperIdealFunctional.java` line 295-298 declares
+> ```java
+> @Override
+> public boolean hasHessian() {
+>     return false;
+> }
+> ```
+> i.e. the upstream Java implementation supplies **no** HyperIdeal Hessian at
+> all — neither analytic nor numerical.  The chain rule above is the
+> *mathematical formulation* (from Springborn 2020 §4 + Schläfli 1858), not
+> something the Java code implements.
 
-```
-(bᵢ, aₑ) → lᵢⱼ → ζ₁₃/ζ₁₄/ζ₁₅ → αᵢⱼ / βᵢ
-```
+### Actual state of HyperIdeal Hessian in conformallab++
 
-conformallab++ uses a **symmetric finite-difference approximation**:
+| Variant | Status | Notes |
+|---|---|---|
+| Phase 4a — full FD `H[i,j] = (G(x+εeⱼ)[i] − G(x−εeⱼ)[i]) / (2ε)` | ✅ implemented | O(n·F) cost; PSD by Springborn 2020 strict convexity |
+| Phase 9b — block-FD (per-face 6×6 local block, scatter to global) | ✅ implemented (PR #9) | O(F·36) cost; ~96× speed-up over Phase 4a measured on V=200 mesh |
+| Phase 9b-analytic — Schläfli identity + chain rule through ζ₁₃/ζ₁₄/ζ₁₅ | 🔲 planned (research) | See `doc/roadmap/research-track.md` for the formal plan and citations |
 
-```
-H[i,j] = ( G(x + ε·eⱼ)[i] − G(x − ε·eⱼ)[i] ) / (2ε),   ε = 1e-5
-```
-
-Accuracy: O(ε²) ≈ 10⁻¹⁰ relative error. PSD guaranteed by strict convexity (Springborn 2020).
-Cost: n extra gradient evaluations per Newton step.
-Impact: negligible for meshes < 500 DOFs; measurable for larger meshes.
-
-The analytic Hessian is deferred to Phase 9b.
+All three are **new research beyond the Java port**.  Java parity for
+HyperIdeal stops at the gradient.