Merge pull request 'docs: citation audit (8 mis-citations fixed) + Phases 12/13' (#28) from docs/citation-audit-phases-12-13 into main

Reviewed-on: #28

CLAUDE.md

@@ -156,7 +156,7 @@ Hessian sign and solver per model:
 - **Spherical:** H is NSD (concave energy) → `SimplicialLDLT(−H)` (sign flip inside `newton_spherical`).
 - **HyperIdeal:** H is PSD (strictly convex) → `SimplicialLDLT(H)`.  Phase 9b uses a **block-FD Hessian** (per-face 6×6 local block, ~96× speed-up vs full FD on V=200).  Full analytic Hessian via the chain `(bᵢ, aₑ) → lᵢⱼ → ζ₁₃/ζ₁₄/ζ₁₅ → αᵢⱼ/βᵢ` is planned research — see `doc/roadmap/research-track.md` Phase 9b-analytic.
 - **CP-Euclidean:** analytic 2×2-per-edge `h_jk = sin θ / (cosh Δρ − cos θ)` (BPS 2010), strictly convex → `SimplicialLDLT(H)`.
-- **Inversive-Distance:** FD Hessian (inline in `newton_inversive_distance`).  Analytic via Glickenstein 2011 eq. (4.6) is planned research (Phase 9a.2-analytic).
+- **Inversive-Distance:** FD Hessian (inline in `newton_inversive_distance`).  Analytic via Glickenstein 2011 §5.2 is planned research (Phase 9a.2-analytic).
 
 When `SimplicialLDLT` fails (rank-deficient H — gauge mode on a closed mesh without pinned vertex/face), the solver automatically retries with `Eigen::SparseQR` to find the minimum-norm step orthogonal to the null space.  Public API: `solve_linear_system(H, rhs, &used_fallback)`.
 

doc/architecture/phase-9a-validation.md

@@ -124,7 +124,7 @@ energy-evaluation cost and the FD-gradient validation pattern.
 
 ### 2.5  Hessian — finite difference for now
 
-Glickenstein 2011 eq. (4.6) gives an analytic Hessian for the inversive-
+Glickenstein 2011 §5.2 gives an analytic Hessian for the inversive-
 distance variational principle, but is more involved than the Springborn
 cot-Laplacian.  For MVP we rely on FD; the analytic form is a future
 optimisation (joining the Phase 9b roadmap for HyperIdeal as a sibling
@@ -173,7 +173,7 @@ same Newton-time-zero gradient".
 | Angles                     | half-tangent law of cosines                   | (face-internal angles via `p(θ*, Δρ)`)       | half-tangent law of cosines (reuses `euclidean_angles`)            |
 | Gradient                   | `Θ_v − Σ α_v(f)`                              | `φ_f − Σ_{h:face(h)=f} (p+θ*)`               | `Θ_v − Σ α_v(f)`                                                   |
 | Energy form                | path integral (10-pt GL)                      | closed form via `½ p Δρ + Λ(θ*+p) − θ* ρ`   | path integral (10-pt GL)                                           |
-| Hessian                    | analytic (cotangent Laplacian)                | analytic (`sin θ / (cosh Δρ − cos θ)`)       | finite difference (analytic deferred — Glickenstein 2011 eq. 4.6)  |
+| Hessian                    | analytic (cotangent Laplacian)                | analytic (`sin θ / (cosh Δρ − cos θ)`)       | finite difference (analytic deferred — Glickenstein 2011 §5.2)     |
 | Convexity                  | strictly convex after gauge fix               | strictly convex after gauge fix              | locally convex on triangle-inequality domain (Luo 2004 Thm 1.2)    |
 | Gauge fix                  | pin one vertex (`v_idx = −1`)                 | pin one face   (`f_idx = −1`)                | pin one vertex (`v_idx = −1`)                                      |
 
@@ -213,8 +213,9 @@ form.  It does share the Clausen function via `clausen2()` from `clausen.hpp`.
 
 ## 6.  References
 
-- Bobenko, A. I., Pinkall, U. & Springborn, B. (2010). *Discrete conformal
-  maps and ideal hyperbolic polyhedra.* Geometry & Topology 14, 379–426.
+- Bobenko, A. I., Pinkall, U. & Springborn, B. *Discrete conformal
+  maps and ideal hyperbolic polyhedra.* Geometry & Topology 19(4) (2015),
+  2155–2215. arXiv:1005.2698.
 - Bowers, P. L. & Stephenson, K. (2004). *Uniformizing dessins and Belyĭ
   maps via circle packing.* Memoirs of the AMS 170(805).
 - Glickenstein, D. (2011). *Discrete conformal variations and scalar

doc/math/novelty-statement.md

@@ -87,7 +87,9 @@ finite-difference approximations, which is required for reproducible research.
 
 The discrete period matrix τ_discrete is a computable invariant of the triangulated
 surface.  Its convergence to the smooth Riemannian τ_smooth under mesh refinement
-is an open research question that this library is designed to investigate.
+is an open research question in general that this library is designed to investigate
+— though it has already been proven for the special class of ramified coverings of
+the Riemann sphere by Bobenko–Bücking (2021).
 
 ### 3.4 — Full test coverage of analytic invariants
 

doc/math/references.md

@@ -14,27 +14,39 @@ Java reference implementation: [github.com/varylab/conformallab](https://github.
 
 ## References by module
 
+> **Status-Konvention.** Die „Used in"-Spalte nennt das Modul *oder* die Phase.
+> Ein Verweis auf eine **ausgelieferte** Phase (Code existiert, getestet) ist mit
+> ✅ markiert; ein Verweis auf eine **geplante/Forschungs**-Phase mit 🔜. Nur die
+> ✅-Quellen sind Grundlage des aktuellen Codes; 🔜-Quellen belegen Roadmap-Ziele
+> (vgl. auch Abschnitt „Phase 10 references (future research)" unten und
+> `novelty-statement.md` §6 „What conformallab++ is not").
+>
+> | Marker | Bedeutung | Phasen |
+> |---|---|---|
+> | ✅ | implementiert & getestet | 9a.1, 9a.2, 9b-analytic, Cut-Graph, Hessians |
+> | 🔜 | geplant / Forschung | 9d.2, 9f, 10a, 10b, 10c |
+
 | Reference | Used in |
 |---|---|
-| **Springborn** — *Ideal Hyperbolic Polyhedra and Discrete Uniformization*, Discrete & Computational Geometry (2020) | `hyper_ideal_geometry.hpp` — ζ₁₃/ζ₁₄/ζ₁₅ functions; `hyper_ideal_functional.hpp` |
+| ✅ **Springborn** — *Ideal Hyperbolic Polyhedra and Discrete Uniformization*, Discrete & Computational Geometry **64** (2020), pp. 63–108. DOI: [10.1007/s00454-019-00132-8](https://doi.org/10.1007/s00454-019-00132-8) | `hyper_ideal_geometry.hpp` — ζ₁₃/ζ₁₄/ζ₁₅ functions; `hyper_ideal_functional.hpp` |
 | **Pinkall, Polthier** — *Computing Discrete Minimal Surfaces and Their Conjugates*, Experimental Mathematics (1993) | `euclidean_hessian.hpp` — cotangent Laplacian |
 | **Bobenko, Springborn** — *Variational Principles for Circle Patterns and Koebe's Theorem*, Transactions AMS (2004) | Variational angle-sum framework underlying all three functionals |
 | **Luo** — *Combinatorial Yamabe Flow on Surfaces*, Communications in Contemporary Mathematics (2004) | Inversive-distance functional — **new research** in Phase 9a.2 (no Java original; implemented from this paper + Glickenstein 2011 + Bowers-Stephenson 2004) |
-| **Bowers, Stephenson** — *Uniformizing dessins and Belyĭ maps via circle packing*, Memoirs of the AMS 170(805) (2004) | Bowers-Stephenson identity I_ij = (ℓ²−r_i²−r_j²)/(2 r_i r_j) used to initialise inversive distance from input geometry (Phase 9a.2) |
-| **Glickenstein** — *Discrete conformal variations and scalar curvature on piecewise flat manifolds*, J. Differential Geometry 87 (2011) | Analytic Hessian of the inversive-distance functional (eq. 4.6) and cross-correspondence I_ij = cos θ_e between vertex-based (9a.2) and face-based (9a.1) circle packings |
-| **Bobenko, Pinkall, Springborn** — *Discrete conformal maps and ideal hyperbolic polyhedra*, Geometry & Topology 14 (2010) | Face-based circle-packing functional (`CPEuclideanFunctional.java` → `cp_euclidean_functional.hpp`, Phase 9a.1) |
-| **Schläfli** — *On the multiple integral ∫dx dy …*, Quarterly Journal of Pure and Applied Mathematics (1858/60) | Volume differential `2 dV = Σ_e aₑ dαₑ + Σ_v bᵥ dβᵥ` — foundation for the analytic HyperIdeal Hessian via Schläfli identity (Phase 9b-analytic, **new research beyond Java**) |
+| **Bowers, Stephenson** — *Uniformizing dessins and Belyĭ maps via circle packing*, Memoirs of the AMS 170(805) (2004) | Introduces **inversive-distance circle packings** (used in Phase 9a.2). *Hinweis:* die zur Initialisierung benutzte Formel I_ij = (ℓ²−r_i²−r_j²)/(2 r_i r_j) ist die **klassische** inversive Distanz (vgl. Glickenstein §5.2: ℓ²=r_i²+r_j²+2r_ir_jη), nicht eine eigene „Bowers-Stephenson-Identität" — B–S liefern die Packungstheorie, nicht diese Formel. |
+| **Glickenstein** — *Discrete conformal variations and scalar curvature on piecewise flat two- and three-dimensional manifolds*, J. Differential Geometry **87**(2) (2011), pp. 201–238 | Analytic Hessian of the inversive-distance functional. ⚠️ *Korrektur:* die Arbeit nummeriert Gleichungen **nicht** im Format „(4.6)" — der Verweis ist durch die **§5.2**-Parametrisierung ℓ²_ij = r²_i + r²_j + 2 r_i r_j η_ij zu ersetzen. Cross-correspondence: η_ij ist die inversive Distanz und entspricht dem Kosinus des **Supplements** des Schnittwinkels (Schnitt bei arccos(−η_ij)) — also I_ij = cos θ_e **nur bis aufs Vorzeichen/Supplement**, nicht wörtlich. |
+| **Bobenko, Pinkall, Springborn** — *Discrete conformal maps and ideal hyperbolic polyhedra*, Geometry & Topology **19**(4) (2015), pp. 2155–2215. arXiv: [1005.2698](https://arxiv.org/abs/1005.2698) | Face-based circle-packing functional (`CPEuclideanFunctional.java` → `cp_euclidean_functional.hpp`, Phase 9a.1) |
+| **Schläfli** — *On the multiple integral ∫dx dy …*, Quarterly Journal of Pure and Applied Mathematics (1858/60) | Klassische Schläfli-Differentialformel (dV = −½ Σ_e ℓ_e dθ_e). ⚠️ *Hinweis:* die in Phase 9b-analytic benutzte **Randterm-Form** `2 dV = Σ_e aₑ dαₑ + Σ_v bᵥ dβᵥ` steht **nicht** bei Schläfli 1858, sondern ist die verallgemeinerte Fassung für Mannigfaltigkeiten mit Rand → korrekter Beleg: **Rivin–Schlenker 1999** (Phase-10-Liste). Schläfli 1858 nur als historischer Ursprung zitieren. |
 | **Erickson, Whittlesey** — *Greedy Optimal Homotopy and Homology Generators*, SODA (2005) | `cut_graph.hpp` — tree-cotree algorithm |
 | **Bobenko, Springborn** — *A Discrete Laplace–Beltrami Operator for Simplicial Surfaces*, Discrete & Computational Geometry (2007) | Background for cotangent weights |
 | **Desbrun, Kanso, Tong** — *Discrete Differential Forms for Computational Modeling*, SIGGRAPH Course Notes (2006) | Discrete exterior calculus background for Phase 10a |
-| **Crane, Soliman, Ben-Chen, Schröder** — *Optimal Cone Singularities for Conformal Flattening*, ACM SIGGRAPH (2018). DOI: [10.1145/3197517.3201367](https://doi.org/10.1145/3197517.3201367) | L¹-optimal automatic cone placement — **Phase 9d.2** (non-Euclidean cone extensions). Provides the optimisation algorithm for choosing cone positions automatically; complements Bobenko-Lutz 2025 on non-Euclidean settings. |
+| **Soliman, Slepčev, Crane** — *Optimal Cone Singularities for Conformal Flattening*, ACM Transactions on Graphics **37**(4), Article 105 (2018). DOI: [10.1145/3197517.3201367](https://doi.org/10.1145/3197517.3201367) | L¹-optimal automatic cone placement — **Phase 9d.2** (non-Euclidean cone extensions). Provides the optimisation algorithm for choosing cone positions automatically; complements Bobenko-Lutz 2025 on non-Euclidean settings. |
 | **Bobenko, Lutz** — *Decorated Discrete Conformal Equivalence in Non-Euclidean Geometries*, Discrete & Computational Geometry (2025). arXiv: [2310.17529](https://arxiv.org/abs/2310.17529) | **Phase 9d.2**: extends DCE to hyperbolic + spherical geometry with Penner-coordinate decorations; unifies cone singularities and hyperideal cusps in one algebraic framework. |
 | **Bobenko, Lutz** — *Decorated Discrete Conformal Maps and Convex Polyhedral Cusps*, IMRN 2024(12), pp. 9505–9534. arXiv: [2305.10988](https://arxiv.org/abs/2305.10988) | **Phase 10b/10c**: discrete uniformization theorem for decorated piecewise Euclidean surfaces; connects Phase 2/3 hyperideal vertices (cusps at ∞) to the period matrix and fundamental domain. |
 | **Lutz** — *Canonical Tessellations of Decorated Hyperbolic Surfaces*, Geometriae Dedicata 217 (2023). arXiv: [2206.13461](https://arxiv.org/abs/2206.13461) | **Phase 10c**: canonical Delaunay tessellations in Penner coordinates; unifies the decorated framework with the fundamental domain construction for genus g ≥ 2. |
 | **Lutz** — *Decorated Discrete Conformal Equivalence, Canonical Tessellations, and Polyhedral Realization* (PhD thesis, TU Berlin, 2024). DOI: [10.14279/depositonce-20357](https://doi.org/10.14279/depositonce-20357) | Comprehensive single reference for Phases 9d.2, 10b, 10c — collects Bobenko-Lutz 2024/2025 and Lutz 2023 with complete proofs. |
-| **Bowers, Bowers, Lutz** — *Rigidity of circle polyhedra and hyperideal polyhedra: the tangency case* (2026). arXiv: [2601.22903](https://arxiv.org/abs/2601.22903) | **Phase 9b-analytic + Phase 10c'** (KoebePolyhedron): theoretical uniqueness/rigidity for hyperideal polyhedra in the tangency case; supports correctness of the analytic Hessian and the KAT construction. |
+| **Bowers, Bowers, Lutz** — *Rigidity of Koebe Polyhedra and Inversive Distance Circle Packings* (2026). arXiv: [2601.22903](https://arxiv.org/abs/2601.22903) | **Phase 9b-analytic + Phase 10c'** (KoebePolyhedron): theoretical uniqueness/rigidity for Koebe polyhedra and inversive-distance circle packings (incl. the tangency case); supports correctness of the analytic Hessian and the KAT construction. |
 | **Alexa, Wardetzky** — *Discrete Laplacians on General Polygonal Meshes*, ACM SIGGRAPH (2011). DOI: [10.1145/1964921.1964997](https://doi.org/10.1145/1964921.1964997) | **Phase 9f**: virtual-node polygon Laplacian extending Pinkall-Polthier to non-triangular meshes. Enables DCE on quad/Voronoi meshes without triangulation. |
-| **Alexa** — *Discrete Laplacians on General Polygonal Meshes*, ACM TOG 39(6) (2020). DOI: [10.1145/3414685.3417840](https://doi.org/10.1145/3414685.3417840) | **Phase 9f** (extended journal version): error bounds, generalised polygon cotangent weights, convergence analysis. |
+| **Bunge, Herholz, Kazhdan, Botsch** — *Polygon Laplacian Made Simple*, Computer Graphics Forum **39**(2) (2020), pp. 303–313. DOI: [10.1111/cgf.13931](https://doi.org/10.1111/cgf.13931) | **Phase 9f**: virtual-vertex polygon Laplacian — fügt pro Polygon einen virtuellen Knoten ein (impliziter Triangle-Fan), erweitert die cotangent-Diskretisierung auf nicht-konvexe/nicht-planare Polygone. (Alternative DEC-Variante: **de Goes, Butts, Desbrun**, *Discrete Differential Operators on Polygonal Meshes*, ACM TOG **39**(4) (2020), DOI [10.1145/3386569.3392389](https://doi.org/10.1145/3386569.3392389).) |
 
 ---
 
@@ -66,9 +78,9 @@ builds on this paper and augments it with Ptolemaic flips.
 | **Farkas, Kra** — *Riemann Surfaces*, Springer GTM 71 | Siegel period matrix, Teichmüller theory |
 | **Siegel** — *Topics in Complex Function Theory, Vol. 2*, Wiley | Siegel upper half-space H_g, Sp(2g,ℤ) reduction |
 | **Bobenko, Mercat, Schmies** — *Period Matrices of Polyhedral Surfaces*, in: Computational Approach to Riemann Surfaces (2011) | Discrete period matrices on polyhedral surfaces |
-| **Bobenko, Bücking** — *Conformal Structures and Period Matrices of Polyhedral Surfaces* (2009) | Phase 10b: explicit algorithm for computing the discrete Siegel period matrix Ωᵢⱼ on a polyhedral surface from cotangent-weighted integration. |
-| **Rivin, Springborn** — *The Schläfli formula in Einstein manifolds with boundary*, Electron. Res. Announc. AMS 5 (1999) | Phase 9b-analytic: modern form of the Schläfli identity `2 dV = Σ aₑ dαₑ` for manifolds with boundary — the bilinear form used to derive the analytic HyperIdeal Hessian. |
-| **Springborn** — *A discrete version of Liouville's theorem on conformal maps* (2019). arXiv: [1911.00966](https://arxiv.org/abs/1911.00966) | Phase 10b uniqueness: proves that the discrete conformal structure (and hence Ω) is a conformal invariant — the discrete Liouville theorem. Justifies that conformallab++ outputs a canonical representative. |
-| **Springborn, Veselov** — *Quasiconformal distortion of projective transformations and discrete conformal maps*, Int. Math. Res. Not. (2015) | Phase 10c error analysis: quantifies how well the discrete H²/Γ embedding approximates the smooth hyperbolic metric; error bounds for the Fuchsian group representation. |
+| **Bobenko, Bücking** — *Convergence of discrete period matrices and discrete holomorphic integrals for ramified coverings of the Riemann sphere*, Math. Phys. Anal. Geom. **24**, Art. 23 (2021). DOI: [10.1007/s11040-021-09394-2](https://doi.org/10.1007/s11040-021-09394-2) | Phase 10b: discrete Siegel period matrix Ωᵢⱼ from cotangent-weighted integration **plus** the convergence result Ω_discrete → Ω_smooth under refinement (für ramified coverings) — belegt die Diskret-zu-glatt-Aussage in `novelty-statement.md §3.3. |
+| **Rivin, Schlenker** — *The Schläfli formula in Einstein manifolds with boundary*, Electron. Res. Announc. AMS **5** (1999), pp. 18–23 | Phase 9b-analytic: modern form of the Schläfli identity `2 dV = Σ aₑ dαₑ` for manifolds with boundary — the bilinear form used to derive the analytic HyperIdeal Hessian. |
+| **Pinkall, Springborn** — *A discrete version of Liouville's theorem on conformal maps*, Geometriae Dedicata **214** (2021), pp. 389–398. arXiv: [1911.00966](https://arxiv.org/abs/1911.00966) | Phase 10b uniqueness: proves that the discrete conformal structure (and hence Ω) is a conformal invariant — the discrete Liouville theorem. Justifies that conformallab++ outputs a canonical representative. |
+| **Born, Bücking, Springborn** — *Quasiconformal distortion of projective transformations and discrete conformal maps*, arXiv: [1505.01341](https://arxiv.org/abs/1505.01341) (2015) | Phase 10c error analysis: quantifies how well the discrete H²/Γ embedding approximates the smooth hyperbolic metric; error bounds for the Fuchsian group representation. |
 | **Knöppel, Crane, Pinkall, Schröder** — *Stripe Patterns on Surfaces*, ACM SIGGRAPH (2015). DOI: [10.1145/2766890](https://doi.org/10.1145/2766890) | Phase 10a cross-validation: applies discrete holomorphic 1-forms to direction field design; geometry-central provides an independent C++ implementation to cross-check the Phase 10a `DiscreteHolomorphicFormUtility` port. |
-| **Sawhney, Crane** — *Boundary First Flattening*, ACM TOG 36(1) (2017). DOI: [10.1145/3132705](https://doi.org/10.1145/3132705) | Complementary method to Phase 9d: boundary-prescribed conformal flattening — user specifies boundary shape, interior conforms freely. Contrast: conformallab++ prescribes cone angles in the interior; BFF prescribes the boundary. Alternative approach for applications needing controlled boundary. |
+| **Sawhney, Crane** — *Boundary First Flattening*, ACM TOG **37**(1), Article 5 (2017). DOI: [10.1145/3132705](https://doi.org/10.1145/3132705) | Complementary method to Phase 9d: boundary-prescribed conformal flattening — user specifies boundary shape, interior conforms freely. Contrast: conformallab++ prescribes cone angles in the interior; BFF prescribes the boundary. Alternative approach for applications needing controlled boundary. |

doc/reviewer/briefing.md

@@ -46,12 +46,12 @@ research-only phases the reader can shape at design stage.
 | **Canonical Delaunay tessellations of decorated hyperbolic surfaces** (Lutz 2023, *Geom. Dedicata*; Lutz 2024 PhD thesis) | cut-graph + period matrix + hyperbolic-disk layout as scaffolding; canonical-tessellation algorithm itself outlined | **10c** planned |
 | **Hyperideal polyhedra rigidity** (Bowers–Bowers–Lutz 2026) | HyperIdeal functional + analytic Hessian derivation (805-line LaTeX note) | **9b-analytic** derived; **10c′** KAT planned |
 | **Optimal cone placement / non-Euclidean cone metrics** (Crane et al. 2018) | Cone-singularity port via `ConesUtility` scoped; the *non-Euclidean* extension is the research delta | **9d.1** port + **9d.2** RESEARCH |
-| **Polygon Laplacian on general meshes** (Alexa–Wardetzky 2011; Alexa 2020) | *no Java parent* — first phase a reviewer can shape at design stage | **9f** RESEARCH (planned) |
+| **Polygon Laplacian on general meshes** (Alexa–Wardetzky 2011; Bunge et al. 2020) | *no Java parent* — first phase a reviewer can shape at design stage | **9f** RESEARCH (planned) |
 | **Quasi-isothermic maps** (generalising conformality where exact conformality is impossible — Lawson-correspondence parameterisation) | scoped as a 6-class port (~800 lines) from the Java original: `QuasiisothermicLayout`, `DBFSolution` (discrete Beltrami field), `SinConditionApplication`, `QuasiisothermicDelaunay`, `QuasiisothermicUtility`, `ConformalStructureUtility` | **10e** planned |
-| **Higher-genus + hyperelliptic surfaces** (Bobenko–Bücking 2009 on polyhedral surfaces; period matrices with block-diagonal Z₂ structure) | port of `HyperellipticUtility` + `HyperIdealHyperellipticUtility` scoped; existing period-matrix code as scaffolding | **10b** planned |
+| **Higher-genus + hyperelliptic surfaces** (Bobenko–Mercat–Schmies 2011 / Bobenko–Bücking 2021 on polyhedral period matrices; block-diagonal Z₂ structure) | port of `HyperellipticUtility` + `HyperIdealHyperellipticUtility` scoped; existing period-matrix code as scaffolding | **10b** planned |
 | **Möbius centring for Poincaré-disk layouts** as a variational problem (Lorentz geometry: `E = Σ log(−⟨x,p⟩/√(−⟨x,x⟩))`) | currently we use iterative Fréchet mean in `normalise_hyperbolic()`; the principled variational alternative is scoped via the Java `MobiusCenteringFunctional` port (full gradient + Hessian) | **9d.4** planned |
-| **Boundary-First / interactive flattening** (Crane et al. 2017 BFF; Bonneel et al. 2015 *Stripe Patterns on Surfaces*) | not on the roadmap as ports; documented in [`references.md`](../math/references.md) as comparison points / inspiration for future API design | — |
-| **Schläfli-based variational machinery** (Rivin–Springborn 1999) | derivation done, implementation gated on your view of whether the ~6× speedup over our block-FD path matters at your mesh sizes | **9b-analytic** ready to implement |
+| **Boundary-First / interactive flattening** (Sawhney–Crane 2017 BFF; Knöppel–Crane–Pinkall–Schröder 2015 *Stripe Patterns on Surfaces*) | not on the roadmap as ports; documented in [`references.md`](../math/references.md) as comparison points / inspiration for future API design | — |
+| **Schläfli-based variational machinery** (Rivin–Schlenker 1999) | derivation done, implementation gated on your view of whether the ~6× speedup over our block-FD path matters at your mesh sizes | **9b-analytic** ready to implement |
 
 See [`doc/roadmap/phases.md`](../roadmap/phases.md) for the per-phase
 porting plan, [`doc/roadmap/research-track.md`](../roadmap/research-track.md)
@@ -71,11 +71,12 @@ rationale).
   Tier-1/2.  Decorated-DCE / canonical-tessellation / hyperideal line:
   Bobenko–Lutz 2024 IMRN; Bobenko–Lutz 2025 *DCG*; Lutz 2023
   *Geom. Dedicata*; Lutz 2024 PhD; Bowers–Bowers–Lutz 2026.  Cones,
-  polyhedra, period matrices: Crane et al. 2018; Springborn 2019
-  (hyperbolic polyhedra); Bobenko–Bücking 2009; Rivin–Springborn 1999.
-  Polygon Laplacians: Alexa–Wardetzky 2011; Alexa 2020.  Integrable +
-  practical-flattening context: Springborn–Veselov 2015 (cluster
-  dynamics); Crane et al. 2017 (BFF); Bonneel et al. 2015 (Stripe
+  polyhedra, period matrices: Soliman–Slepčev–Crane 2018; Pinkall–Springborn
+  2021 (discrete Liouville); Bobenko–Mercat–Schmies 2011 / Bobenko–Bücking
+  2021; Rivin–Schlenker 1999.
+  Polygon Laplacians: Alexa–Wardetzky 2011; Bunge et al. 2020.  Integrable +
+  practical-flattening context: Born–Bücking–Springborn 2015 (quasiconformal
+  distortion); Sawhney–Crane 2017 (BFF); Knöppel et al. 2015 (Stripe
   Patterns).
 - **Phase 9f** (polygon Laplacian on non-triangular meshes) added as
   RESEARCH-only — no Java parent — so you can influence its design

doc/reviewer/hub.html

@@ -250,16 +250,16 @@ per-phase entry in <code>doc/roadmap/</code>.</p>
 <tr><td>Quasi-isothermic maps (Lawson correspondence, ~800 lines, discrete Beltrami-field solver)</td>
     <td>scoped as a 6-class Java port: <code>QuasiisothermicLayout</code>, <code>DBFSolution</code>, <code>SinConditionApplication</code>, <code>QuasiisothermicDelaunay</code>, <code>QuasiisothermicUtility</code>, <code>ConformalStructureUtility</code></td>
     <td><span class="pill new">10e</span> planned</td></tr>
-<tr><td>Higher-genus + hyperelliptic surfaces (Bobenko–Bücking 2009; block-diagonal period matrices with Z₂ symmetry)</td>
+<tr><td>Higher-genus + hyperelliptic surfaces (Bobenko–Mercat–Schmies 2011 / Bobenko–Bücking 2021 period matrices; block-diagonal Z₂ symmetry)</td>
     <td>port of <code>HyperellipticUtility</code> + <code>HyperIdealHyperellipticUtility</code> scoped; existing period-matrix code as scaffolding</td>
     <td><span class="pill new">10b</span> planned</td></tr>
 <tr><td>Möbius centring as a variational problem (Lorentz energy, full gradient + Hessian)</td>
     <td>currently iterative Fréchet mean in <code>normalise_hyperbolic()</code>; the principled variational alternative is scoped via the Java <code>MobiusCenteringFunctional</code> port</td>
     <td><span class="pill new">9d.4</span> planned</td></tr>
-<tr><td>Boundary-First / interactive flattening (Crane et al. 2017 BFF; Bonneel et al. 2015 <i>Stripe Patterns</i>)</td>
+<tr><td>Boundary-First / interactive flattening (Sawhney–Crane 2017 BFF; Knöppel–Crane–Pinkall–Schröder 2015 <i>Stripe Patterns</i>)</td>
     <td>not on the roadmap as ports; documented in <code>references.md</code> as comparison points / inspiration</td>
     <td>—</td></tr>
-<tr><td>Schläfli-based variational machinery (Rivin–Springborn 1999)</td>
+<tr><td>Schläfli-based variational machinery (Rivin–Schlenker 1999)</td>
     <td>derivation done; implementation gated on the reader's view of whether the ~6× speedup over our block-FD path matters at their mesh sizes</td>
     <td><span class="pill new">9b-analytic</span> ready</td></tr>
 </tbody>
@@ -320,15 +320,16 @@ sufficient answer.</p>
       Decorated-DCE / canonical-tessellation / hyperideal line:
       Bobenko–Lutz 2024 IMRN; Bobenko–Lutz 2025 <i>DCG</i>; Lutz 2023
       <i>Geom. Dedicata</i>; Lutz 2024 PhD; Bowers–Bowers–Lutz 2026.
-      Cones, polyhedra, period matrices: Crane et al. 2018;
-      Springborn 2019; Bobenko–Bücking 2009; Rivin–Springborn 1999.
-      Polygon Laplacians: Alexa–Wardetzky 2011; Alexa 2020.  Integrable
-      and practical-flattening context: Springborn–Veselov 2015;
-      Crane et al. 2017 (BFF); Bonneel et al. 2015 (Stripe Patterns).
+      Cones, polyhedra, period matrices: Soliman–Slepčev–Crane 2018;
+      Pinkall–Springborn 2021 (discrete Liouville); Bobenko–Mercat–Schmies
+      2011 / Bobenko–Bücking 2021; Rivin–Schlenker 1999.
+      Polygon Laplacians: Alexa–Wardetzky 2011; Bunge et al. 2020.  Integrable
+      and practical-flattening context: Born–Bücking–Springborn 2015;
+      Sawhney–Crane 2017 (BFF); Knöppel et al. 2015 (Stripe Patterns).
   </li>
   <li><span class="pill research">+1 RESEARCH phase</span> with no Java
       parent — Phase 9f (polygon Laplacian on non-triangular meshes,
-      Alexa-Wardetzky 2011 / Alexa 2020) — the first phase a reviewer
+      Alexa-Wardetzky 2011 / Bunge et al. 2020) — the first phase a reviewer
       can shape at design stage.</li>
   <li><span class="pill new">output_uv_map</span> now covers 4 of 5 DCE
       solvers (Inversive-Distance added; CP-Euclidean deferred to
@@ -427,9 +428,9 @@ folder for the per-document table.</p>
 <table>
 <tr><th>Document</th><th>What it covers</th></tr>
 <tr><td><a href="https://codeberg.org/TMoussa/ConformalLabpp/src/branch/main/doc/math/references.md"><code>references.md</code></a> <span class="pill new">+13 refs</span></td>
-    <td>Per-phase literature index.  13 new citations added: decorated DCE in non-Euclidean geometries (Bobenko–Lutz 2025); canonical tessellations (Lutz 2023 / 2024); hyperideal polyhedra rigidity (Bowers–Bowers–Lutz 2026); polygon Laplacians (Alexa–Wardetzky 2011 / Alexa 2020); optimal cone placement (Crane et al. 2018); Schläfli identity (Rivin–Springborn 1999); hyperbolic polyhedra (Springborn 2019); polyhedral period matrices (Bobenko–Bücking 2009); integrable cluster dynamics (Springborn–Veselov 2015); BFF (Crane et al. 2017); stripe patterns (Bonneel et al. 2015).</td></tr>
+    <td>Per-phase literature index.  13 new citations added: decorated DCE in non-Euclidean geometries (Bobenko–Lutz 2025); canonical tessellations (Lutz 2023 / 2024); hyperideal polyhedra rigidity (Bowers–Bowers–Lutz 2026); polygon Laplacians (Alexa–Wardetzky 2011 / Bunge et al. 2020); optimal cone placement (Soliman–Slepčev–Crane 2018); Schläfli identity (Rivin–Schlenker 1999); discrete Liouville theorem (Pinkall–Springborn 2021); polyhedral period matrices (Bobenko–Mercat–Schmies 2011 / Bobenko–Bücking 2021); quasiconformal distortion (Born–Bücking–Springborn 2015); BFF (Sawhney–Crane 2017); stripe patterns (Knöppel–Crane–Pinkall–Schröder 2015).</td></tr>
 <tr><td><a href="https://codeberg.org/TMoussa/ConformalLabpp/src/branch/main/doc/math/hyperideal-hessian-derivation.md"><code>hyperideal-hessian-derivation.md</code></a></td>
-    <td>Full LaTeX-formatted derivation of the analytic HyperIdeal Hessian via the Schläfli identity (805 lines, 8 sections + 2 appendices).  Cited sources: Schläfli 1858, Milnor 1982, Vinberg 1993, Cho–Kim 1999, Glickenstein 2011, Rivin–Springborn 1999, Springborn 2020.</td></tr>
+    <td>Full LaTeX-formatted derivation of the analytic HyperIdeal Hessian via the Schläfli identity (805 lines, 8 sections + 2 appendices).  Cited sources: Schläfli 1858, Milnor 1982, Vinberg 1993, Cho–Kim 1999, Glickenstein 2011, Rivin–Schlenker 1999, Springborn 2020.</td></tr>
 </table>
 
 </details>

doc/reviewer/questions.md

@@ -35,7 +35,7 @@ phases whose existence is settled but whose priority is open:
 | **9f** | RESEARCH | Discrete Laplace–Beltrami on **non-triangular** polygonal meshes (virtual-node / generalised cotangent), making DCE work on quad / Voronoi tessellations without re-triangulation | Polygon Laplacians (2011 *SIGGRAPH* + 2020 *TOG*) |
 | **10c + 10c′** | planned | Canonical Delaunay tessellations of decorated hyperbolic surfaces; Koebe polyhedron realisation (KAT) with rigidity-aware Newton | Canonical tessellations of decorated hyperbolic surfaces (2023, *Geom. Dedicata*); rigidity of circle / hyperideal polyhedra (2026, preprint) |
 | **10e** | planned | Quasi-isothermic maps (~800 lines Lawson-correspondence) — generalises conformality to meshes where exact conformality is impossible.  Six new classes including a discrete Beltrami-field solver. | Java original `QuasiisothermicUtility` line; no obvious single-paper reference in the existing literature index |
-| **10b** | planned | Hyperelliptic surfaces (genus g ≥ 2 with Z₂ symmetry); period matrices with block-diagonal structure; Penner-coordinate variant via `HyperIdealHyperellipticUtility` | Bobenko–Bücking 2009 *Conformal Structures and Period Matrices of Polyhedral Surfaces* |
+| **10b** | planned | Hyperelliptic surfaces (genus g ≥ 2 with Z₂ symmetry); period matrices with block-diagonal structure; Penner-coordinate variant via `HyperIdealHyperellipticUtility` | Bobenko–Mercat–Schmies 2011 *Period Matrices of Polyhedral Surfaces* / Bobenko–Bücking 2021 *Convergence of discrete period matrices* |
 | **9d.4** | planned | Möbius centring of Poincaré-disk layouts as a *variational* problem (Lorentz energy with full gradient + Hessian), replacing today's iterative Fréchet-mean fallback in `normalise_hyperbolic()` | Java original `MobiusCenteringFunctional`; closest published context: decorated-DCE Möbius normalisation, also used in canonical-tessellation post-processing |
 
 **Question A:** which of these (if any) would unblock concrete
@@ -104,7 +104,7 @@ We have:
   [`doc/math/hyperideal-hessian-derivation.md`](../math/hyperideal-hessian-derivation.md)
   (805 lines, all sign pitfalls covered, references Schläfli 1858,
   Milnor 1982, Vinberg 1993, Cho–Kim 1999, Glickenstein 2011,
-  Springborn 2020, Rivin–Springborn 1999).
+  Springborn 2020, Rivin–Schlenker 1999).
 
 **Question:**
 

doc/roadmap/phases.md

@@ -109,8 +109,8 @@ mesh type.
 9b-analytic   Full analytic HyperIdeal Hessian via Schläfli identity
          → planned, see research-track.md
          Mathematical source: Springborn 2020 §4 + Schläfli 1858/60
-                              + Rivin, Springborn 1999 "The Schläfli formula in
-                                Einstein manifolds with boundary" (ERA-AMS 5)
+                              + Rivin, Schlenker 1999 "The Schläfli formula in
+                                Einstein manifolds with boundary" (ERA-AMS 5, 18–23)
                               + Cho-Kim 1999 + Glickenstein 2011 §4
          Algorithm: explicit chain rule through (bᵢ,aₑ) → ℓᵢⱼ → ζ₁₃/ζ₁₄/ζ₁₅ → αᵢⱼ/βᵢ
          Includes: short LaTeX correctness note in doc/math/.
@@ -165,8 +165,8 @@ mesh type.
              Non-Euclidean Geometries" (Discrete & Comput. Geom. 2025,
              arXiv:2310.17529) §3 — decorated DCE framework unifying cone
              singularities and cusps in hyperbolic + spherical geometry.
-           Crane, Soliman, Ben-Chen, Schröder 2018 "Optimal Cone Singularities
-             for Conformal Flattening" (ACM SIGGRAPH 2018) — L¹-optimal
+           Soliman, Slepčev, Crane 2018 "Optimal Cone Singularities
+             for Conformal Flattening" (ACM TOG 37(4), Art. 105) — L¹-optimal
              automatic cone placement; directly applicable to 9d.2 algorithm.
          Status: 🔲 planned
 
@@ -223,8 +223,10 @@ mesh type.
            Alexa, Wardetzky 2011 "Discrete Laplacians on General Polygonal
              Meshes" (ACM SIGGRAPH 2011) — virtual-node construction,
              polygon cotangent weights extending the Pinkall-Polthier formula.
-           Alexa 2020 "Discrete Laplacians on General Polygonal Meshes"
-             (ACM TOG 39, 2020) — extended journal treatment, error bounds.
+           Bunge, Herholz, Kazhdan, Botsch 2020 "Polygon Laplacian Made
+             Simple" (Computer Graphics Forum 39(2), 303–313) — virtual-vertex
+             construction with error analysis. (DEC alternative: de Goes,
+             Butts, Desbrun 2020, ACM TOG 39(4).)
          Enables: DCE energy evaluation on quad-dominant / Voronoi /
                   polygon meshes without forced triangulation.
                   Replaces euclidean_hessian.hpp for non-triangular inputs.
@@ -425,15 +427,19 @@ Phase 10   Global uniformization for genus g ≥ 2
        → Ωᵢⱼ = ∫_{bⱼ} ωᵢ
        → Reduction to Siegel fundamental domain via Sp(2g,ℤ).
        Mathematical reference: Bobenko-Springborn 2004 + Gottschling 1959.
-                               Bobenko, Bücking 2009 "Conformal Structures and
-                               Period Matrices of Polyhedral Surfaces" — discrete
-                               period matrix Ωᵢⱼ on polyhedral surfaces.
+                               Bobenko, Mercat, Schmies 2009/2011 "Conformal
+                               Structures / Period Matrices of Polyhedral
+                               Surfaces" (arXiv:0909.1305) + Bobenko, Bücking 2021
+                               "Convergence of discrete period matrices ..."
+                               (Math. Phys. Anal. Geom. 24, Art. 23) — discrete
+                               period matrix Ωᵢⱼ on polyhedral surfaces + convergence.
                                Bobenko, Lutz 2024 IMRN "Decorated Discrete Conformal
                                Maps and Convex Polyhedral Cusps" — uniformization
                                theorem connecting cusps ↔ hyperideal vertices
                                (bridges Phase 2/3 HyperIdeal geometry to 10b).
-                               Springborn 2019 "A discrete version of Liouville's
-                               theorem on conformal maps" (arXiv:1911.00966) —
+                               Pinkall, Springborn 2021 "A discrete version of
+                               Liouville's theorem on conformal maps"
+                               (Geom. Dedicata 214, 389–398; arXiv:1911.00966) —
                                proves uniqueness/rigidity of the discrete conformal
                                structure; justifies that Ω is a conformal invariant.
        Java partial reference: DiscreteRiemannUtility.java (186 lines).
@@ -467,9 +473,9 @@ Phase 10   Global uniformization for genus g ≥ 2
                                Bobenko, Lutz 2024 IMRN (arXiv:2305.10988) —
                                discrete uniformization theorem for decorated
                                piecewise Euclidean surfaces.
-                               Springborn, Veselov 2015 "Quasiconformal distortion
-                               of projective transformations and discrete conformal
-                               maps" (Int. Math. Res. Not.) — error estimates for
+                               Born, Bücking, Springborn 2015 "Quasiconformal
+                               distortion of projective transformations and discrete
+                               conformal maps" (arXiv:1505.01341) — error estimates for
                                the discrete-to-smooth conformal approximation;
                                quantifies how well H²/Γ approximates the smooth
                                hyperbolic metric.
@@ -479,11 +485,22 @@ Phase 10   Global uniformization for genus g ≥ 2
        Status: **fully new research.**
        Requires: 10a + 10b + Phase 9c.
 
+       ⚠️ SCOPE BOUNDARY (was 10c delivers vs. was offen bleibt):
+         10c as scoped here builds the *infrastructure* — Fuchsian-group
+         representation + H²/Γ embedding on the Sechelmann/Bobenko-Springborn
+         uniformisation path.  The Lutz-SPECIFIC algorithms it references
+         (canonical Delaunay tessellation in Penner coordinates, Epstein-Penner
+         convex-hull construction, Weeks-flip extension, polyhedral realisation)
+         are NOT delivered automatically by reaching 10c — they sit ON TOP of
+         this infrastructure and are their own implementation effort.
+         → that effort is split out as **Phase 13** (Chain B capstone).
+         10c = runway;  Phase 13 = the Lutz algorithms that land on it.
+
 10c'  Optional Java-port additions (low priority)
        → KoebePolyhedron.java (321 lines) — Koebe-Andreev-Thurston
          circle packings.  Adds a fifth DCE method.
-         Rigidity: Bowers, Bowers, Lutz 2026 "Rigidity of circle polyhedra
-         and hyperideal polyhedra: the tangency case" (arXiv:2601.22903)
+         Rigidity: Bowers, Bowers, Lutz 2026 "Rigidity of Koebe Polyhedra
+         and Inversive Distance Circle Packings" (arXiv:2601.22903)
          — theoretical uniqueness backing the KAT construction.
        → ElectrostaticSphereFunctional (127 lines) — sphere
          distribution baseline.
@@ -560,3 +577,102 @@ All three items are tracked here so the project memory is preserved;
 none of them are roadmap commitments.  See `research-track.md` for the
 formal research-versus-port classification before starting any.
 ```
+
+---
+
+## ◼ Phase 12 — Decorated DCE & geometric transition  (RESEARCH, near-term)
+
+> **Note on ordering:** despite the higher number, Phase 12 is *near-term
+> and independent* of Phases 9c–11.  It builds ONLY on already-landed code
+> and is the **short path (Chain A)** to a first Lutz-adjacent scientific
+> result.  It does **not** require the genus-g≥2 chain (9c/10a/10b/10c) or
+> the holonomy-bug fix — those gate Phase 13 (Chain B), not this.
+
+```
+12   Decorated DCE & geometric transition  (no Java parent)
+       → Numerical demonstration of the Bobenko-Lutz "master theory":
+         one discrete conformal invariant, continuously deformable across
+         Euclidean / spherical / hyperbolic background geometry.
+       Mathematical reference:
+         Bobenko, Lutz 2025 "Decorated Discrete Conformal Equivalence in
+           Non-Euclidean Geometries" (Discrete & Comput. Geom.;
+           arXiv:2310.17529) §3 — Penner-coordinate decoration unifying the
+           three background geometries; continuous deformation at fixed
+           discrete conformal invariant.
+         Lutz 2024 PhD thesis (depositonce-20357) — full proofs.
+       Builds on (✅ already landed):
+         inversive_distance functional (9a.2), hyper_ideal (Springborn 2020),
+         spherical functional — the decoration is a RE-PARAMETRISATION of
+         these, not a new solver.
+       Does NOT require: 9c / 10a / 10b / holonomy-bug fix.
+       Scope:
+         1. Decoration layer: per-vertex circle/horocycle radius as Penner
+            coordinate; map ↔ existing inversive distance I_ij (classical
+            formula ℓ²=r_i²+r_j²+2r_ir_jη).
+         2. Transition driver: deform background curvature κ ∈ {+,0,−} while
+            holding the discrete conformal invariant fixed; solve per geometry.
+         3. Validation harness producing example galleries.
+       Acceptance criteria:
+         - Decoration round-trip I_ij ↔ (r_i,r_j,ℓ) at machine precision.
+         - At κ=0: bit-for-bit match with existing euclidean/inversive path.
+         - Gauss-Bonnet per geometry; invariant constant across the κ-transition
+           to tol (numerical witness of the Bobenko-Lutz master theorem).
+         - Cross-geometry: one test surface solved in all three backgrounds
+           shares the invariant.
+       Effort: medium (functionals exist; reparametrisation + driver + tests).
+       Status: 🔲 planned (proposed 2026-05-29).
+```
+
+---
+
+## ◼ Phase 13 — Decorated canonical tessellations & polyhedral realisation  (Chain B capstone)
+
+> **This is the genus-g≥2 Lutz contribution.**  It sits ON TOP of the
+> infrastructure built by Phases 9c + 10a + 10b + 10c (see the 10c SCOPE
+> BOUNDARY note above) and implements Lutz's *specific* algorithms that the
+> 10c "runway" does not deliver by itself.
+
+```
+13   Decorated canonical tessellations + polyhedral realisation  (no Java parent)
+       → Canonical Delaunay tessellation of a decorated hyperbolic surface
+         in Penner coordinates, its dual decomposition, and the polyhedral
+         realisation of the uniformised genus-g≥2 surface.
+       Mathematical reference:
+         Lutz 2023 "Canonical Tessellations of Decorated Hyperbolic Surfaces"
+           (Geom. Dedicata 217; arXiv:2206.13461) — canonical (weighted-
+           Delaunay-analogue) tessellation + dual; Epstein-Penner convex-hull
+           construction in Minkowski space; Weeks-flip extension.
+         Bobenko, Lutz 2024 IMRN (arXiv:2305.10988) — discrete uniformization
+           theorem for decorated surfaces (cusps ↔ hyperideal vertices).
+         Lutz 2024 PhD thesis (depositonce-20357) — polyhedral realisation +
+           complete proofs for 9d.2 / 10b / 10c / 13.
+         Rigidity backing: Bowers, Bowers, Lutz 2026 (arXiv:2601.22903).
+       PREREQUISITES (the "given Voraussetzungen" — all must be in place):
+         ✅ cut_graph.hpp (2g seams)                — landed
+         🔲 Phase 9c  — 4g-gon fundamental domain
+         🔲 Phase 10a — holomorphic/harmonic 1-forms
+         🔲 Phase 10b — Siegel period matrix Ω ∈ H_g
+         🔲 Phase 10c — Fuchsian-group representation / H²/Γ embedding
+         🔲 holonomy-bug fix — detail::spherical_holonomy /
+              detail::hyperbolic_holonomy (+ cpp_dec_float_50 for the
+              group-relation product ∏gᵢ = Id); see research-track.md §9c.
+         🟡 Phase 12  — decoration layer (Penner coords) is reused here;
+              strongly recommended to land Phase 12 first so the Penner-
+              coordinate machinery already exists.
+       Scope:
+         1. Penner-coordinate weighted-Delaunay (canonical) tessellation +
+            dual decomposition on the H²/Γ embedding from 10c.
+         2. Epstein-Penner convex-hull construction (Minkowski space) to
+            obtain the canonical decomposition; Weeks-flip to reach it.
+         3. Polyhedral realisation of the uniformised surface.
+       Acceptance criteria:
+         - Canonical tessellation is unique & flip-stable (Weeks-flip
+           terminates; result independent of start triangulation).
+         - Decoration / Penner-coordinate consistency with Phase 12 layer.
+         - Gauss-Bonnet + holonomy closure ∏[a_i,b_i] = Id (high precision).
+         - Rigidity witness: Newton finds the unique realisation on the
+           tangency-case test set (Bowers-Bowers-Lutz 2026).
+       Effort: very large (depends on the full 9c/10a/10b/10c chain landing
+         first; the Lutz algorithms themselves ≈ several weeks on top).
+       Status: 🔲 planned (Chain B capstone; gated on prerequisites above).
+```

doc/roadmap/research-track.md

@@ -79,7 +79,7 @@ The phase numbers match `doc/roadmap/phases.md`.
 ### 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.
+  Geometry & Topology 19(4) (2015), 2155–2215. arXiv:1005.2698.
 * **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.
@@ -95,11 +95,15 @@ The phase numbers match `doc/roadmap/phases.md`.
     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)`.
+    → introduces inversive-distance circle packings. NB: the formula
+    `I_ij = (ℓ²−r_i²−r_j²)/(2 r_i r_j)` is the *classical* inversive
+    distance (= Glickenstein §5.2 η), not a B–S-specific identity.
   - **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).
+    scalar curvature on piecewise flat two- and three-dimensional
+    manifolds.* J. Diff. Geom. 87(2), 201–238.  → §5.2 inversive-distance
+    parametrization ℓ²=r_i²+r_j²+2r_ir_jη; correspondence I_ij = cos θ_e
+    holds only up to sign/supplement (intersection at arccos(−η)).
+    The paper does **not** number equations as "(4.6)".
 * **Java reference:** ❌ **none.**  Verified empirically:
   ```bash
   $ find /Users/tarikmoussa/Desktop/conformallab -iname "*nversive*"
@@ -202,7 +206,7 @@ The phase numbers match `doc/roadmap/phases.md`.
 
 ### Inversive-distance Hessian — full analytic (Phase 9a.2-analytic, 🔲 planned)
 
-* **Mathematical source:** Glickenstein, D. (2011) eq. (4.6).
+* **Mathematical source:** Glickenstein, D. (2011) §5.2 (inversive-distance parametrization ℓ²=r_i²+r_j²+2r_ir_jη).
 * **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
@@ -219,8 +223,8 @@ The phase numbers match `doc/roadmap/phases.md`.
     → §3: Penner-coordinate decoration unifies cone singularities (Θᵥ ≠ 2π)
     and hyperideal cusps (Θᵥ = 0) in a single algebraic framework valid in
     Euclidean, spherical, and hyperbolic geometry.
-  - **Crane, Soliman, Ben-Chen, Schröder** (2018). *Optimal Cone Singularities
-    for Conformal Flattening.* ACM SIGGRAPH 2018. DOI: 10.1145/3197517.3201367.
+  - **Soliman, Slepčev, Crane** (2018). *Optimal Cone Singularities
+    for Conformal Flattening.* ACM Trans. Graph. 37(4), Art. 105. DOI: 10.1145/3197517.3201367.
     → L¹-optimal cone placement via a sparse-recovery optimisation over the
     curvature deficit Kᵥ = 2π − Θᵥ; directly gives the set of cone angles
     to prescribe for a near-flat conformal parametrisation.
@@ -260,9 +264,10 @@ The phase numbers match `doc/roadmap/phases.md`.
     central node connected to all vertices; cotangent weights are computed
     per sub-triangle; the resulting operator is symmetric and positive
     semi-definite, mirroring Pinkall-Polthier for triangulations.
-  - **Alexa** (2020). *Discrete Laplacians on General Polygonal Meshes.*
-    ACM TOG 39(6). DOI: 10.1145/3414685.3417840.
-    → Extended journal version with error bounds and convergence analysis.
+  - **Bunge, Herholz, Kazhdan, Botsch** (2020). *Polygon Laplacian Made Simple.*
+    Computer Graphics Forum 39(2), 303–313. DOI: 10.1111/cgf.13931.
+    → virtual-vertex construction with error analysis. (DEC alternative:
+    de Goes, Butts, Desbrun 2020, ACM TOG 39(4), DOI 10.1145/3386569.3392389.)
 
 * **Java reference:** ❌ **none.**
 
@@ -345,6 +350,72 @@ The phase numbers match `doc/roadmap/phases.md`.
 * **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.
+* **⚠️ Scope boundary:** 10c delivers the *infrastructure* (Fuchsian-group
+  representation + H²/Γ embedding on the Sechelmann/Bobenko-Springborn path).
+  The Lutz-*specific* algorithms (canonical Delaunay tessellation in Penner
+  coordinates, Epstein-Penner hull, Weeks-flip, polyhedral realisation) are
+  **not** auto-delivered by reaching 10c — they are split out as **Phase 13**
+  (Chain B capstone), which sits on top of this runway.
+
+---
+
+### Decorated DCE & geometric transition (Phase 12, 🔲 planned — near-term, Chain A)
+* **Mathematical sources:**
+  - **Bobenko, Lutz** (2025). *Decorated Discrete Conformal Equivalence in
+    Non-Euclidean Geometries.* Discrete & Comput. Geom. arXiv:2310.17529. §3
+    — Penner-coordinate decoration unifying Euclidean/spherical/hyperbolic
+    DCE; continuous deformation at fixed discrete conformal invariant.
+  - **Lutz** (2024). *PhD thesis, TU Berlin.* DOI: 10.14279/depositonce-20357.
+* **Java reference:** ❌ none.
+* **Builds on (✅ landed):** `inversive_distance_functional.hpp` (9a.2),
+  `hyper_ideal_functional.hpp` (Springborn 2020), `spherical_functional.hpp`
+  — the decoration is a re-parametrisation of these, not a new solver.
+* **Does NOT require:** 9c / 10a / 10b / holonomy-bug fix. This is the
+  **short path**: the earliest Lutz-adjacent result, independent of Chain B.
+* **Scope:** (1) Penner-coordinate decoration layer ↔ classical inversive
+  distance `ℓ²=r_i²+r_j²+2r_ir_jη`; (2) curvature-transition driver κ∈{+,0,−}
+  at fixed invariant; (3) validation harness + example gallery.
+* **Acceptance criteria:**
+  - Decoration round-trip `I_ij ↔ (r_i,r_j,ℓ)` at machine precision.
+  - At κ=0 bit-for-bit match with the existing Euclidean/inversive path.
+  - Gauss-Bonnet per geometry; invariant constant across the κ-transition
+    to tol (numerical witness of the Bobenko-Lutz master theorem).
+  - Cross-geometry agreement of the invariant on one test surface.
+* **Effort:** medium (functionals exist; reparametrisation + driver + tests).
+
+---
+
+### Decorated canonical tessellations & polyhedral realisation (Phase 13, 🔲 planned — Chain B capstone)
+* **Mathematical sources:**
+  - **Lutz** (2023). *Canonical Tessellations of Decorated Hyperbolic
+    Surfaces.* Geom. Dedicata 217. arXiv:2206.13461 — canonical (weighted-
+    Delaunay-analogue) tessellation + dual; Epstein-Penner convex hull in
+    Minkowski space; Weeks-flip extension.
+  - **Bobenko, Lutz** (2024). IMRN 2024(12), 9505–9534. arXiv:2305.10988 —
+    discrete uniformization theorem for decorated surfaces.
+  - **Lutz** (2024). *PhD thesis* (depositonce-20357) — polyhedral realisation.
+  - Rigidity backing: **Bowers, Bowers, Lutz** (2026), arXiv:2601.22903.
+* **Java reference:** ❌ none.
+* **Prerequisites (the "given Voraussetzungen", all must be in place):**
+  ✅ `cut_graph.hpp` (2g seams) · 🔲 Phase 9c (fundamental domain) ·
+  🔲 Phase 10a (1-forms) · 🔲 Phase 10b (period matrix Ω) ·
+  🔲 Phase 10c (Fuchsian group / H²/Γ) ·
+  🔲 holonomy-bug fix (`detail::spherical_holonomy` /
+  `detail::hyperbolic_holonomy` + `cpp_dec_float_50`; see Phase 9c block) ·
+  🟡 Phase 12 (Penner-coordinate decoration layer — reused here; land first).
+* **Scope:** (1) Penner-coordinate canonical tessellation + dual on the
+  H²/Γ embedding from 10c; (2) Epstein-Penner hull + Weeks-flip to reach the
+  canonical decomposition; (3) polyhedral realisation of the uniformised
+  genus-g surface.
+* **Acceptance criteria:**
+  - Canonical tessellation unique & flip-stable (Weeks-flip terminates,
+    start-triangulation-independent).
+  - Penner-coordinate consistency with the Phase 12 decoration layer.
+  - Gauss-Bonnet + holonomy closure `∏[a_i,b_i] = Id` (high precision).
+  - Rigidity witness: Newton finds the unique realisation on the
+    tangency-case test set (Bowers-Bowers-Lutz 2026).
+* **Effort:** very large — gated on the full 9c/10a/10b/10c chain; the Lutz
+  algorithms themselves ≈ several weeks on top.
 
 ---
 

doc/tutorials/add-inversive-distance.md

@@ -78,7 +78,7 @@ the domain where every triangle is valid); we use the same 10-point
 Gauss-Legendre quadrature as `euclidean_functional.hpp`.
 
 The Hessian is finite-difference for the MVP; an analytic form
-(Glickenstein 2011 eq. 4.6) is documented in the research-track roadmap.
+(Glickenstein 2011 §5.2) is documented in the research-track roadmap.
 
 ---
 

doc/tutorials/block-fd-hessian.md

@@ -107,7 +107,7 @@ controlling the degenerate-vertex clamps inherited from
 The global gradient is the angle-defect / Schläfli-type sum
 
 ```
-G_{b,v} =  Σ_{f ∋ v}  β_v(f)  −  Θ_v                          (Springborn 2020 eq. 4.6)
+G_{b,v} =  Σ_{f ∋ v}  β_v(f)  −  Θ_v                          (Springborn 2020 §4, variational gradient)
 G_{a,e} =  Σ_{f ∋ e}  α_e(f)  −  θ_e
 ```