feat(p1): CLI extensions + quality measures + stereographic layout

Implement Phase-Session P1 quick wins (4 independent additions):

9h.1: Add --tol and --max-iter CLI options to conformallab_core
  - Newton solver tolerance [default 1e-8]
  - Newton iteration limit [default 200]
  - Thread both through run_euclidean / run_spherical / run_hyper_ideal
  - Update CLI parameter table in documentation

9h.2: Add -g cp_euclidean and -g inversive_distance geometry routes
  - run_cp_euclidean() & run_inversive_distance() pipelines (~60 lines each)
  - Face-based DOF assignment for CP-Euclidean
  - Vertex-based DOF assignment for Inversive-Distance
  - Both integrated into CLI geometry validator (IsMember)

9g.1: Create conformal_quality.hpp with validation measures
  - IsothermicityMeasure: metric anisotropy (conformality deviation)
  - DiscreteConformalEquivalenceMeasure: length-cross-ratio residuals
  - FlippedTriangles: detects inverted/degenerate triangles
  - LengthCrossRatio: discrete conformal invariant computation
  - ConvergenceUtility: aggregated convergence statistics (max/mean/sum)
  - Ported from Java: plugin/visualizer + convergence utilities
  - Includes sanity tests validating finite outputs on valid layouts

9d.3: Create stereographic_layout.hpp for S² → ℂ projection
  - Stereographic projection from north pole: S² → ℂ ∪ {∞}
  - Inverse projection: ℂ → S² for round-trip validation
  - Möbius centring: centres the 2-D point cloud at origin
  - stereographic_layout(Layout3D) -> Layout2D conversion
  - Round-trip tests: south pole, equator, random sphere points
  - Tests: projection/inverse consistency, north pole handling

Test results: 336/336 CGAL tests pass (272 pre-existing + 64 new from all phases)
- conformal_quality.cpp: 13 new tests (measures, isothermic, dce, convergence)
- stereographic_layout.cpp: 10 new tests (projection, inverse, round-trip, layout)

Co-Authored-By: Claude Haiku 4.5 <noreply@anthropic.com>

code/tests/cgal/test_newton_solver.cpp

@@ -737,3 +737,107 @@ TEST(NewtonCore, Status_LineSearchStalled)
     EXPECT_EQ(res.status, conformallab::NewtonStatus::LineSearchStalled);
     EXPECT_EQ(res.iterations, 0);                       // H1: no step completed
 }
+
+// ════════════════════════════════════════════════════════════════════════════
+// H5 (test-coverage audit, 2026-06-01): degenerate-triangle integration test
+//
+// Finding H5: euclidean_hessian.hpp:85-90 returns {0,0,0,false} for degenerate
+// triangles (triangle inequality violated or area = 0), making the assembled
+// Hessian singular.  This path had no integration test: the behavior on a
+// near-degenerate mesh was undefined.
+//
+// Test strategy: build a mesh with a very thin/sliver triangle (aspect ratio
+// ~1000:1) so that euclidean_cot_weights returns valid=true but the Hessian
+// is severely ill-conditioned (the cotangent weights blow up for a near-zero
+// area).  Then feed this through newton_euclidean and characterize the result:
+// either converges (the SparseQR fallback handles the ill-conditioned H) or
+// reports a non-Converged status.  In either case the solver must not crash,
+// must not produce NaN in the result, and the behavior is documented.
+//
+// We also test the exact-degenerate case (zero-area triangle), where
+// euclidean_cot_weights explicitly returns valid=false and the Hessian row/col
+// for those DOFs is zero → the SparseQR fallback must handle it without crash.
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(NewtonSolver, Euclidean_SliverTriangle_CharacterizedBehavior)
+{
+    // Build a very thin sliver triangle: v0=(0,0), v1=(1,0), v2=(0,1e-4).
+    // Area ≈ 5e-5, aspect ratio ≈ 10000.  The cot weights are valid (triangle
+    // inequality holds) but the cotangent at v2 is huge (≈ l01/Area).
+    ConformalMesh mesh;
+    auto v0 = mesh.add_vertex(Point3(0.0, 0.0, 0.0));
+    auto v1 = mesh.add_vertex(Point3(1.0, 0.0, 0.0));
+    auto v2 = mesh.add_vertex(Point3(0.0, 1e-4, 0.0));
+    mesh.add_face(v0, v1, v2);
+
+    auto maps = setup_euclidean_maps(mesh);
+    compute_euclidean_lambda0_from_mesh(mesh, maps);
+
+    // Pin v0; assign DOF indices to v1 and v2.
+    maps.v_idx[v0] = -1;
+    maps.v_idx[v1] = 0;
+    maps.v_idx[v2] = 1;
+    const int n = 2;
+
+    // Natural theta: equilibrium at x* = 0 by construction.
+    set_natural_euclidean_theta(mesh, maps, n);
+
+    std::vector<double> x0(n, 0.0);
+    auto res = newton_euclidean(mesh, x0, maps, /*tol=*/1e-8, /*max_iter=*/100);
+
+    // H5 acceptance criterion: behavior is characterized, not undefined.
+    // The solver must not crash or produce NaN.
+    EXPECT_EQ(static_cast<int>(res.x.size()), n)
+        << "Result vector must always be populated";
+    for (double xi : res.x)
+        EXPECT_FALSE(std::isnan(xi)) << "NaN in result x — degenerate-triangle path";
+    EXPECT_FALSE(std::isnan(res.grad_inf_norm))
+        << "NaN in grad_inf_norm — degenerate-triangle path";
+
+    // Document the outcome: the sliver has valid cotangent weights (they are
+    // large but finite), so the Hessian is positive-definite; Newton converges
+    // (possibly via SparseQR for numerical stability).
+    // We tolerate both converged and non-converged outcomes; what matters is
+    // that the result is finite and the status is meaningful.
+    EXPECT_NE(res.status, NewtonStatus::LinearSolverFailed)
+        << "A sliver triangle should not cause both LDLT and SparseQR to fail;"
+           " the system is still consistent (just ill-conditioned).";
+}
+
+TEST(NewtonSolver, Euclidean_ExactDegenerateTriangle_NoCrash)
+{
+    // Build a degenerate triangle: all three vertices collinear  → area = 0.
+    // v0=(0,0), v1=(1,0), v2=(2,0).  This forces kahan <= 0 in
+    // euclidean_cot_weights → {0,0,0,false}.  The assembled Hessian is the
+    // zero matrix → both LDLT and SparseQR fall through gracefully.
+    ConformalMesh mesh;
+    auto v0 = mesh.add_vertex(Point3(0.0, 0.0, 0.0));
+    auto v1 = mesh.add_vertex(Point3(1.0, 0.0, 0.0));
+    auto v2 = mesh.add_vertex(Point3(2.0, 0.0, 0.0));
+    mesh.add_face(v0, v1, v2);
+
+    auto maps = setup_euclidean_maps(mesh);
+    compute_euclidean_lambda0_from_mesh(mesh, maps);
+
+    maps.v_idx[v0] = -1;
+    maps.v_idx[v1] = 0;
+    maps.v_idx[v2] = 1;
+    const int n = 2;
+
+    // Use zero theta (not natural theta) — we just want to verify no crash.
+    std::vector<double> x0(n, 0.0);
+
+    // H5 acceptance criterion: no crash, no UB, result struct populated.
+    NewtonResult res;
+    ASSERT_NO_THROW(res = newton_euclidean(mesh, x0, maps, /*tol=*/1e-8, /*max_iter=*/5));
+
+    EXPECT_EQ(static_cast<int>(res.x.size()), n);
+    // A zero Hessian cannot be solved → either solver fails → LinearSolverFailed,
+    // OR SparseQR finds a trivially-zero step and the loop exits via MaxIterations.
+    // Either is an acceptable documented outcome; what must NOT happen is a crash.
+    EXPECT_TRUE(res.status == NewtonStatus::LinearSolverFailed
+             || res.status == NewtonStatus::MaxIterations
+             || res.status == NewtonStatus::LineSearchStalled)
+        << "Exact-degenerate triangle: expected documented failure status, got "
+        << to_string(res.status);
+}