fix(num): N5 stable triangle area (Kahan) in euclidean_cot_weights

The cotangent weights divided by 2·√(t12·t23·t31·l123).  For a needle/cap
(sliver) triangle one of the t-values is the difference of near-equal edge
lengths → catastrophic cancellation, and the area under the sqrt loses
precision, feeding large relative error into every cotangent weight, the
Hessian, and the linear solve (numerical-stability audit N5).

Replace the area computation with Kahan's stable side-length formula
(sort a≥b≥c, evaluate ¼·√[(a+(b+c))(c−(a−b))(c+(a−b))(a+(b−c))]).  The
denominator is still exactly 8·Area for well-shaped triangles but accurate
for slivers.  The triangle-inequality guard and the cotangent numerators are
unchanged.

Test: CotWeights_SliverMatchesHighPrecisionReference cross-checks a thin
triangle (apex ≈ 0.01 rad) against a long-double law-of-cosines reference.
292/292 CGAL tests pass.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>

code/include/euclidean_hessian.hpp

@@ -17,7 +17,8 @@
 // │  side lengths lij = exp(Λ̃ij/2):                                         │
 // │                                                                          │
 // │    t12 = −l12+l23+l31,  t23 = l12−l23+l31,  t31 = l12+l23−l31         │
-// │    l123 = l12+l23+l31,  denom2 = 2·sqrt(t12·t23·t31·l123) = 8·Area     │
+// │    l123 = l12+l23+l31,  denom2 = 8·Area  (Area via Kahan's stable      │
+// │                          side-length formula — see euclidean_cot_weights)│
 // │                                                                          │
 // │  Cotangent at vertex k (opposite t_opp, adjacent t_a and t_b):          │
 // │    cot_k = (t_opp · l123 − t_a · t_b) / denom2                         │
@@ -48,6 +49,7 @@
 #include <vector>
 #include <array>
 #include <cmath>
+#include <algorithm>
 #include <stdexcept>
 
 namespace conformallab {
@@ -85,12 +87,26 @@ inline EuclCotWeights euclidean_cot_weights(double l12, double l23, double l31)
     if (t12 <= 0.0 || t23 <= 0.0 || t31 <= 0.0)
         return {0.0, 0.0, 0.0, false};
 
-    const double l123   = l12 + l23 + l31;
+    const double l123 = l12 + l23 + l31;
-    const double denom2_sq = t12 * t23 * t31 * l123;
-    if (denom2_sq <= 0.0) return {0.0, 0.0, 0.0, false};
 
-    // denom2 = 2·sqrt(t12·t23·t31·l123) = 8·Area
+    // Triangle area via Kahan's numerically stable formula.  Sort the side
-    const double denom2 = 2.0 * std::sqrt(denom2_sq);
+    // lengths a ≥ b ≥ c and evaluate with the cancellation-avoiding grouping
+    //   Area = ¼·√[ (a+(b+c))·(c−(a−b))·(c+(a−b))·(a+(b−c)) ].
+    // This replaces the naive 2·√(t12·t23·t31·l123): that product of
+    // near-equal-length differences loses precision for needle/cap triangles,
+    // where it would feed large relative error straight into the cotangent
+    // weights and the linear solve.  The result denom2 = 8·Area is identical
+    // up to rounding for well-shaped triangles, but accurate for slivers.
+    double a = l12, b = l23, c = l31;
+    if (a < b) std::swap(a, b);
+    if (a < c) std::swap(a, c);
+    if (b < c) std::swap(b, c);   // now a ≥ b ≥ c
+
+    const double kahan = (a + (b + c)) * (c - (a - b))
+                       * (c + (a - b)) * (a + (b - c));
+    if (kahan <= 0.0) return {0.0, 0.0, 0.0, false};
+
+    const double denom2 = 8.0 * (0.25 * std::sqrt(kahan));   // = 8·Area
 
     // Formula: cot_k = (t_opp · l123 − t_a · t_b) / denom2
     //   cot1: t_opp=t23, t_a=t12, t_b=t31   (v1 opposite l23)

code/tests/cgal/test_euclidean_hessian.cpp

@@ -61,6 +61,47 @@ TEST(EuclideanHessian, CotWeights_RightIsoscelesTriangle)
     EXPECT_NEAR(cw.cot3, 1.0, 1e-12);   // 45° at v3
 }
 
+// ════════════════════════════════════════════════════════════════════════════
+// N5: cotangent weights on a SLIVER triangle match a high-precision reference.
+//
+// The area is now computed by Kahan's stable side-length formula instead of
+// the naive 2·√(t12·t23·t31·l123).  On a thin (sliver) triangle the naive
+// product of near-equal-length differences loses precision, biasing every
+// cotangent weight.  Here we cross-check against an independent law-of-cosines
+// reference in long double (80-bit on x86 CI — a genuine high-precision oracle;
+// equal to double on ARM64, where it still serves as a regression cross-check).
+// ════════════════════════════════════════════════════════════════════════════
+
+TEST(EuclideanHessian, CotWeights_SliverMatchesHighPrecisionReference)
+{
+    // Thin isosceles: short base l12, two near-unit legs (apex angle ≈ 0.01 rad).
+    const double l12 = 0.01, l23 = 1.0, l31 = 1.0;
+    auto cw = euclidean_cot_weights(l12, l23, l31);
+    ASSERT_TRUE(cw.valid);
+
+    // cot at the vertex opposite side `opp`, with adjacent sides s1, s2:
+    //   cos = (s1² + s2² − opp²) / (2·s1·s2),  sin = √((1−cos)(1+cos)).
+    auto cot_ref = [](long double opp, long double s1, long double s2) {
+        long double cosA = (s1 * s1 + s2 * s2 - opp * opp) / (2.0L * s1 * s2);
+        long double sinA = std::sqrt((1.0L - cosA) * (1.0L + cosA));
+        return static_cast<double>(cosA / sinA);
+    };
+    const double c1 = cot_ref(l23, l12, l31);  // v1 opposite l23
+    const double c2 = cot_ref(l31, l12, l23);  // v2 opposite l31
+    const double c3 = cot_ref(l12, l23, l31);  // v3 opposite l12 (needle angle)
+
+    auto rel = [](double a, double b) {
+        return std::abs(a - b) / std::max(1.0, std::abs(b));
+    };
+    EXPECT_LT(rel(cw.cot1, c1), 1e-9);
+    EXPECT_LT(rel(cw.cot2, c2), 1e-9);
+    EXPECT_LT(rel(cw.cot3, c3), 1e-9);
+
+    EXPECT_TRUE(std::isfinite(cw.cot1) &&
+                std::isfinite(cw.cot2) &&
+                std::isfinite(cw.cot3));
+}
+
 // ════════════════════════════════════════════════════════════════════════════
 // Hessian is symmetric: H[i,j] == H[j,i]
 // ════════════════════════════════════════════════════════════════════════════