feat(phase3d+3e): port EuclideanCyclicFunctional + add SphericalFunctional gauge-fix
Phase 3d — EuclideanCyclicFunctional:
• euclidean_geometry.hpp: t-value / atan2 corner-angle formula with
centering trick (μ = (Λ̃₁₂+Λ̃₂₃+Λ̃₃₁)/6) for numerical stability
• euclidean_functional.hpp: EuclideanMaps bundle, gradient (G_v = Θ_v − Σα_v,
G_e = α_opp⁺ + α_opp⁻ − φ_e), 10-point GL path-integral energy,
gradient_check_euclidean — identical halfedge convention to SphericalFunctional
• test_euclidean_functional.cpp: 11 tests (1 skip) covering angle formula,
right-isosceles triangle, angle sum = π, degenerate detection, gradient
checks on triangle/quad-strip/tetrahedron/fan-5/mixed-pinned, NaN check
Phase 3e — Spherical gauge-fix:
• spherical_gauge_shift(): Newton + backtracking line search to find t*
where ΣG_v(x + t·1) = 0 (maximises E along the global scale direction);
bisection used when sign change is detectable, Newton+backtrack otherwise
• apply_spherical_gauge(): in-place wrapper
• 3 new tests: GaugeFix_SpherTetVertexZerosSumGv, GaugeFix_ApplyInPlace,
GaugeFix_AlreadyAtGaugeReturnsTNearZero
Total: 45 cgal tests pass, 3 skipped (@Ignore Hessian stubs, one per functional)
Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>
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