Phase 9b: Hyper-ideal Hessian — block-FD optimisation (96× speed-up)

Replaces the O(n·F) full-FD Hessian with an O(F·36) block-local variant
that exploits the per-face locality of the hyper-ideal functional.  Both
variants are kept (full-FD as correctness reference, block-FD as default)
and proven to match to FD rounding tolerance on all test configurations.

Java parity note
────────────────
HyperIdealFunctional.java line 295-298 declares:
    public boolean hasHessian() { return false; }
i.e. the upstream Java functional has NO Hessian implementation, analytic
or numerical.  Both Hessian variants in this file are conformallab++
extensions beyond the Java port.  Analytic Hessian via Schläfli-type
differentiation through (b_i, a_e) → l_ij → ζ_13/ζ_14/ζ_15 → α_ij/β_i
is deferred to a future PR.

Implementation
──────────────
* code/include/hyper_ideal_functional.hpp
  - New pure-math helper face_angles_from_local_dofs() takes 6 input DOFs
    (b1, b2, b3, a12, a23, a31) + variability flags and returns the 6
    output angles (β1, β2, β3, α12, α23, α31).
  - Used by block-FD Hessian as the inner loop; identical semantics to
    the existing compute_face_angles().

* code/include/hyper_ideal_hessian.hpp
  - hyper_ideal_hessian_block_fd()  — new, default production path
  - hyper_ideal_hessian_block_fd_sym() — symmetrised variant
  - hyper_ideal_hessian()  — full-FD baseline, kept for cross-validation
  - hyper_ideal_hessian_sym() — symmetrised baseline
  - Header docblock documents speed-up curve: ~33× at cathead.obj scale,
    ~1166× at brezel.obj scale.

Tests (7 new in test_hyper_ideal_hessian.cpp)
─────────────────────────────────────────────
* PureHelperMatchesMeshHelper — refactor sanity
* BlockFD_MatchesFullFD_ClosedTetrahedron
* BlockFD_MatchesFullFD_Open3FaceMesh (boundary edge path)
* BlockFD_MatchesFullFD_PinnedDOFs    (partial-DOF path)
* BlockFD_IsPSD                       (Springborn 2020 convexity)
* BlockFD_SparsityMatchesFaceAdjacency (structural correctness)
* BlockFD_FasterThanFullFD           (performance assertion: ≥ 3×)

Measured speed-up on the 200-face tet strip (603 DOFs):
    full-FD:   226 591 µs
    block-FD:    2 347 µs
    ratio:        96.5×
The assertion uses ≥ 3× to leave wide CI-hardware tolerance.

Test count
──────────
CGAL suite: 184 → 191 (+7).  Zero skips.

Why not full analytic now
─────────────────────────
Full analytic Hessian via the chain rule
    (b_i, a_e) → l_ij → ζ_{13,14,15} → α_ij / β_i
requires Schläfli-type differentiation with multiple cases for the
ideal / hyper-ideal vertex mix.  It would add another ~6× over
block-FD but at significantly higher implementation and verification
cost.  Block-FD already removes the practical bottleneck for meshes
up to ~10k faces; analytic optimisation can land later when justified
by a concrete profiling result.

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

code/include/hyper_ideal_functional.hpp

@@ -105,6 +105,76 @@ static inline std::size_t hidx(Halfedge_index h)
     return static_cast<std::size_t>(static_cast<std::uint32_t>(h));
 }
 
+// ── Pure-math face-angle kernel ──────────────────────────────────────────────
+//
+// Computes the six per-face angle outputs (β₁, β₂, β₃, α₁₂, α₂₃, α₃₁) from
+// the six local DOF inputs (b₁, b₂, b₃, a₁₂, a₂₃, a₃₁) and the variability
+// flags (vᵢb).  This is the pure functional core of `compute_face_angles`
+// — no mesh, no property maps, no global x vector.
+//
+// Why exposed as a free function (Phase 9b):
+// ─────────────────────────────────────────
+// The block-FD Hessian (`hyper_ideal_hessian_block_fd`) perturbs only the
+// 6 DOFs adjacent to a single face at a time, recomputes the 6 angle
+// outputs of that face, and uses the local 6×6 Jacobian to scatter into
+// the global Hessian.  Working through a pure 6→6 function (instead of
+// perturbing the full x and re-running the gradient over all faces)
+// reduces the cost of the Hessian from O(F·n) to O(F·36).
+//
+// The clamping logic (negative b → 0.01, negative a → 0) mirrors
+// HyperIdealFunctional.java's defensive behaviour (lines 122-127 of the
+// Java original); this keeps the FD perturbation regime well-defined.
+
+struct FaceAngleOutputs {
+    double beta1, beta2, beta3;       ///< interior angles at v₁,v₂,v₃
+    double alpha12, alpha23, alpha31; ///< dihedral angles at e₁₂,e₂₃,e₃₁
+};
+
+inline FaceAngleOutputs face_angles_from_local_dofs(
+    double b1, double b2, double b3,
+    double a12, double a23, double a31,
+    bool   v1b, bool v2b, bool v3b)
+{
+    // Same defensive clamps as compute_face_angles.
+    if (v1b && v2b && a12 < 0.0) a12 = 0.0;
+    if (v2b && v3b && a23 < 0.0) a23 = 0.0;
+    if (v3b && v1b && a31 < 0.0) a31 = 0.0;
+    if (v1b && b1  < 0.0) b1 = 0.01;
+    if (v2b && b2  < 0.0) b2 = 0.01;
+    if (v3b && b3  < 0.0) b3 = 0.01;
+
+    double l12 = lij(b1, b2, a12, v1b, v2b);
+    double l23 = lij(b2, b3, a23, v2b, v3b);
+    double l31 = lij(b3, b1, a31, v3b, v1b);
+
+    if (l12 < 1E-12 && l23 < 1E-12 && l31 < 1E-12)
+        l12 = l23 = l31 = 1E-12;
+
+    FaceAngleOutputs o;
+
+    if (l12 > l23 + l31) {
+        o.beta1 = 0.0; o.beta2 = 0.0; o.beta3 = PI;
+        o.alpha12 = PI; o.alpha23 = 0.0; o.alpha31 = 0.0;
+    } else if (l23 > l12 + l31) {
+        o.beta1 = PI; o.beta2 = 0.0; o.beta3 = 0.0;
+        o.alpha12 = 0.0; o.alpha23 = PI; o.alpha31 = 0.0;
+    } else if (l31 > l12 + l23) {
+        o.beta1 = 0.0; o.beta2 = PI; o.beta3 = 0.0;
+        o.alpha12 = 0.0; o.alpha23 = 0.0; o.alpha31 = PI;
+    } else {
+        o.beta1 = zeta(l12, l31, l23);
+        o.beta2 = zeta(l23, l12, l31);
+        o.beta3 = zeta(l31, l23, l12);
+        o.alpha12 = alpha_ij(a12, a23, a31, b1, b2, b3,
+                             o.beta1, o.beta2, o.beta3, v1b, v2b, v3b);
+        o.alpha23 = alpha_ij(a23, a31, a12, b2, b3, b1,
+                             o.beta2, o.beta3, o.beta1, v2b, v3b, v1b);
+        o.alpha31 = alpha_ij(a31, a12, a23, b3, b1, b2,
+                             o.beta3, o.beta1, o.beta2, v3b, v1b, v2b);
+    }
+    return o;
+}
+
 // ── Per-face angle kernel ─────────────────────────────────────────────────────
 
 struct FaceAngles {