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feat(euclidean): block-FD edge-DOF Hessian → cyclic Newton + Java convergence oracle

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Implements the edge-DOF (cyclic) Euclidean Hessian, unblocking the full cyclic
Newton solve, and enables the Java EuclideanCyclicConvergenceTest cross-validation.

- euclidean_hessian.hpp: `euclidean_hessian_block_fd` / `_sym` — per-face 6×6
  block FD over (u1,u2,u3,λ12,λ23,λ31), mirroring hyper_ideal_hessian_block_fd.
  Per-face outputs carry the gradient signs (−α vertex, +α_opp edge), so the
  result equals ∂G/∂x by construction (locality lemma). Analytic vertex-only
  cotangent Hessian unchanged (still used for vertex-only layouts).
- newton_solver.hpp: newton_euclidean routes cyclic layouts (edge DOFs present)
  through the block-FD Hessian; vertex-only path unchanged.
- tests:
  * CyclicCircularEdge_CatHead_JavaXVal (now GREEN) — prescribe φ=π−0.1 on one
    interior edge, solve, assert realised α_opp+α_opp = π−0.1 @1e-9.
  * CyclicCircularEdge_PhiEntersGradient_CatHead — solver-free φ-wiring check.
  * CyclicHessian_BlockFD_MatchesGradientFD_Tetrahedron — Hessian correctness.

243/243 cgal tests pass; vertex-only Euclidean Newton unaffected.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
This commit is contained in:
Tarik Moussa
2026-05-30 00:09:37 +02:00
committed by Tarik Moussa
parent 5d74a94b78
commit ea5f01d73d
4 changed files with 210 additions and 52 deletions
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111
code/include/euclidean_hessian.hpp
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@@ -43,6 +43,7 @@
#include "euclidean_functional.hpp"
#include <Eigen/Sparse>
#include <vector>
#include <array>
#include <cmath>
#include <stdexcept>
@@ -187,6 +188,116 @@ inline Eigen::SparseMatrix<double> euclidean_hessian(
return H;
}
// ── Block-FD Hessian (cyclic: vertex + edge DOFs) ────────────────────────────
//
// The analytic `euclidean_hessian` above covers only the vertex-block cotangent
// Laplacian. The *cyclic* functional adds edge DOFs λ_e, giving vertex-edge and
// edge-edge Hessian blocks. We obtain the full Hessian by per-face block FD,
// mirroring `hyper_ideal_hessian_block_fd`.
//
// Locality lemma: the gradient decomposes by face,
// G_v = Θ_v − Σ_{f∋v} α_v(f) (vertex output = −α_v)
// G_e = Σ_{f∋e} α_opp(f) − φ_e (edge output = +α_opp)
// and α at face f depends ONLY on f's 6 local DOFs (u₁,u₂,u₃,λ₁₂,λ₂₃,λ₃₁), so
// ∂G_x/∂y = Σ_{f: x,y ∈ local(f)} ∂(output)/∂y at f.
// Accumulating per-face 6×6 blocks therefore reproduces the full Hessian and is
// consistent with `euclidean_gradient` by construction (it is its FD).
/// Per-face contributions to the cyclic gradient, in slot order
/// (v₁, v₂, v₃, e₁₂, e₂₃, e₃₁). Vertex slots carry −α (since G_v = Θ−Σα),
/// edge slots carry +α_opp (since G_e = Σα_opp − φ).
inline std::array<double, 6> eucl_face_local_outputs(
double u1, double u2, double u3,
double d12, double d23, double d31,
double lam0_12, double lam0_23, double lam0_31)
{
const double lam12 = lam0_12 + u1 + u2 + d12;
const double lam23 = lam0_23 + u2 + u3 + d23;
const double lam31 = lam0_31 + u3 + u1 + d31;
const auto fa = euclidean_angles(lam12, lam23, lam31);
// edge slot e12 ↔ opposite corner α3, e23 ↔ α1, e31 ↔ α2 (see gradient Pass 3).
return { -fa.alpha1, -fa.alpha2, -fa.alpha3,
+fa.alpha3, +fa.alpha1, +fa.alpha2 };
}
/// Block-FD Euclidean cyclic Hessian (vertex + edge DOFs), sparse.
/// Supports both vertex-only and cyclic DOF layouts; cost `F·12` face-angle
/// evaluations. Consistent with `euclidean_gradient` to `O(ε²)`.
inline Eigen::SparseMatrix<double> euclidean_hessian_block_fd(
ConformalMesh& mesh,
const std::vector<double>& x,
const EuclideanMaps& m,
double eps = 1e-5)
{
const int n = euclidean_dimension(mesh, m);
std::vector<Eigen::Triplet<double>> trips;
trips.reserve(static_cast<std::size_t>(36) * mesh.number_of_faces());
for (auto f : mesh.faces()) {
Halfedge_index h0 = mesh.halfedge(f);
Halfedge_index h1 = mesh.next(h0);
Halfedge_index h2 = mesh.next(h1);
Vertex_index v1 = mesh.source(h0);
Vertex_index v2 = mesh.source(h1);
Vertex_index v3 = mesh.source(h2);
Edge_index e12 = mesh.edge(h0);
Edge_index e23 = mesh.edge(h1);
Edge_index e31 = mesh.edge(h2);
const int idx[6] = {
m.v_idx[v1], m.v_idx[v2], m.v_idx[v3],
m.e_idx[e12], m.e_idx[e23], m.e_idx[e31]
};
const double lam0_12 = m.lambda0[e12];
const double lam0_23 = m.lambda0[e23];
const double lam0_31 = m.lambda0[e31];
const double vals[6] = {
eucl_dof_val(idx[0], x), eucl_dof_val(idx[1], x), eucl_dof_val(idx[2], x),
eucl_dof_val(idx[3], x), eucl_dof_val(idx[4], x), eucl_dof_val(idx[5], x)
};
for (int j = 0; j < 6; ++j) {
if (idx[j] < 0) continue; // pinned: no column
double vp[6], vm[6];
for (int k = 0; k < 6; ++k) { vp[k] = vm[k] = vals[k]; }
vp[j] += eps;
vm[j] -= eps;
auto Op = eucl_face_local_outputs(
vp[0], vp[1], vp[2], vp[3], vp[4], vp[5], lam0_12, lam0_23, lam0_31);
auto Om = eucl_face_local_outputs(
vm[0], vm[1], vm[2], vm[3], vm[4], vm[5], lam0_12, lam0_23, lam0_31);
for (int i = 0; i < 6; ++i) {
if (idx[i] < 0) continue;
const double val = (Op[static_cast<std::size_t>(i)]
- Om[static_cast<std::size_t>(i)]) / (2.0 * eps);
if (std::abs(val) > 1e-15)
trips.emplace_back(idx[i], idx[j], val);
}
}
}
Eigen::SparseMatrix<double> H(n, n);
H.setFromTriplets(trips.begin(), trips.end());
return H;
}
/// Symmetrised block-FD Euclidean cyclic Hessian: `(H + Hᵀ)/2`.
inline Eigen::SparseMatrix<double> euclidean_hessian_block_fd_sym(
ConformalMesh& mesh,
const std::vector<double>& x,
const EuclideanMaps& m,
double eps = 1e-5)
{
auto H = euclidean_hessian_block_fd(mesh, x, m, eps);
Eigen::SparseMatrix<double> Ht = H.transpose();
return (H + Ht) * 0.5;
}
// ── Finite-difference Hessian check ──────────────────────────────────────────
/// FD Hessian check for the Euclidean functional. Compares analytic
/// `H` column-by-column to `(G(x+εeⱼ) − G(x−εeⱼ)) / (2ε)`; returns

9
code/include/newton_solver.hpp
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@@ -258,7 +258,14 @@ inline NewtonResult newton_euclidean(
}
// ── Hessian + solve H·Δx = −G (SparseQR fallback for singular H) ──
auto H = euclidean_hessian(mesh, x, m);
// Cyclic layout (edge DOFs present) → block-FD Hessian, which covers the
// vertex-edge / edge-edge blocks the analytic cotangent Laplacian omits.
// Vertex-only layout → analytic cotangent Laplacian (cheaper, exact).
bool has_edge_dof = false;
for (auto e : mesh.edges())
if (m.e_idx[e] >= 0) { has_edge_dof = true; break; }
auto H = has_edge_dof ? euclidean_hessian_block_fd_sym(mesh, x, m)
: euclidean_hessian(mesh, x, m);
bool ok = false;
Eigen::VectorXd dx = detail::solve_with_fallback(H, -G, ok);
if (!ok) break;

90
code/tests/cgal/test_euclidean_functional.cpp
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@@ -485,18 +485,10 @@ TEST(EuclideanFunctional, CyclicCircularEdge_PhiEntersGradient_CatHead)
// is the prescribed φ); the test therefore FAILS if the solver ignores a
// non-default φ (the sum would stay at its natural value, not π − 0.1).
//
// ⚠️ DISABLED (build-verified 2026-05-29): blocked by a missing feature, not a
// bug. `newton_euclidean` uses `euclidean_hessian`, which throws
// "euclidean_hessian: edge DOFs are not supported
// (only the vertex-block cotangent Laplacian is implemented)".
// The cyclic functional needs vertex+edge DOFs, so the full Newton solve is not
// yet possible in C++ (the gradient supports edge DOFs; the analytic Hessian
// does not). PREREQUISITE: edge-DOF Euclidean Hessian — see
// `doc/roadmap/research-track.md` and `doc/reviewer/java-ignore-crossvalidation.md`.
// The golden semantics (φ = π−0.1 ⇒ realised α_opp+α_opp = π−0.1) are kept here
// so this test auto-activates once that Hessian lands; just drop the DISABLED_.
// Enabled 2026-05-30: `newton_euclidean` now uses the block-FD edge-DOF Hessian
// (`euclidean_hessian_block_fd_sym`) for cyclic layouts, so the full solve runs.
// ════════════════════════════════════════════════════════════════════════════
TEST(EuclideanFunctional, DISABLED_CyclicCircularEdge_CatHead_JavaXVal)
TEST(EuclideanFunctional, CyclicCircularEdge_CatHead_JavaXVal)
{
const std::string path = std::string(CONFORMALLAB_DATA_DIR) + "/obj/cathead.obj";
ConformalMesh mesh;
@@ -505,26 +497,13 @@ TEST(EuclideanFunctional, DISABLED_CyclicCircularEdge_CatHead_JavaXVal)
auto maps = setup_euclidean_maps(mesh);
compute_euclidean_lambda0_from_mesh(mesh, maps);
// Cyclic DOFs: interior vertices (border pinned) + all edges.
// DOFs: interior vertices (border pinned) + exactly ONE edge DOF on the
// "circular" edge (Java marks a single circularHoleEdge). Giving every edge
// a DOF would make λ_e redundant with u_i+u_j (a V-dim null space) and stall
// Newton; the single extra edge variable keeps the system well-posed.
int idx = 0;
for (auto v : mesh.vertices())
maps.v_idx[v] = mesh.is_border(v) ? -1 : idx++;
for (auto e : mesh.edges())
maps.e_idx[e] = idx++;
const int n = idx;
ASSERT_GT(n, 0);
// Natural targets: set Θ_v / φ_e so that x = 0 is the equilibrium (G(0)=0).
std::vector<double> x0(static_cast<std::size_t>(n), 0.0);
auto G0 = euclidean_gradient(mesh, x0, maps);
for (auto v : mesh.vertices()) {
int iv = maps.v_idx[v];
if (iv >= 0) maps.theta_v[v] -= G0[static_cast<std::size_t>(iv)];
}
for (auto e : mesh.edges()) {
int ie = maps.e_idx[e];
if (ie >= 0) maps.phi_e[e] += G0[static_cast<std::size_t>(ie)];
}
// Pick one interior edge: both incident faces present, both endpoints interior.
Edge_index circular{};
@@ -537,8 +516,21 @@ TEST(EuclideanFunctional, DISABLED_CyclicCircularEdge_CatHead_JavaXVal)
circular = e; found = true; break;
}
ASSERT_TRUE(found) << "no interior edge found on cathead";
maps.e_idx[circular] = idx++; // the single circular-edge DOF
const int n = idx;
ASSERT_GT(n, 0);
const std::size_t ie = static_cast<std::size_t>(maps.e_idx[circular]);
// Natural targets: set Θ_v / φ_e so that x = 0 is the equilibrium (G(0)=0),
// so the only deviation is the prescribed circular φ below.
std::vector<double> x0(static_cast<std::size_t>(n), 0.0);
auto G0 = euclidean_gradient(mesh, x0, maps);
for (auto v : mesh.vertices()) {
int iv = maps.v_idx[v];
if (iv >= 0) maps.theta_v[v] -= G0[static_cast<std::size_t>(iv)];
}
maps.phi_e[circular] += G0[ie];
// Prescribe the circular edge turn angle φ = π − 0.1 (Java CustomEdgeInfo.phi).
const double phi_target = PI - 0.1;
maps.phi_e[circular] = phi_target;
@@ -553,3 +545,45 @@ TEST(EuclideanFunctional, DISABLED_CyclicCircularEdge_CatHead_JavaXVal)
EXPECT_NEAR(phi_target, realised, 1e-9)
<< "prescribed circular edge turn angle π−0.1 not realised at the solution";
}
// ════════════════════════════════════════════════════════════════════════════
// Correctness of the block-FD edge-DOF (cyclic) Hessian
//
// Validates `euclidean_hessian_block_fd` directly: on a tetrahedron with the
// full cyclic DOF layout (4 vertex + 6 edge), every entry must match the
// column-wise finite difference of `euclidean_gradient` (the true Jacobian of
// G). This pins the per-face output sign mapping (−α vertex / +α_opp edge) and
// the scatter independently of the convergence test above.
// ════════════════════════════════════════════════════════════════════════════
TEST(EuclideanFunctional, CyclicHessian_BlockFD_MatchesGradientFD_Tetrahedron)
{
auto mesh = make_tetrahedron();
auto maps = setup_euclidean_maps(mesh);
compute_euclidean_lambda0_from_mesh(mesh, maps);
const int n = assign_euclidean_all_dof_indices(mesh, maps); // 4 + 6 = 10
std::vector<double> x(static_cast<std::size_t>(n), 0.0);
for (int i = 0; i < 4; ++i) x[static_cast<std::size_t>(i)] = -0.15; // vertices
for (int i = 4; i < n; ++i) x[static_cast<std::size_t>(i)] = 0.05; // edges
auto H = euclidean_hessian_block_fd(mesh, x, maps);
const double eps = 1e-6;
std::vector<double> xp = x, xm = x;
double max_err = 0.0;
for (int j = 0; j < n; ++j) {
const std::size_t sj = static_cast<std::size_t>(j);
xp[sj] = x[sj] + eps;
xm[sj] = x[sj] - eps;
auto Gp = euclidean_gradient(mesh, xp, maps);
auto Gm = euclidean_gradient(mesh, xm, maps);
xp[sj] = xm[sj] = x[sj];
for (int i = 0; i < n; ++i) {
const double fd = (Gp[static_cast<std::size_t>(i)]
- Gm[static_cast<std::size_t>(i)]) / (2.0 * eps);
max_err = std::max(max_err, std::abs(H.coeff(i, j) - fd));
}
}
EXPECT_LT(max_err, 1e-5)
<< "block-FD cyclic Hessian disagrees with the gradient finite difference";
}

52
doc/reviewer/java-ignore-crossvalidation.md
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@@ -121,33 +121,39 @@ Assertion strategy: because the values are symmetric per DOF-class, assert
XML reader; strongest deterministic pipeline oracle that maps to landed C++).
4. **Tier 3 — Lawson generator** (only after Tiers 1–2).
### Build-verification finding (2026-05-29) — Tier 1 cyclic is blocked
### Build-verification finding (2026-05-29) → resolved (2026-05-30)
Attempting the Tier-1 `EuclideanCyclicConvergenceTest` port **build-verified** a
blocker: `newton_euclidean` calls `euclidean_hessian`, which throws
*"edge DOFs are not supported (only the vertex-block cotangent Laplacian is
implemented)"*. The cyclic functional needs **vertex + edge** DOFs, so the full
Newton solve is **not yet possible** in C++ — the *gradient* supports edge DOFs,
the *analytic Hessian* does not. (This is also why PR #27 cross-validated only
the cyclic *evaluation*, never a solve.)
blocker first: `newton_euclidean` → `euclidean_hessian` threw *"edge DOFs are
not supported"*. The cyclic functional needs **vertex + edge** DOFs (the
*gradient* supported them; the *analytic cotangent Hessian* did not). That is
why PR #27 cross-validated only the cyclic *evaluation*, never a solve.
**Landed in this PR instead** (`test_euclidean_functional.cpp`):
- ✅ `CyclicCircularEdge_PhiEntersGradient_CatHead` (**GREEN**) — solver-free
cross-validation that the circular-edge φ target enters the cyclic gradient
exactly (`ΔG_e = −Δφ_e` at 1e-12, no other component moves). This is the
evaluation-level prerequisite of the convergence oracle.
- ⏸️ `DISABLED_CyclicCircularEdge_CatHead_JavaXVal` — the full Java convergence
assertion (`α_opp+α_opp = π−0.1`), kept in-code with the golden semantics; it
auto-activates (drop `DISABLED_`) once the edge-DOF Hessian lands.
**Resolved (2026-05-30):** implemented a **block-FD edge-DOF Euclidean Hessian**
(`euclidean_hessian_block_fd` / `_sym` in `euclidean_hessian.hpp`, mirroring
`hyper_ideal_hessian_block_fd`); `newton_euclidean` now routes cyclic layouts
(edge DOFs present) through it, vertex-only layouts still use the analytic
cotangent Laplacian.
**New prerequisite for Tier-1 cyclic:** an **edge-DOF Euclidean Hessian** (the
cyclic Hessian over vertex + edge variables, cf. Java `getNonZeroPattern`). Spherical
convergence (Tier-1 #2) turned out to be **already covered** by
`test_newton_solver` (`Spherical_ConvergesFromPerturbation` et al. on the
spherical tetrahedron), and `make_octahedron_face()` is a single face, not a
closed octahedron — so Tier-1 #2 adds little. → the next genuinely-new
cross-validation target is **Tier 2 (Wente uniformization XML)** or implementing
the edge-DOF Hessian to unblock the cyclic convergence oracle.
**Landed (`test_euclidean_functional.cpp`, all GREEN, 243/243 cgal tests pass):**
- ✅ `CyclicCircularEdge_PhiEntersGradient_CatHead` — solver-free check that the
circular-edge φ enters the gradient exactly (`ΔG_e = −Δφ_e` @1e-12).
- ✅ `CyclicCircularEdge_CatHead_JavaXVal` — **the full Java convergence
oracle**: prescribe φ = π−0.1 on one interior ("circular") edge of cathead,
solve the cyclic Newton, assert the realised `α_opp+α_opp = π−0.1` @1e-9.
- ✅ `CyclicHessian_BlockFD_MatchesGradientFD_Tetrahedron` — the block-FD edge-DOF
Hessian matches the column-wise gradient FD on the full cyclic layout.
> DOF note: only the **single** circular edge gets an edge DOF (as in Java's
> one `circularHoleEdge`). Giving *every* edge a DOF makes λ_e redundant with
> u_i+u_j (a V-dim null space) and stalls Newton.
Spherical convergence (Tier-1 #2) was found **already covered** by
`test_newton_solver` (`Spherical_ConvergesFromPerturbation` et al.), and
`make_octahedron_face()` is a single face, not a closed octahedron — so it adds
little. → next genuinely-new target: **Tier 2 (Wente uniformization XML)**. A
full *analytic* edge-DOF Hessian (vs the current block-FD) remains a future
optimisation.
### Other generators in the Java tree
`HyperellipticCurveGenerator` (→ Phase 13), `SchottkyGenerator` (→ Phase 11a),