Port GradientCheck_Hessian tests: replace GTEST_SKIP stubs with real cross-module checks

Implements the two GTEST_SKIP stubs that tracked the missing analytic
Hessian gradient checks (Java @Ignore ports). Both are now replaced with
live cross-module consistency tests that verify euclidean_gradient() ↔
euclidean_hessian() and spherical_gradient() ↔ spherical_hessian() via
finite-difference comparison.

Result: 176 tests from 35 test suites — 176 PASSED, 0 SKIPPED.

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

code/tests/cgal/test_euclidean_functional.cpp

@@ -6,7 +6,7 @@
 //
 // Test map (Java → C++)
 // ──────────────────────
-//   testHessian (Ignored)       → GradientCheck_Hessian           (SKIPPED)
+//   testHessian (Ignored)       → GradientCheck_Hessian           (ported)
 //   testGradient…Triangle       → GradientCheck_TriangleVertex     (ported)
 //   testGradient…QuadStrip      → GradientCheck_QuadStripVertex    (ported)
 //   testGradient…Tetrahedron    → GradientCheck_TetrahedronVertex  (ported)
@@ -22,6 +22,7 @@
 #include "mesh_builder.hpp"
 #include "euclidean_geometry.hpp"
 #include "euclidean_functional.hpp"
+#include "euclidean_hessian.hpp"
 #include <gtest/gtest.h>
 #include <cmath>
 #include <vector>
@@ -29,12 +30,31 @@
 using namespace conformallab;
 
 // ════════════════════════════════════════════════════════════════════════════
-// @Ignore in Java: no Hessian implemented yet
+// Cross-module Hessian check: euclidean_gradient() ↔ euclidean_hessian()
+//
+// Java @Ignore reason: "no Hessian implemented yet" — the Java functional
+// test was written before the Hessian existed.  In C++ the analytic
+// cotangent-Laplace Hessian (euclidean_hessian.hpp, Phase 3f) is complete.
+//
+// This test verifies cross-module consistency:
+//   H[i,j] ≈ (G_i(x+ε·eⱼ) − G_i(x−ε·eⱼ)) / (2ε)
+// using the gradient from euclidean_functional.hpp and the Hessian from
+// euclidean_hessian.hpp.  A bug in DOF-index mapping or sign convention
+// that affects both modules independently would only be caught here.
 // ════════════════════════════════════════════════════════════════════════════
 
 TEST(EuclideanFunctional, GradientCheck_Hessian)
 {
-    GTEST_SKIP() << "@Ignore in Java – Hessian not yet implemented";
+    auto mesh = make_triangle();
+    auto maps = setup_euclidean_maps(mesh);
+    compute_euclidean_lambda0_from_mesh(mesh, maps);
+    int n = assign_euclidean_vertex_dof_indices(mesh, maps);
+
+    std::vector<double> x(static_cast<std::size_t>(n), -0.1);
+
+    // hessian_check_euclidean: H[i,j] ≈ FD(G)[i,j] using euclidean_gradient()
+    EXPECT_TRUE(hessian_check_euclidean(mesh, x, maps))
+        << "Cross-module: euclidean_gradient() and euclidean_hessian() are inconsistent";
 }
 
 // ════════════════════════════════════════════════════════════════════════════

code/tests/cgal/test_spherical_functional.cpp

@@ -6,7 +6,7 @@
 //
 // Test map (Java → C++)
 // ──────────────────────
-//   testHessian (Ignored)         → GradientCheck_Hessian     (SKIPPED)
+//   testHessian (Ignored)         → GradientCheck_Hessian     (ported)
 //   testGradientWithHyperIdeal…   → GradientCheck_OctaFaceVertex   (ported)
 //   testGradientInExtendedDomain  → GradientCheck_SpherTetVertex    (ported)
 //   testGradientWithHyperelliptic → GradientCheck_SpherTetAllDofs   (ported)
@@ -23,6 +23,7 @@
 #include "conformal_mesh.hpp"
 #include "mesh_builder.hpp"
 #include "spherical_functional.hpp"
+#include "spherical_hessian.hpp"
 #include <gtest/gtest.h>
 #include <cmath>
 #include <vector>
@@ -30,12 +31,29 @@
 using namespace conformallab;
 
 // ════════════════════════════════════════════════════════════════════════════
-// @Ignore in Java: no Hessian implemented
+// Cross-module Hessian check: spherical_gradient() ↔ spherical_hessian()
+//
+// Java @Ignore reason: "no Hessian implemented" — the Java functional test
+// was written before the Hessian existed.  In C++ the analytic spherical
+// Hessian (spherical_hessian.hpp, Phase 3f) is complete.
+//
+// This test verifies cross-module consistency between the functional and
+// the Hessian module.  The spherical Hessian is NSD (negative semi-definite)
+// because the spherical energy is concave — hessian_check_spherical() uses
+// the sign-corrected FD check appropriate for the spherical case.
 // ════════════════════════════════════════════════════════════════════════════
 
 TEST(SphericalFunctional, GradientCheck_Hessian)
 {
-    GTEST_SKIP() << "@Ignore in Java – Hessian not implemented";
+    auto mesh = make_spherical_tetrahedron();
+    auto maps = setup_spherical_maps(mesh);
+    compute_lambda0_from_mesh(mesh, maps);
+    int n = assign_vertex_dof_indices(mesh, maps);
+
+    std::vector<double> x(static_cast<std::size_t>(n), -0.2);
+
+    EXPECT_TRUE(hessian_check_spherical(mesh, x, maps))
+        << "Cross-module: spherical_gradient() and spherical_hessian() are inconsistent";
 }
 
 // ════════════════════════════════════════════════════════════════════════════