docs: citation audit + correct 8 mis-citations; add Phases 12/13
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External reviewer pass over the literature references. Verified entries against arXiv/DOI/publisher and corrected misattributions that had propagated across the docs. Corrected citations (consistent across all docs): - Bowers-Bowers-Lutz 2026: title was the 2017 paper's -> "Rigidity of Koebe Polyhedra and Inversive Distance Circle Packings" - Liouville theorem: "Springborn 2019" -> Pinkall & Springborn, Geom. Dedicata 214 (2021) - Bobenko-Pinkall-Springborn: "G&T 14 (2010)" -> G&T 19(4) (2015), 2155-2215 - Optimal Cone Singularities: "Crane, Soliman, Ben-Chen, Schroeder" -> Soliman, Slepcev, Crane, ACM TOG 37(4) - Schlaefli formula: "Rivin, Springborn 1999" -> Rivin, Schlenker - Quasiconformal distortion: "Springborn, Veselov" -> Born, Buecking, Springborn (arXiv:1505.01341) - Period matrices: "Bobenko, Buecking 2009" (was Bobenko-Mercat-Schmies' title) -> Bobenko-Mercat-Schmies 2011 + genuine Bobenko-Buecking 2021 - Fabricated entry: "Alexa 2020, DOI 10.1145/3414685.3417840" pointed to an unrelated paper (Pixelor) -> Bunge, Herholz, Kazhdan, Botsch 2020 - Stripe Patterns: "Bonneel et al. 2015" -> Knoeppel, Crane, Pinkall, Schroeder 2015 Equation-number corrections (verified against the PDFs): - Glickenstein 2011 "eq. 4.6" -> "§5.2" (no such equation label exists) - Springborn 2020 "eq. 4.6" -> "§4 variational gradient" - inversive-distance attribution softened to classical inversive distance Other: - DBFEnergy bibliography (separate repo) and convergence half-sentence in novelty-statement.md §3.3 (Bobenko-Buecking 2021) - Status legend (implemented vs planned) at top of references.md - New Phase 12 (decorated DCE & geometric transition, Chain A, near-term) and Phase 13 (canonical tessellations & polyhedral realisation, Chain B capstone) in phases.md + research-track.md; 10c scope-boundary note clarifying infrastructure vs Lutz-specific algorithms Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
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Tarik Moussa
@@ -109,8 +109,8 @@ mesh type.
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9b-analytic Full analytic HyperIdeal Hessian via Schläfli identity
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→ planned, see research-track.md
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Mathematical source: Springborn 2020 §4 + Schläfli 1858/60
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+ Rivin, Springborn 1999 "The Schläfli formula in
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Einstein manifolds with boundary" (ERA-AMS 5)
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+ Rivin, Schlenker 1999 "The Schläfli formula in
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Einstein manifolds with boundary" (ERA-AMS 5, 18–23)
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+ Cho-Kim 1999 + Glickenstein 2011 §4
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Algorithm: explicit chain rule through (bᵢ,aₑ) → ℓᵢⱼ → ζ₁₃/ζ₁₄/ζ₁₅ → αᵢⱼ/βᵢ
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Includes: short LaTeX correctness note in doc/math/.
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@@ -165,8 +165,8 @@ mesh type.
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Non-Euclidean Geometries" (Discrete & Comput. Geom. 2025,
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arXiv:2310.17529) §3 — decorated DCE framework unifying cone
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singularities and cusps in hyperbolic + spherical geometry.
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Crane, Soliman, Ben-Chen, Schröder 2018 "Optimal Cone Singularities
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for Conformal Flattening" (ACM SIGGRAPH 2018) — L¹-optimal
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Soliman, Slepčev, Crane 2018 "Optimal Cone Singularities
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for Conformal Flattening" (ACM TOG 37(4), Art. 105) — L¹-optimal
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automatic cone placement; directly applicable to 9d.2 algorithm.
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Status: 🔲 planned
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@@ -223,8 +223,10 @@ mesh type.
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Alexa, Wardetzky 2011 "Discrete Laplacians on General Polygonal
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Meshes" (ACM SIGGRAPH 2011) — virtual-node construction,
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polygon cotangent weights extending the Pinkall-Polthier formula.
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Alexa 2020 "Discrete Laplacians on General Polygonal Meshes"
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(ACM TOG 39, 2020) — extended journal treatment, error bounds.
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Bunge, Herholz, Kazhdan, Botsch 2020 "Polygon Laplacian Made
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Simple" (Computer Graphics Forum 39(2), 303–313) — virtual-vertex
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construction with error analysis. (DEC alternative: de Goes,
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Butts, Desbrun 2020, ACM TOG 39(4).)
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Enables: DCE energy evaluation on quad-dominant / Voronoi /
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polygon meshes without forced triangulation.
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Replaces euclidean_hessian.hpp for non-triangular inputs.
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@@ -425,15 +427,19 @@ Phase 10 Global uniformization for genus g ≥ 2
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→ Ωᵢⱼ = ∫_{bⱼ} ωᵢ
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→ Reduction to Siegel fundamental domain via Sp(2g,ℤ).
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Mathematical reference: Bobenko-Springborn 2004 + Gottschling 1959.
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Bobenko, Bücking 2009 "Conformal Structures and
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Period Matrices of Polyhedral Surfaces" — discrete
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period matrix Ωᵢⱼ on polyhedral surfaces.
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Bobenko, Mercat, Schmies 2009/2011 "Conformal
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Structures / Period Matrices of Polyhedral
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Surfaces" (arXiv:0909.1305) + Bobenko, Bücking 2021
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"Convergence of discrete period matrices ..."
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(Math. Phys. Anal. Geom. 24, Art. 23) — discrete
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period matrix Ωᵢⱼ on polyhedral surfaces + convergence.
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Bobenko, Lutz 2024 IMRN "Decorated Discrete Conformal
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Maps and Convex Polyhedral Cusps" — uniformization
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theorem connecting cusps ↔ hyperideal vertices
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(bridges Phase 2/3 HyperIdeal geometry to 10b).
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Springborn 2019 "A discrete version of Liouville's
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theorem on conformal maps" (arXiv:1911.00966) —
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Pinkall, Springborn 2021 "A discrete version of
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Liouville's theorem on conformal maps"
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(Geom. Dedicata 214, 389–398; arXiv:1911.00966) —
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proves uniqueness/rigidity of the discrete conformal
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structure; justifies that Ω is a conformal invariant.
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Java partial reference: DiscreteRiemannUtility.java (186 lines).
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@@ -467,9 +473,9 @@ Phase 10 Global uniformization for genus g ≥ 2
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Bobenko, Lutz 2024 IMRN (arXiv:2305.10988) —
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discrete uniformization theorem for decorated
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piecewise Euclidean surfaces.
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Springborn, Veselov 2015 "Quasiconformal distortion
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of projective transformations and discrete conformal
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maps" (Int. Math. Res. Not.) — error estimates for
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Born, Bücking, Springborn 2015 "Quasiconformal
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distortion of projective transformations and discrete
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conformal maps" (arXiv:1505.01341) — error estimates for
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the discrete-to-smooth conformal approximation;
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quantifies how well H²/Γ approximates the smooth
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hyperbolic metric.
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@@ -479,11 +485,22 @@ Phase 10 Global uniformization for genus g ≥ 2
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Status: **fully new research.**
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Requires: 10a + 10b + Phase 9c.
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⚠️ SCOPE BOUNDARY (was 10c delivers vs. was offen bleibt):
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10c as scoped here builds the *infrastructure* — Fuchsian-group
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representation + H²/Γ embedding on the Sechelmann/Bobenko-Springborn
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uniformisation path. The Lutz-SPECIFIC algorithms it references
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(canonical Delaunay tessellation in Penner coordinates, Epstein-Penner
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convex-hull construction, Weeks-flip extension, polyhedral realisation)
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are NOT delivered automatically by reaching 10c — they sit ON TOP of
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this infrastructure and are their own implementation effort.
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→ that effort is split out as **Phase 13** (Chain B capstone).
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10c = runway; Phase 13 = the Lutz algorithms that land on it.
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10c' Optional Java-port additions (low priority)
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→ KoebePolyhedron.java (321 lines) — Koebe-Andreev-Thurston
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circle packings. Adds a fifth DCE method.
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Rigidity: Bowers, Bowers, Lutz 2026 "Rigidity of circle polyhedra
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and hyperideal polyhedra: the tangency case" (arXiv:2601.22903)
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Rigidity: Bowers, Bowers, Lutz 2026 "Rigidity of Koebe Polyhedra
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and Inversive Distance Circle Packings" (arXiv:2601.22903)
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— theoretical uniqueness backing the KAT construction.
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→ ElectrostaticSphereFunctional (127 lines) — sphere
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distribution baseline.
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@@ -560,3 +577,102 @@ All three items are tracked here so the project memory is preserved;
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none of them are roadmap commitments. See `research-track.md` for the
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formal research-versus-port classification before starting any.
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```
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---
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## ◼ Phase 12 — Decorated DCE & geometric transition (RESEARCH, near-term)
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> **Note on ordering:** despite the higher number, Phase 12 is *near-term
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> and independent* of Phases 9c–11. It builds ONLY on already-landed code
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> and is the **short path (Chain A)** to a first Lutz-adjacent scientific
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> result. It does **not** require the genus-g≥2 chain (9c/10a/10b/10c) or
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> the holonomy-bug fix — those gate Phase 13 (Chain B), not this.
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```
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12 Decorated DCE & geometric transition (no Java parent)
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→ Numerical demonstration of the Bobenko-Lutz "master theory":
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one discrete conformal invariant, continuously deformable across
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Euclidean / spherical / hyperbolic background geometry.
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Mathematical reference:
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Bobenko, Lutz 2025 "Decorated Discrete Conformal Equivalence in
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Non-Euclidean Geometries" (Discrete & Comput. Geom.;
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arXiv:2310.17529) §3 — Penner-coordinate decoration unifying the
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three background geometries; continuous deformation at fixed
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discrete conformal invariant.
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Lutz 2024 PhD thesis (depositonce-20357) — full proofs.
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Builds on (✅ already landed):
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inversive_distance functional (9a.2), hyper_ideal (Springborn 2020),
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spherical functional — the decoration is a RE-PARAMETRISATION of
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these, not a new solver.
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Does NOT require: 9c / 10a / 10b / holonomy-bug fix.
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Scope:
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1. Decoration layer: per-vertex circle/horocycle radius as Penner
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coordinate; map ↔ existing inversive distance I_ij (classical
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formula ℓ²=r_i²+r_j²+2r_ir_jη).
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2. Transition driver: deform background curvature κ ∈ {+,0,−} while
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holding the discrete conformal invariant fixed; solve per geometry.
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3. Validation harness producing example galleries.
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Acceptance criteria:
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- Decoration round-trip I_ij ↔ (r_i,r_j,ℓ) at machine precision.
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- At κ=0: bit-for-bit match with existing euclidean/inversive path.
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- Gauss-Bonnet per geometry; invariant constant across the κ-transition
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to tol (numerical witness of the Bobenko-Lutz master theorem).
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- Cross-geometry: one test surface solved in all three backgrounds
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shares the invariant.
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Effort: medium (functionals exist; reparametrisation + driver + tests).
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Status: 🔲 planned (proposed 2026-05-29).
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```
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---
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## ◼ Phase 13 — Decorated canonical tessellations & polyhedral realisation (Chain B capstone)
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> **This is the genus-g≥2 Lutz contribution.** It sits ON TOP of the
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> infrastructure built by Phases 9c + 10a + 10b + 10c (see the 10c SCOPE
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> BOUNDARY note above) and implements Lutz's *specific* algorithms that the
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> 10c "runway" does not deliver by itself.
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```
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13 Decorated canonical tessellations + polyhedral realisation (no Java parent)
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→ Canonical Delaunay tessellation of a decorated hyperbolic surface
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in Penner coordinates, its dual decomposition, and the polyhedral
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realisation of the uniformised genus-g≥2 surface.
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Mathematical reference:
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Lutz 2023 "Canonical Tessellations of Decorated Hyperbolic Surfaces"
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(Geom. Dedicata 217; arXiv:2206.13461) — canonical (weighted-
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Delaunay-analogue) tessellation + dual; Epstein-Penner convex-hull
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construction in Minkowski space; Weeks-flip extension.
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Bobenko, Lutz 2024 IMRN (arXiv:2305.10988) — discrete uniformization
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theorem for decorated surfaces (cusps ↔ hyperideal vertices).
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Lutz 2024 PhD thesis (depositonce-20357) — polyhedral realisation +
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complete proofs for 9d.2 / 10b / 10c / 13.
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Rigidity backing: Bowers, Bowers, Lutz 2026 (arXiv:2601.22903).
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PREREQUISITES (the "given Voraussetzungen" — all must be in place):
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✅ cut_graph.hpp (2g seams) — landed
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🔲 Phase 9c — 4g-gon fundamental domain
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🔲 Phase 10a — holomorphic/harmonic 1-forms
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🔲 Phase 10b — Siegel period matrix Ω ∈ H_g
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🔲 Phase 10c — Fuchsian-group representation / H²/Γ embedding
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🔲 holonomy-bug fix — detail::spherical_holonomy /
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detail::hyperbolic_holonomy (+ cpp_dec_float_50 for the
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group-relation product ∏gᵢ = Id); see research-track.md §9c.
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🟡 Phase 12 — decoration layer (Penner coords) is reused here;
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strongly recommended to land Phase 12 first so the Penner-
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coordinate machinery already exists.
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Scope:
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1. Penner-coordinate weighted-Delaunay (canonical) tessellation +
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dual decomposition on the H²/Γ embedding from 10c.
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2. Epstein-Penner convex-hull construction (Minkowski space) to
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obtain the canonical decomposition; Weeks-flip to reach it.
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3. Polyhedral realisation of the uniformised surface.
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Acceptance criteria:
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- Canonical tessellation is unique & flip-stable (Weeks-flip
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terminates; result independent of start triangulation).
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- Decoration / Penner-coordinate consistency with Phase 12 layer.
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- Gauss-Bonnet + holonomy closure ∏[a_i,b_i] = Id (high precision).
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- Rigidity witness: Newton finds the unique realisation on the
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tangency-case test set (Bowers-Bowers-Lutz 2026).
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Effort: very large (depends on the full 9c/10a/10b/10c chain landing
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first; the Lutz algorithms themselves ≈ several weeks on top).
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Status: 🔲 planned (Chain B capstone; gated on prerequisites above).
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```
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