docs(euclidean-hessian): fix wrong cotangent formula in two comment blocks

Finding-C from doc/reviewer/external-audit-2026-05-30.md.

The box comment at the top and the function-level comment above
euclidean_cot_weights() both stated:

  cot_k = (t_adj1·l123 − t_adj2·t_opp) / denom2   ← WRONG

The correct formula (verified numerically on a 3-4-5 right triangle,
expected cot1=4/3, cot2=3/4, cot3=0) is:

  cot_k = (t_opp · l123 − t_a · t_b) / denom2

where t_opp is the t-value of the edge OPPOSITE vertex k, and t_a/t_b
are the t-values of the two edges ADJACENT to vertex k.

The implementation in euclidean_cot_weights() was already correct;
only the documentation was wrong.

Changes (documentation only, zero code changes):
- Box comment: rewritten with correct formula and explicit per-vertex
  assignment (cot1: t_opp=t23, t_a=t12, t_b=t31; etc.)
- Box comment: added missing ½ factor to Hessian contribution lines
- Function-level comment: corrected to (t_opp·l123 − t_a·t_b)/(8·Area)
  with a pointer to the box comment for the full assignment
- Inline return comment in euclidean_cot_weights(): now shows the
  mapping (cot1: t_opp=t23, t_a=t12, t_b=t31) directly at the formula

263/263 CGAL tests pass, 0 failed.

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

code/include/euclidean_hessian.hpp

@@ -17,18 +17,21 @@
 // │  side lengths lij = exp(Λ̃ij/2):                                         │
 // │                                                                          │
 // │    t12 = −l12+l23+l31,  t23 = l12−l23+l31,  t31 = l12+l23−l31         │
-// │    denom2 = 2·sqrt(t12·t23·t31·l123) = 8·Area                           │
+// │    l123 = l12+l23+l31,  denom2 = 2·sqrt(t12·t23·t31·l123) = 8·Area     │
 // │                                                                          │
-// │    cot_k = (t_adj1·l123 − t_adj2·t_opp) / denom2                        │
+// │  Cotangent at vertex k (opposite t_opp, adjacent t_a and t_b):          │
-// │          = cotangent of the angle αk at vertex k                         │
+// │    cot_k = (t_opp · l123 − t_a · t_b) / denom2                         │
 // │                                                                          │
-// │  Hessian contributions per face:                                         │
+// │  Assignment (k ↔ opposite edge ↔ opposite t-value):                     │
-// │    H[vi, vi] += cot_vj + cot_vk         (diagonal, both non-opp angles) │
+// │    cot1 (v1, opp l23): t_opp=t23, t_a=t12, t_b=t31                     │
-// │    H[vi, vj] -= cot_vk                  (off-diagonal, for variable vi,vj│
+// │    cot2 (v2, opp l31): t_opp=t31, t_a=t12, t_b=t23                     │
+// │    cot3 (v3, opp l12): t_opp=t12, t_a=t23, t_b=t31                     │
 // │                                                                          │
-// │  This is exactly the cotangent-Laplace operator from Pinkall–Polthier.   │
+// │  Hessian contributions per face (Pinkall–Polthier ½ factor):            │
+// │    H[vi, vi] += (cot_vj + cot_vk) / 2   (diagonal)                     │
+// │    H[vi, vj] −= cot_vk / 2              (off-diagonal, variable pairs)  │
 // │                                                                          │
-// │  Pinned vertices (v_idx = −1) contribute to diagonal of neighbours but   │
+// │  Pinned vertices (v_idx = −1) contribute to diagonal of neighbours but  │
 // │  do not create a column/row in H themselves.                             │
 // └──────────────────────────────────────────────────────────────────────────┘
 //
@@ -54,7 +57,11 @@ namespace conformallab {
 // Given three Euclidean SIDE LENGTHS l12, l23, l31 (already exp(Λ̃/2)),
 // return the three cotangent weights (cot1, cot2, cot3).
 //
-// cot_k = (t_adj·l123 − t_opp·t_other) / (8·Area)
+// cot_k = (t_opp · l123 − t_a · t_b) / (8·Area)
+//
+// where t_opp is the t-value of the edge OPPOSITE vertex k, and t_a, t_b are
+// the t-values of the two edges ADJACENT to vertex k.  See the box comment at
+// the top of this file for the explicit assignment of t_opp/t_a/t_b per vertex.
 //
 // Returns {0,0,0} for degenerate faces (triangle inequality violated or Area=0).
 /// Three Euclidean cotangent weights `(cot1, cot2, cot3)` for the
@@ -85,9 +92,10 @@ inline EuclCotWeights euclidean_cot_weights(double l12, double l23, double l31)
     // denom2 = 2·sqrt(t12·t23·t31·l123) = 8·Area
     const double denom2 = 2.0 * std::sqrt(denom2_sq);
 
-    // cot at v1 (opposite l23): adjacent t-values are t12 and t31.
+    // Formula: cot_k = (t_opp · l123 − t_a · t_b) / denom2
-    // cot at v2 (opposite l31): adjacent t-values are t12 and t23.
+    //   cot1: t_opp=t23, t_a=t12, t_b=t31   (v1 opposite l23)
-    // cot at v3 (opposite l12): adjacent t-values are t23 and t31.
+    //   cot2: t_opp=t31, t_a=t12, t_b=t23   (v2 opposite l31)
+    //   cot3: t_opp=t12, t_a=t23, t_b=t31   (v3 opposite l12)
     return {
         (t23 * l123 - t31 * t12) / denom2,  // cot1
         (t31 * l123 - t12 * t23) / denom2,  // cot2

doc/reviewer/external-audit-2026-05-30.md

@@ -701,7 +701,7 @@ index. Add a comment:
 |----|------|-------|------|----------|--------|
 | A | `hyper_ideal_functional.hpp` | 319–344 | Bug | Critical (mixed config) | ✅ Fixed 2026-05-30 |
 | B | `gauss_bonnet.hpp` | 87–88, 128–134 | API error | Medium | ✅ Fixed 2026-05-31 |
-| C | `euclidean_hessian.hpp` | 26–27, 57–58 | Doc error | Medium | 🟡 Open |
+| C | `euclidean_hessian.hpp` | 26–27, 57–58 | Doc error | Medium | ✅ Fixed 2026-05-31 |
 | D | `euclidean_functional.hpp` + 2 others | 97–107 | Doc error | Medium | 🟡 Open |
 | E | `cp_euclidean_functional.hpp` | 338–349, 373–387 | Inconsistency | Medium | 🟡 Open |
 | F | test files | — | Test gap | Medium | 🟠 Open |