diff --git a/README.md b/README.md
index 1b94fa8..04d24f1 100644
--- a/README.md
+++ b/README.md
@@ -81,6 +81,7 @@ Layout2D layout = euclidean_layout(mesh, res.x, maps);
 | **Processing unit contracts** — preconditions / provides table | [doc/api/contracts.md](doc/api/contracts.md) |
 | **CGAL package design** — Phase 8 target, YAML pipeline | [doc/api/cgal-package.md](doc/api/cgal-package.md) |
 | **Architecture & pipeline diagram** | [doc/architecture/overall_pipeline.md](doc/architecture/overall_pipeline.md) |
+| **geometry-central comparison** — shared core, demarcation, adoption candidates, scientific added value | [doc/architecture/geometry-central-comparison.md](doc/architecture/geometry-central-comparison.md) |
 | **Design decisions** — key architectural choices + rationale | [doc/architecture/design-decisions.md](doc/architecture/design-decisions.md) |
 | **Project structure** — directory tree + build targets | [doc/architecture/project-structure.md](doc/architecture/project-structure.md) |
 | **Discrete conformal theory** — mathematical background for collaborators | [doc/math/discrete-conformal-theory.md](doc/math/discrete-conformal-theory.md) |
diff --git a/doc/architecture/geometry-central-comparison.md b/doc/architecture/geometry-central-comparison.md
new file mode 100644
index 0000000..40a78b4
--- /dev/null
+++ b/doc/architecture/geometry-central-comparison.md
@@ -0,0 +1,281 @@
+# conformallab++ vs. geometry-central — Detailed Comparison
+
+> **Purpose of this document.**  
+> conformallab++ and geometry-central (CMU, Keenan Crane's group) both implement
+> discrete conformal equivalence of triangulated surfaces.  They share the same
+> mathematical core but diverge in algorithmic strategy, scope, and target audience.
+> This document maps the overlap precisely, identifies what cannot and should not be
+> adopted, and explains where a side-by-side study creates scientific added value.
+
+---
+
+## 1 — Shared mathematical foundation
+
+Both libraries implement the following chain:
+
+```
+Input mesh M  →  edge lengths ℓᵢⱼ  →  solve for u ∈ ℝᵛ
+  such that  ℓ̃ᵢⱼ = e^{(uᵢ+uⱼ)/2} · ℓᵢⱼ  satisfies  Σα_v(u) = Θᵥ  ∀v
+```
+
+The variational framework that makes this a well-posed optimisation problem goes
+back to **Bobenko & Springborn (2004)**.  The hyperbolic (HyperIdeal) geometry is
+from **Springborn (2020)**.  The geometry-central implementation (Gillespie,
+Springborn & Crane, SIGGRAPH 2021) is an explicit extension of Springborn 2020
+to intrinsic triangulations.
+
+**Key consequence:** the *mathematical problem* is identical.  Any difference in
+output is either a normalization convention or a bug in one of the two libraries —
+making cross-validation directly meaningful.
+
+---
+
+## 2 — Algorithmic comparison
+
+| Dimension | conformallab++ | geometry-central (Gillespie 2021) |
+|---|---|---|
+| **Solver** | Newton–Raphson, analytical Hessian | Newton or Yamabe gradient flow (user choice) |
+| **Convergence** | Quadratic (8–20 iterations on typical meshes) | Newton: quadratic; Yamabe: linear (~hundreds of steps) |
+| **Triangulation** | Fixed throughout — operates on original `Surface_mesh` | Ptolemaic flips applied before/during solve to reach intrinsic Delaunay |
+| **Hessian** | SimplicialLDLT + SparseQR fallback; analytical for Euclidean/Spherical, FD for HyperIdeal | Assembled on the current (possibly flipped) triangulation |
+| **Mesh backend** | CGAL `Surface_mesh<Point_3>` | geometry-central `ManifoldSurfaceMesh` |
+| **Geometry modes** | Euclidean ✓ · Spherical ✓ · HyperIdeal ✓ | Euclidean ✓ · Hyperbolic ✓ · Spherical ✗ |
+| **Open meshes** | ✓ (boundary DOFs pinned) | ✓ |
+
+### Why Newton on a fixed triangulation works well
+
+The Euclidean and HyperIdeal energies are strictly convex after gauge-fixing.
+Newton therefore converges from u=0 in 8–20 iterations for any reasonable mesh.
+The Hessian is the cotangent Laplacian (Euclidean) or its hyperbolic analog —
+well-conditioned on Delaunay meshes, but can degrade on strongly non-Delaunay inputs.
+
+### What Ptolemaic flips add
+
+A Ptolemaic flip replaces diagonal AC with BD in a quadrilateral under the constraint
+that the Ptolemy relation
+
+```
+AC · BD = AB · CD + AD · BC
+```
+
+holds.  This is *conformal-class-preserving* — the new λ₀ values represent the same
+discrete conformal structure.  The gain: the flipped triangulation is intrinsic
+Delaunay, which bounds the off-diagonal Hessian entries and prevents ill-conditioning
+on pathological inputs.
+
+**This is the only genuine algorithmic advantage geometry-central has for the shared
+sub-problem.** It costs nothing mathematically and buys robustness on bad meshes.
+
+---
+
+## 3 — Feature matrix: what exists where
+
+| Feature | conformallab++ | geometry-central | Notes |
+|---|---|---|---|
+| Discrete conformal equivalence (Euclidean) | ✓ | ✓ | Shared core |
+| Discrete conformal equivalence (Hyperbolic/HyperIdeal) | ✓ (Springborn 2020 formulation) | ✓ (Gillespie 2021 extension) | Mathematically equivalent |
+| Discrete conformal equivalence (Spherical) | ✓ | ✗ | Unique to conformallab++ |
+| Analytical Hessian (Euclidean & Spherical) | ✓ | ✓ | |
+| Analytical Hessian (HyperIdeal) | FD (Phase 9b: analytical planned) | ✓ | gc has analytical version |
+| Gauss–Bonnet check & enforce | ✓ | implicit in solver | |
+| Tree-cotree cut graph (2g seam edges) | ✓ | ✗ | Required for period matrix |
+| Priority-BFS layout in ℝ²/S²/Poincaré disk | ✓ | partial (conformal param only) | |
+| Möbius holonomy SU(1,1) | ✓ | ✗ | Unique to conformallab++ |
+| Period matrix τ ∈ ℍ + SL(2,ℤ) reduction | ✓ | ✗ | Unique to conformallab++ |
+| Fundamental domain + tiling | ✓ | ✗ | Unique to conformallab++ |
+| Intrinsic Delaunay triangulation | ✗ | ✓ | gc has via SignpostIntrinsicTriangulation |
+| Ptolemaic flips | ✗ | ✓ | gc's robustness mechanism |
+| Heat method (geodesic distances) | ✗ | ✓ | Auxiliary tool in gc |
+| YAML/declarative pipeline | ✓ (Phase 8e spec) | ✗ | |
+| CGAL-package submission | ✓ (Phase 8 target) | ✗ | |
+| JSON/XML serialisation | ✓ | ✗ | |
+| CLI app | ✓ | ✗ | |
+| Inversive distance functional (Luo 2004) | ✗ (Phase 9a) | ✗ | Neither has it yet |
+| Siegel period matrix Ω (genus g≥2) | ✗ (Phase 10b) | ✗ | |
+
+---
+
+## 4 — What should be adopted — and what should not
+
+### Adopt: Ptolemaic pre-conditioning (GC-2, after Phase 8)
+
+**What:** a single preprocessing pass that Delaunay-izes the input triangulation
+via Ptolemaic flips, updates λ₀ accordingly, then hands off to the existing
+Newton pipeline unchanged.
+
+**Why it fits:**
+- Preserves the conformal class — mathematically sound
+- Drop-in before `compute_euclidean_lambda0_from_mesh()`, no interface change
+- Does not touch cut graph, holonomy, period matrix
+- ~200 lines of code, one new test suite
+
+**Interface sketch:**
+```cpp
+// include/preprocessing_delaunay.hpp  (Phase GC-2)
+void ptolemy_delaunay(ConformalMesh& mesh, EuclideanMaps& maps);
+// Flips edges until all faces satisfy the Delaunay condition.
+// Updates maps.lambda0[e] via the Ptolemy relation after each flip.
+// Prerequisite: compute_euclidean_lambda0_from_mesh() already called.
+// Postcondition: mesh is an intrinsic Delaunay triangulation of the same surface.
+```
+
+### Adopt partially: analytical HyperIdeal Hessian
+
+geometry-central has a closed-form Hessian for the hyperbolic energy.
+conformallab++ currently uses finite differences (Phase 9b plans analytical).
+The geometry-central implementation can serve as a reference for Phase 9b —
+not a code copy, but a mathematical cross-check.
+
+### Do not adopt: full intrinsic triangulations as architecture
+
+**SignpostIntrinsicTriangulation** is geometry-central's core data structure.
+It tracks vertex positions as (face, barycentric coordinates) rather than 3D points.
+Replacing `CGAL::Surface_mesh` with this would require:
+
+1. Rewriting the cut graph algorithm (which works on halfedges of the *original* mesh
+   and must survive across flips — non-trivial bookkeeping)
+2. Tracking seam edges through flip events for holonomy computation
+3. Abandoning the CGAL package target (Phase 8) — CGAL's mesh concepts are extrinsic
+4. Losing the 3D layout output (Poincaré disk, sphere) which clients depend on
+
+**Verdict:** the architecture incompatibility is fundamental, not incidental.
+The period matrix pipeline requires a stable topological cut that does not survive
+arbitrary flip sequences. This is not a solvable engineering problem within the
+current project scope — it would be a different project.
+
+### Do not adopt: Yamabe flow
+
+Newton converges in 8–20 iterations; Yamabe flow needs hundreds.
+The only reason to use Yamabe flow is when the Hessian is indefinite (which happens
+in the Spherical case — and conformallab++ already handles this with the correct
+sign flip in the energy). There is no mesh type where Yamabe flow beats Newton
+on metrics that conformallab++ targets.
+
+---
+
+## 5 — Where cross-comparison creates scientific added value
+
+### 5.1 — Independent cross-validation of the shared core
+
+The discrete conformal equivalence problem for Euclidean and HyperIdeal geometry
+is implemented independently in two codebases, by different groups, with different
+algorithms.  Agreement on:
+
+- the u-vector (after normalization)
+- UV coordinates (up to Möbius transformation)
+- the residual ‖G(u*)‖ at convergence
+
+would constitute **mutual validation without ground truth**.  This is the same
+methodology used in numerical PDE literature to validate independent solvers.
+
+**Concrete protocol:**
+```
+For each test mesh (cathead.obj, brezel.obj, torus_4x4.off, torus_hex_6x6.off):
+  1. Load into both libraries with identical vertex ordering
+  2. Run conformallab++ Newton solver  →  u_clab, UV_clab
+  3. Run geometry-central solver        →  u_gc,   UV_gc
+  4. Normalize both (subtract mean, divide by scale)
+  5. Report  max|u_clab[v] - u_gc[v]|  and  mean conformal distortion difference
+```
+
+Expected: agreement to ≤ 1e-8 on well-conditioned meshes.  Discrepancy would
+indicate a bug or a normalization mismatch worth investigating.
+
+### 5.2 — Convergence study: Newton with vs. without Ptolemaic pre-conditioning
+
+Hypothesis: on non-Delaunay meshes (e.g. a torus refined by subdivision without
+re-meshing), Ptolemaic Delaunay pre-conditioning reduces Newton iteration count.
+
+**Measurable quantities:**
+- Newton iterations to ‖G‖ < 1e-8
+- Hessian condition number κ(H) at u = 0
+- Wall-clock time
+
+This comparison requires only GC-2 to be implemented in conformallab++ and would
+answer the question: *how bad does a mesh have to be before Ptolemaic pre-conditioning
+pays off?*
+
+**Publication potential:** a short note or conference contribution comparing the
+two approaches on a systematic mesh quality benchmark would be self-contained and
+novel — neither library has published this comparison.
+
+### 5.3 — Period matrix and holonomy as differentiating contribution
+
+geometry-central deliberately stops at the conformal parameterization.  The period
+matrix τ and Möbius holonomy computation in conformallab++ extend the pipeline into
+Teichmüller theory territory that geometry-central does not address.
+
+This is the strongest scientific differentiator: conformallab++ can compute
+
+```
+τ = ω_b / ω_a  ∈ ℍ,  SL(2,ℤ)-reduced
+```
+
+for any closed genus-1 surface, and (in Phase 10) the Siegel matrix Ω for genus g≥2.
+No other open-source C++ library does this.  Cross-comparison with geometry-central
+makes this gap explicit and positions conformallab++ as the more complete tool for
+Teichmüller-theoretic applications.
+
+### 5.4 — Spherical geometry as unique contribution
+
+The Spherical geometry mode (angle sums → 4π on a sphere, NSD Hessian with sign
+flip) has no counterpart in geometry-central.  A mathematician interested in
+conformal maps on surfaces of positive curvature (constant curvature +1) has no
+alternative in open-source C++.
+
+### 5.5 — Validation of Springborn 2020 in two independent implementations
+
+Springborn 2020 ("Ideal Hyperbolic Polyhedra and Discrete Uniformization") is the
+shared theoretical reference for both the HyperIdeal geometry mode in conformallab++
+(Phase 2/3) and the hyperbolic component of Gillespie 2021 in geometry-central.
+Cross-checking the ζ-function values (ζ₁₃, ζ₁₄, ζ₁₅) and the resulting angle sums
+on the same meshes would validate both implementations of the paper — a service to
+the discrete geometry community.
+
+---
+
+## 6 — Demarcation: where the comparison ends
+
+| Topic | conformallab++ | geometry-central | Comparable? |
+|---|---|---|---|
+| u-vector at convergence | ✓ | ✓ | ✓ after normalization |
+| UV parameterization | ✓ | ✓ | ✓ up to Möbius |
+| Convergence speed (iterations) | ✓ | ✓ (Newton mode) | ✓ direct |
+| HyperIdeal angle sums | ✓ | ✓ | ✓ |
+| Spherical angle sums | ✓ | ✗ | ✗ |
+| Period matrix τ | ✓ | ✗ | ✗ |
+| Holonomy T_a, T_b | ✓ | ✗ | ✗ |
+| Fundamental domain | ✓ | ✗ | ✗ |
+| Intrinsic Delaunay quality | not tracked | ✓ | — |
+| Mesh topology handling | CGAL halfedge | gc manifold mesh | not comparable |
+| Scalability (large meshes) | not benchmarked yet | benchmarked in paper | comparable if same mesh |
+
+The comparison is meaningful and complete for the **shared conformal core**.
+It ends where conformallab++ continues into Teichmüller theory (holonomy, τ,
+fundamental domain) — that region has no counterpart in geometry-central and
+must be validated by analytic invariants alone (→ `doc/math/validation.md`).
+
+---
+
+## 7 — Practical roadmap for the comparison
+
+| Step | When | What | Effort |
+|---|---|---|---|
+| **GC-1a** | Now | Manual UV comparison on cathead.obj — run both, diff u-vectors | 1 day |
+| **GC-1b** | Now | Add normalization utility to conformallab++ (`normalize_u_vector()`) | 2h |
+| **GC-1c** | After Phase 8 | Automated comparison script (Python or small C++ binary) | 2 days |
+| **GC-2** | After Phase 8 | `ptolemy_delaunay()` preprocessing pass | 1 week |
+| **GC-bench** | After GC-2 | Convergence study: Newton ± Ptolemaic pre-conditioning on 10 meshes | 1 week |
+| **GC-paper** | Phase 10 | Short note on the comparison — period matrix as differentiator | — |
+
+---
+
+## 8 — References
+
+| Reference | Role in this comparison |
+|---|---|
+| **Bobenko, Springborn** — *Variational Principles for Circle Patterns*, Trans. AMS (2004) | Shared variational foundation for all three geometry modes |
+| **Springborn** — *Ideal Hyperbolic Polyhedra and Discrete Uniformization*, DCG (2020) | Mathematical basis for HyperIdeal in conformallab++ AND for Gillespie 2021 |
+| **Gillespie, Springborn, Crane** — *Discrete Conformal Equivalence of Polyhedral Surfaces*, SIGGRAPH (2021) | geometry-central implementation; introduces Ptolemaic flips |
+| **Sharp, Soliman, Crane** — *Navigating Intrinsic Triangulations*, SIGGRAPH (2019) | geometry-central `SignpostIntrinsicTriangulation` — basis for GC-2 |
+| **Sechelmann** — doctoral thesis, TU Berlin (2016) | conformallab++ primary source; covers period matrix, holonomy, all three modes |