docs: restructure documentation into focused files

README.md: reduced from 703 to ~75 lines — what/why, status, quick
start, minimal usage example, navigation table to doc/ files.

doc/architecture/overall_pipeline.md: trimmed — roadmap, extension
points, declarative pipeline YAML, and references sections removed
(each now has its own dedicated file). Replaced with a link table.

New files:
  doc/getting-started.md       — build modes, single-test invocation, CLI
  doc/api/pipeline.md          — full pipeline API with code for all 3 geometries
  doc/api/extending.md         — new functionals, geometry modes, Java porting guide
  doc/api/contracts.md         — processing unit preconditions/provides table
  doc/api/cgal-package.md      — Phase 8 CGAL package design + YAML pipeline (TODO)
  doc/math/geometry-modes.md   — Euclidean/Spherical/HyperIdeal comparison
  doc/math/references.md       — all papers by module
  doc/roadmap/phases.md        — Phases 1–10 with porting/research boundary
  doc/roadmap/java-parity.md   — Java vs C++ feature parity table
  doc/contributing.md          — language policy, test standards, release flow

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

doc/math/geometry-modes.md

@@ -0,0 +1,112 @@
+# The Three Geometry Modes
+
+conformallab++ implements discrete conformal geometry in three model spaces.
+All three share the same algorithmic structure — only the angle formula,
+the trilateration geometry, and the Hessian sign differ.
+
+---
+
+## Comparison
+
+```
+                  Euclidean            Spherical            Hyper-ideal
+                  ─────────────────────────────────────────────────────
+Space             ℝ²                   S²                   H² (Poincaré disk)
+Curvature         K = 0                K = +1               K = −1
+Typical genus     any (cone metrics)   0                    ≥ 1
+Angle sum         Σαᵥ = Θᵥ             Σαᵥ = Θᵥ             Σβᵥ = Θᵥ
+Gradient sign     G_v = actual − Θᵥ    G_v = Θᵥ − actual    G_v = actual − Θᵥ
+Hessian           PSD (cotangent-Lap.) NSD (sign-flip)       PSD (FD, strict conv.)
+Newton solve      SimplicialLDLT(H)    SimplicialLDLT(−H)    SimplicialLDLT(H)
+Holonomy          translations ωᵢ      rotations (2-D)       Möbius maps Tᵢ ∈ SU(1,1)
+Period            τ = ω₂/ω₁ ∈ ℍ        —                     axis of Tᵢ
+Normalise         PCA centring         Rodrigues to N pole   weighted Möbius centring
+Layout            Layout2D (ℝ²)        Layout3D (unit sphere) Layout2D (Poincaré disk)
+```
+
+---
+
+## Euclidean (ℝ²)
+
+**Energy:** the cotangent-weighted angle defect energy (Pinkall & Polthier 1993).
+
+**Effective log-length:**
+```
+Λ̃ᵢⱼ = λ°ᵢⱼ + uᵢ + uⱼ
+```
+
+**Gradient:** `G_v = Σ_{faces adj. v} α_v(face) − Θᵥ` — the actual angle sum minus the target.
+
+**Hessian:** cotangent Laplacian, assembled in `euclidean_hessian.hpp`. PSD with one zero
+eigenvalue (constant function) for closed meshes — handled by pinning one vertex or SparseQR.
+
+**Normalisation:** centroid → origin, major axis → x-axis via PCA (`normalise_euclidean`).
+
+**Holonomy:** lattice translations ω₁, ω₂ ∈ ℂ for closed surfaces.
+Period ratio τ = ω₂/ω₁ ∈ ℍ is the conformal modulus of the torus.
+
+---
+
+## Spherical (S²)
+
+**Typical use:** genus-0 (sphere-like) surfaces.
+
+**Gradient:** `G_v = Θᵥ − Σ α_v(face)` — note the **sign reversal** vs. Euclidean.
+
+**Hessian:** NSD (the spherical energy is concave). Newton solves `(−H)·Δx = G`, i.e.
+`SimplicialLDLT` is called on `−H`. This is handled transparently in `newton_spherical()`.
+
+**Gauge fix:** for a closed spherical surface, one additional DOF must be pinned to remove
+the rotational gauge mode. `SphericalMaps` has a dedicated `gauge_vertex` field.
+
+**Normalisation:** `normalise_spherical()` rotates the layout so the area centroid
+maps to the north pole (Rodrigues rotation formula).
+
+---
+
+## Hyper-ideal (H²)
+
+**Typical use:** genus-g surfaces (g ≥ 1), hyperbolic cone metrics.
+
+**DOFs:** both vertex variables `bᵢ` (hyper-ideal radius) and edge variables `aₑ`
+(intersection angle). `assign_all_dof_indices(mesh, maps)` assigns both automatically.
+No vertex needs to be pinned — the functional is strictly convex.
+
+**Geometry functions** (in `hyper_ideal_geometry.hpp`):
+```
+ζ₁₃(b, a)  ζ₁₄(b, a)  ζ₁₅(b, a)   — the three fundamental Springborn functions
+lᵢⱼ(ζ)                             — edge length from ζ value
+αᵢⱼ(l₁, l₂, l₃)                   — interior angle
+βᵢ(l₁, l₂, l₃)                    — vertex angle sum contribution
+```
+
+**Hessian:** currently a symmetric finite-difference approximation (see `hyper_ideal_hessian.hpp`).
+The analytic Hessian through the `(bᵢ, aₑ) → lᵢⱼ → ζ → αᵢⱼ/βᵢ` chain is Phase 9b.
+
+**Holonomy:** Möbius isometries Tᵢ ∈ SU(1,1) (orientation-preserving isometries of the Poincaré disk).
+`MobiusMap` in `layout.hpp`: T(z) = (az+b)/(cz+d).
+
+**Normalisation:** `normalise_hyperbolic()` performs iterative face-area-weighted Möbius centring
+(Fréchet mean, 30 iterations) to map the weighted centroid to the disk origin.
+
+---
+
+## Gauss–Bonnet constraint
+
+Before calling any Newton solver, the target angles must satisfy the Gauss–Bonnet equation:
+
+```
+Σᵥ (2π − Θᵥ)  =  2π · χ(M)    (Euclidean / flat)
+Σᵥ (2π − Θᵥ)  >  0             (spherical, χ > 0)
+Σᵥ (2π − Θᵥ)  <  0             (hyperbolic, χ < 0)
+```
+
+If this is violated, no conformal factor can realise the target angles and Newton will
+fail to converge without a visible error.
+
+```cpp
+check_gauss_bonnet(mesh, maps);      // throws if violated
+enforce_gauss_bonnet(mesh, maps);    // redistributes residual uniformly across all vertices
+```
+
+**This is the most common source of silent non-convergence.** Always call one of these before solving.