fix(gauss-bonnet): delete HyperIdeal overloads — wrong identity for hyperbolic metrics

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

The Euclidean Gauss–Bonnet identity Σ(2π−Θ_v) = 2π·χ is ONLY valid for
flat/Euclidean and spherical metrics.  For hyperbolic metrics (HyperIdeal)
the correct identity is Σ(2π−Θ_v) − Area(M) = 2π·χ.  The previously
provided gauss_bonnet_sum(HyperIdealMaps) overload would silently pass the
wrong LHS to check_gauss_bonnet, which would always throw "deficit = ±Area"
for valid hyperbolic targets.

Fix:
- gauss_bonnet_sum(mesh, HyperIdealMaps)        → = delete + explanation comment
- enforce_gauss_bonnet(mesh, HyperIdealMaps&)   → = delete + explanation comment
- Header box comment rewritten with the correct hyperbolic Gauss–Bonnet
  identity and a clear "HyperIdeal: NOT SUPPORTED" section

New test in test_phase6.cpp:
- HyperIdeal_EuclideanSumDiscrepancy_DocumentsWhyCheckIsDeleted
  verifies numerically that the Euclidean sum = 0 but 2π·χ = 4π for a
  regular tetrahedron, documenting the −4π discrepancy that motivated
  the deletion
- Three compile-time static_asserts (SFINAE) confirm the overload is not
  invocable with HyperIdealMaps but remains so with Euclidean/SphericalMaps

263/263 CGAL tests pass, 0 failed.

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

code/tests/cgal/test_phase6.cpp

@@ -137,6 +137,78 @@ TEST(GaussBonnet, ManuallySetAnalyticalTheta_PassesCheck)
     EXPECT_NO_THROW(check_gauss_bonnet(m, maps));
 }
 
+// ════════════════════════════════════════════════════════════════════════════
+// GaussBonnet — HyperIdeal API guard (Finding-B from external-audit-2026-05-30)
+//
+// gauss_bonnet_sum(mesh, HyperIdealMaps) and
+// enforce_gauss_bonnet(mesh, HyperIdealMaps) are intentionally DELETED.
+//
+// Reason: the correct hyperbolic Gauss–Bonnet identity is
+//   Σ_v (2π − Θ_v) − Area(M) = 2π · χ(M)
+// not the Euclidean form Σ(2π−Θ_v) = 2π·χ.  For a regular (Θ_v=2π) genus-2
+// surface: Σ(2π−2π)=0 but 2π·χ=−4π, so the Euclidean check would always
+// throw "deficit = 4π" for a perfectly valid HyperIdeal target.
+//
+// Compile-time enforcement: gauss_bonnet_sum / enforce_gauss_bonnet with
+// HyperIdealMaps are = delete, so any accidental call is a compile error.
+// The static_asserts below confirm this is wired correctly.
+//
+// The runtime test shows the discrepancy numerically: even for a regular
+// tetrahedron where all Θ_v=2π (a valid all-hyper-ideal starting point),
+// the "Euclidean sum" is 0 but the correct RHS for χ=2 is 4π — the
+// Euclidean check would be off by 4π.
+// ════════════════════════════════════════════════════════════════════════════
+
+// Invocability check via SFINAE: tries to form the call expression in an
+// unevaluated context; a deleted function causes substitution failure.
+namespace {
+    template <typename Maps>
+    auto try_gb_sum(int) -> decltype(gauss_bonnet_sum(
+        std::declval<const ConformalMesh&>(),
+        std::declval<const Maps&>()), std::true_type{});
+    template <typename>
+    std::false_type try_gb_sum(...);
+}  // namespace
+
+static_assert(!decltype(try_gb_sum<HyperIdealMaps>(0))::value,
+    "gauss_bonnet_sum must NOT be invocable with HyperIdealMaps");
+static_assert( decltype(try_gb_sum<EuclideanMaps>(0))::value,
+    "gauss_bonnet_sum must still be invocable with EuclideanMaps");
+static_assert( decltype(try_gb_sum<SphericalMaps>(0))::value,
+    "gauss_bonnet_sum must still be invocable with SphericalMaps");
+
+TEST(GaussBonnet, HyperIdeal_EuclideanSumDiscrepancy_DocumentsWhyCheckIsDeleted)
+{
+    // On a tetrahedron (χ=2) with all Θ_v = 2π:
+    //   Euclidean sum Σ(2π−2π) = 0
+    //   but 2π·χ = 4π
+    //   deficit (Euclidean formula) = 0 − 4π = −4π  ← WRONG check for HyperIdeal
+    //
+    // The HyperIdeal identity is Σ(2π−Θ_v) − Area = 2π·χ.
+    // Area > 0 for any non-degenerate hyperbolic metric, so the real deficit
+    // would be much smaller.  This test documents the mismatch numerically
+    // so any future re-introduction of the HyperIdeal overload is caught.
+    auto m = make_tetrahedron();
+    auto hi_maps = setup_hyper_ideal_maps(m);
+    // All theta_v default to 2π (regular vertex target).
+
+    // Access the raw property map directly (not via the deleted bundle overload)
+    // to compute the Euclidean-style sum — just for documentation purposes.
+    double euclid_sum = gauss_bonnet_sum(m, hi_maps.theta_v);  // raw map: OK
+    double rhs        = gauss_bonnet_rhs(m);                   // 2π·χ = 4π
+
+    EXPECT_NEAR(euclid_sum, 0.0, 1e-12)   // Σ(2π−2π) = 0
+        << "Expected Euclidean sum = 0 for all-regular HyperIdeal targets";
+    EXPECT_NEAR(rhs, 4.0 * M_PI, 1e-12)  // 2π·χ(tetrahedron) = 4π
+        << "Expected RHS = 4π for tetrahedron (χ=2)";
+
+    // The Euclidean deficit would be 0 − 4π = −4π: completely wrong for HyperIdeal.
+    // If check_gauss_bonnet were called with HyperIdealMaps it would ALWAYS throw
+    // here, even though Θ_v=2π is a valid regular-vertex HyperIdeal target.
+    EXPECT_NEAR(euclid_sum - rhs, -4.0 * M_PI, 1e-10)
+        << "Euclidean deficit for HyperIdeal target = −4π: confirms the deleted API is correct";
+}
+
 // ════════════════════════════════════════════════════════════════════════════
 // CutGraph — tree-cotree algorithm
 // ════════════════════════════════════════════════════════════════════════════