{
  "schema_version": "1.0",
  "chapter_source": "book/chapters/09-volume-hessians.tex",
  "chapter_label": "sec:ipolybridge",
  "review_date": "2026-10-31",
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  "status_definitions": {
    "proved_in_chapter": "A general derivation is written in the canonical chapter and independently audited; finite computations are supplementary.",
    "external_theorem_statement_checked": "The specific statement and hypotheses were checked against the cited primary material, with access limits recorded; complete proofs are not claimed re-audited.",
    "conditional_algorithmic_consequence": "The complexity or iteration consequence holds under the explicit starting-point, arithmetic, accuracy and solver hypotheses.",
    "exact_finite_check": "An exact integer/rational calculation for the named finite fixture; not a general algorithmic certificate.",
    "numerical_check": "Finite floating-point or multiprecision comparisons with recorded residuals and tolerances; not exact or interval arithmetic.",
    "open": "A question retained as unresolved, without a claimed theorem or implementation."
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  "claims": [
    {
      "id": "s9.fem",
      "labels": [
        "eq:ipolyFEM"
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      "status": "proved_in_chapter",
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        "Finite connected simplicial disk with a nondegenerate geometric Euclidean realization.",
        "Piecewise affine nodal function; count each undirected edge once; omit the second opposite angle at a boundary edge."
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        "description": "Barycentric-gradient proof is general. Independent rational FEM assembly is exact only for the integer-grid fixture; maintained assembly uses floating point."
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    },
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      "id": "s9.delaunay_weights",
      "labels": [
        "eq:ipolyDelaunayweight",
        "eq:ipolyWeightLower"
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      "status": "proved_in_chapter",
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        "Positive opposite angles; fixed interior sum Delta in (0,pi].",
        "Boundary conductances require their own sign condition for a passive full network."
      ],
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        "status": "numerical_implementation_and_exact_scope_check",
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          "docs/execution/section9-chapter-2026-10-31/math-audit-checks.json"
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        "description": "Numerical cotangent assembly; exact obtuse-triangle matrix demonstrates signed boundary weights. The trigonometric lower bound is proved independently."
      },
      "coverage_gap": "No general interval certification of Delaunay inequalities or boundary-weight signs."
    },
    {
      "id": "s9.coordinates",
      "labels": [
        "eq:ipolyAngleCoordinates"
      ],
      "status": "proved_in_chapter",
      "hypotheses": [
        "Finite connected consistently oriented simplicial disk, nonempty boundary, three distinct vertices per face.",
        "Nonempty positive angle domain, triangle sums pi, fixed opposite-edge sums, u=0 on all boundary vertices."
      ],
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        "docs/execution/section9-chapter-2026-10-31/math-audit.md"
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        "status": "numerical_map_and_exact_finite_rank_checks",
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        "description": "leading_trailing_matrix and deformed_angles implement the map. General completeness follows from dual incidence rank and Euler characteristic. Exact disk/annulus calculations illustrate why topology matters."
      },
      "coverage_gap": "The maintained operator is not a complete topology validator or a coordinate basis for arbitrary surfaces."
    },
    {
      "id": "s9.gradient_hessian",
      "labels": [
        "prop:ipolyHessian",
        "eq:ipolyHessian"
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      "status": "proved_in_chapter",
      "hypotheses": [
        "The disk coordinate hypotheses and positive angles.",
        "Dilation uses counterclockwise radial development; arbitrary feasible angles need not form an embedded mesh."
      ],
      "review_evidence": [
        "docs/execution/section9-chapter-2026-10-31/math-audit.md"
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      "computation": {
        "status": "numerical_implementation_with_independent_derivative_check",
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          "packages/volume-toolkit/ideal_poly_volume_toolkit/variational.py",
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        "evidence_paths": [
          "experiments/section9/fixture.json",
          "artifacts/section9/result.json"
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        "description": "angle_volume_derivatives computes the gradient and negative Hessian. The fixture's exact matrix is compared with rational FEM; the off-origin comparison uses 50-digit numerical differentiation."
      },
      "coverage_gap": "The driver does not construct a global developing map or certify holonomy closure."
    },
    {
      "id": "s9.strictness",
      "labels": [
        "prop:ipolyHessian"
      ],
      "status": "proved_in_chapter",
      "hypotheses": [
        "Positive triangle angles, connected triangulation, grounded nonempty boundary.",
        "For no interior vertices use the empty-matrix determinant-one convention."
      ],
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        "docs/execution/section9-chapter-2026-10-31/math-audit.md"
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      "computation": {
        "status": "exact_finite_scope_checks",
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          "docs/execution/section9-chapter-2026-10-31/math-audit-checks.py",
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          "docs/execution/section9-chapter-2026-10-31/math-audit-checks.json",
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        "description": "The general proof uses the triangle cotangent-form determinant one and injectivity, including zero Delaunay weights. Exact finite matrices illustrate the conclusion."
      },
      "coverage_gap": "No claim that sampled eigenvalues prove general strictness or certify numerical conditioning."
    },
    {
      "id": "s9.newton_equation",
      "labels": [
        "eq:ipolyNewton"
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      "status": "proved_in_chapter",
      "hypotheses": [
        "Positive disk angle domain; L_II is the grounded Hessian matrix; the chosen holonomy orientation."
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        "docs/execution/section9-chapter-2026-10-31/math-audit.md"
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        "status": "operators_available_no_iteration_driver",
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          "packages/volume-toolkit/ideal_poly_volume_toolkit/variational.py"
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        "evidence_paths": [
          "artifacts/section9/result.json"
        ],
        "description": "The gradient and Hessian needed for the displayed local Newton equation are implemented."
      },
      "coverage_gap": "No fixed-dihedral Newton realization driver, feasibility line search, or convergence certificate is supplied by this example."
    },
    {
      "id": "s9.geometric_realization",
      "labels": [
        "subsec:ipolyCoordinates",
        "prop:ipolyHessian"
      ],
      "status": "external_theorem_statement_checked",
      "hypotheses": [
        "Use the precise prescribed-angle, cone and boundary hypotheses of the cited realization theorem.",
        "For the stated planar convex interpretation: flat interior total angles 2pi, boundary angles strictly between 0 and pi, closure and a global realization argument."
      ],
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        "docs/execution/section9-chapter-2026-10-31/math-audit.md",
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      "computation": {
        "status": "not_implemented",
        "code_paths": [],
        "evidence_paths": [],
        "description": "The chapter proves abstract differential identities and refers separately to realization theorems."
      },
      "coverage_gap": "No constructive embedding certificate, convexity certificate, or general realization solver. A unique closed angle maximum need not be positive.",
      "source_evidence": [
        {
          "source": "RivinOptimization",
          "url": "https://arxiv.org/pdf/math/9907032v1",
          "pin": "Definitions 1.1 and Lemma 1.2; Theorems 1.4, 1.6 and 4.11",
          "access": "Primary theorem statements read; complete proofs not independently audited."
        },
        {
          "source": "RivinEuclidean",
          "url": "https://annals.math.princeton.edu/1994/139-3/p02",
          "pin": "Annals 139 (1994), 553-580, DOI metadata",
          "access": "Metadata verified; original 1994 full theorem/proof text unavailable in this audit."
        }
      ]
    },
    {
      "id": "s9.dtn",
      "labels": [
        "eq:ipolyDtN"
      ],
      "status": "proved_in_chapter",
      "hypotheses": [
        "Symmetric cotangent energy with constants as its only kernel and a grounded interior block.",
        "Full passive resistor interpretation additionally requires nonnegative boundary conductances."
      ],
      "review_evidence": [
        "docs/execution/section9-chapter-2026-10-31/math-audit.md"
      ],
      "computation": {
        "status": "numerical_dense_schur_and_exact_fixture",
        "code_paths": [
          "packages/volume-toolkit/ideal_poly_volume_toolkit/variational.py",
          "experiments/section9/run.py",
          "packages/volume-toolkit/tests/test_variational.py"
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        "evidence_paths": [
          "artifacts/section9/result.json",
          "docs/execution/section9-chapter-2026-10-31/math-audit-checks.json"
        ],
        "description": "schur_complement and the rectangle harmonic-voltage check implement small dense boundary elimination."
      },
      "coverage_gap": "No sparse or nearly-linear boundary-response solver, and no general certified inverse."
    },
    {
      "id": "s9.observable_response",
      "labels": [
        "eq:ipolyBoundaryResponse"
      ],
      "status": "proved_in_chapter",
      "hypotheses": [
        "Smooth function with a nondegenerate local maximum, Hessian -K with K positive definite.",
        "Local maximizing branches for small sources/observables; P has full row rank when chi inverse is used.",
        "Px denotes selected linear observables; any geometric boundary identification is separate."
      ],
      "review_evidence": [
        "docs/execution/section9-chapter-2026-10-31/math-audit.md"
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      "computation": {
        "status": "linear_algebra_only",
        "code_paths": [
          "packages/volume-toolkit/ideal_poly_volume_toolkit/variational.py"
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        "evidence_paths": [
          "artifacts/section9/result.json"
        ],
        "description": "The maintained Schur operator supports the quadratic algebra; the general nonlinear local-response expansion follows from the implicit-function theorem."
      },
      "coverage_gap": "No nonlinear observable-constrained maximizing-branch routine or model-specific boundary-data identification."
    },
    {
      "id": "s9.schur",
      "labels": [
        "eq:ipolySchur"
      ],
      "status": "proved_in_chapter",
      "hypotheses": [
        "Real symmetric positive-definite K, partitioned into eliminated and retained variables."
      ],
      "review_evidence": [
        "docs/execution/section9-chapter-2026-10-31/math-audit.md"
      ],
      "computation": {
        "status": "numerical_implementation_and_exact_fixture",
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          "packages/volume-toolkit/ideal_poly_volume_toolkit/variational.py",
          "experiments/section9/run.py",
          "packages/volume-toolkit/tests/test_variational.py"
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        "evidence_paths": [
          "artifacts/section9/result.json"
        ],
        "description": "Dense Schur complement; the fixture verifies effective coefficient 15/2 and determinant factorization 8*(15/2)=60 exactly."
      }
    },
    {
      "id": "s9.gaussian",
      "labels": [
        "eq:ipolyGaussian"
      ],
      "status": "proved_in_chapter",
      "hypotheses": [
        "Real symmetric positive-definite K, hbar>0, integration over the real eliminated variables."
      ],
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        "docs/execution/section9-chapter-2026-10-31/math-audit.md"
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        "status": "determinant_inputs_only",
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          "packages/volume-toolkit/ideal_poly_volume_toolkit/variational.py",
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        "evidence_paths": [
          "artifacts/section9/result.json"
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        "description": "The determinant and Schur inputs are available; the Gaussian formula is proved by completing the square."
      },
      "coverage_gap": "No Gaussian quadrature validation, oscillatory contour selection, complex determinant-branch routine, or full quantum invariant."
    },
    {
      "id": "s9.rooted_forests",
      "labels": [
        "eq:ipolyForestDeterminant"
      ],
      "status": "proved_in_chapter",
      "hypotheses": [
        "Incidence matrix with boundary rows removed; spanning forests have exactly one boundary vertex in each component.",
        "Signed weights are allowed algebraically; the Delaunay Dirichlet case has nonnegative contributing weights."
      ],
      "review_evidence": [
        "docs/execution/section9-chapter-2026-10-31/math-audit.md"
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      "computation": {
        "status": "exact_finite_enumeration",
        "code_paths": [
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          "packages/volume-toolkit/tests/test_variational.py",
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        "evidence_paths": [
          "artifacts/section9/result.json",
          "docs/execution/section9-chapter-2026-10-31/math-audit-checks.json"
        ],
        "description": "Cauchy-Binet proves the general identity. Enumeration gives fifteen unit-weight terms for the grid and exact determinant sums 196 and 144 for two additional audit graphs."
      },
      "coverage_gap": "Exhaustive enumeration is a small-fixture oracle, not an efficient general determinant algorithm."
    },
    {
      "id": "s9.barrier_compatibility",
      "labels": [
        "eq:ipolyBarrier",
        "prop:ipolyCompatibility",
        "eq:ipolyCompatibility"
      ],
      "status": "proved_in_chapter",
      "hypotheses": [
        "Positive triangle angle structures, each triangle sum pi; tangent coordinates sum to zero per triangle.",
        "Intrinsic affine coordinates after dependencies are removed; t>=0.",
        "The parameter bound m belongs to F; F-tV is a standard self-concordant objective."
      ],
      "review_evidence": [
        "docs/execution/section9-chapter-2026-10-31/math-audit.md",
        "docs/execution/section9-chapter-2026-10-31/companion-audit.md"
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      "computation": {
        "status": "dense_binary64_objective_and_derivatives",
        "code_paths": [
          "packages/volume-toolkit/ideal_poly_volume_toolkit/angle_barrier.py",
          "packages/volume-toolkit/tests/test_angle_barrier.py",
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          "artifacts/section9/barrier-result.json",
          "site/section9/downloads/barrier-reproduction.zip",
          "docs/execution/section9-barrier-2026-10-31/barrier-review.md",
          "docs/execution/section9-barrier-2026-10-31/barrier-publication-review.md",
          "docs/execution/section9-chapter-2026-10-31/companion_reference_probes.json"
        ],
        "description": "The maintained log-sine barrier and affine-angle derivatives are independently checked numerically. The general compatibility inequality is still supported by the unchanged chapter proof and mathematical audits."
      },
      "coverage_gap": "The caller supplies an independent affine tangent basis and a positive feasible start. Rank, trigonometric evaluations and derivatives are numerical; no automatic full peripheral basis or interval certificate is supplied."
    },
    {
      "id": "s9.finite_centering",
      "labels": [
        "subsec:ipolyBarrierComplexity",
        "eq:ipolyStepBound"
      ],
      "status": "proved_in_chapter",
      "hypotheses": [
        "Nonempty positive feasible affine domain; compact closure in angle coordinates; continuous volume.",
        "Intrinsic Hessian H_t and Newton decrement lambda_t; m>=1, t>0.",
        "For the finite certificate lambda_t<=1/4; stopping requires t>=17m/(12*epsilon)."
      ],
      "review_evidence": [
        "docs/execution/section9-chapter-2026-10-31/companion-audit.md"
      ],
      "computation": {
        "status": "dense_numerical_centering_and_gap_estimate",
        "code_paths": [
          "packages/volume-toolkit/ideal_poly_volume_toolkit/angle_barrier.py",
          "packages/volume-toolkit/tests/test_angle_barrier.py",
          "experiments/section9/barrier_run.py"
        ],
        "evidence_paths": [
          "experiments/section9/barrier-fixtures.json",
          "artifacts/section9/barrier-result.json",
          "site/section9/downloads/barrier-reproduction.zip",
          "docs/execution/section9-barrier-2026-10-31/barrier-review.md",
          "docs/execution/section9-barrier-2026-10-31/barrier-publication-review.md",
          "docs/execution/section9-chapter-2026-10-31/companion_reference_probes.json"
        ],
        "description": "The solver separately budgets initial centering, freshly evaluates the endpoint decrement and returns the finite-gap estimate and conservative 17m/(12t) estimate with explicit numerical status. Named examples compare against independent multiprecision volume references."
      },
      "coverage_gap": "Binary64 evaluation of the theorem-derived estimate is not a certified upper bound. Feasibility construction, exact rank, rounding-error enclosure and geometric error remain outside the solver."
    },
    {
      "id": "s9.short_step_iterations",
      "labels": [
        "eq:ipolyStepBound"
      ],
      "status": "conditional_algorithmic_consequence",
      "hypotheses": [
        "Initially centered positive point at t0>0 with decrement<=1/4.",
        "Parameter factor 1+1/(20*sqrt(m)); one full Newton step per change; sufficient terminal parameter 17m/(12*epsilon).",
        "Exact arithmetic or the explicitly controlled relative energy-norm solve error; no cost included for feasibility or initial centering."
      ],
      "review_evidence": [
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      ],
      "computation": {
        "status": "dense_short_step_path_and_trace_checks",
        "code_paths": [
          "packages/volume-toolkit/ideal_poly_volume_toolkit/angle_barrier.py",
          "packages/volume-toolkit/tests/test_angle_barrier.py",
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        ],
        "evidence_paths": [
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          "artifacts/section9/barrier-result.json",
          "site/section9/downloads/barrier-reproduction.zip",
          "docs/execution/section9-barrier-2026-10-31/barrier-review.md",
          "docs/execution/section9-barrier-2026-10-31/barrier-publication-review.md",
          "docs/execution/section9-chapter-2026-10-31/companion_reference_probes.json"
        ],
        "description": "The maintained path uses the specified parameter factor and one full Newton step, checks actual positive angles and fresh decrements, and stops only after the scheduled target and centering condition hold. Iteration traces separate initial centering from path work; exhausted budgets remain unsuccessful."
      },
      "coverage_gap": "The dense implementation supplies neither a nearly-linear complexity result nor an unrestricted plain-Newton bound. Floating-point acceptance checks do not validate exact-arithmetic or controlled energy-error hypotheses.",
      "source_evidence": [
        {
          "source": "BoydVandenbergheSlides",
          "url": "https://web.stanford.edu/~boyd/cvxbook/bv_cvxslides.pdf",
          "pin": "PDF pages 378-382",
          "access": "Primary authors' slides read for short-step analysis and initial centering."
        },
        {
          "source": "BoydVandenberghe",
          "url": "https://stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf",
          "pin": "Chapter 11",
          "access": "Bibliographic background retained; book download timed out, so its full text was not read."
        }
      ]
    },
    {
      "id": "s9.inexact_arithmetic_work",
      "labels": [
        "subsec:ipolyBarrierComplexity",
        "eq:ipolyStepBound"
      ],
      "status": "conditional_algorithmic_consequence",
      "hypotheses": [
        "Grounded disk barrier Hessian has positive planar conductances; n=|I|+|B| counts all mesh vertices, and planarity gives m=O(n) and O(n) matrix entries.",
        "At each new parameter the solve error in the Hessian energy norm is at most one percent of the exact Newton-step norm.",
        "Exact gradient/Hessian evaluation apart from the stated solve error; randomized solver success and precision controls maintained throughout.",
        "Feasibility and initial centering excluded; accuracy factors suppressed only in the tilde notation."
      ],
      "review_evidence": [
        "docs/execution/section9-chapter-2026-10-31/companion-audit.md"
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      "computation": {
        "status": "not_implemented",
        "code_paths": [],
        "evidence_paths": [
          "docs/execution/section9-chapter-2026-10-31/companion_reference_probes.json"
        ],
        "description": "The analytical tolerance implies new decrement<=10420/47961<1/4. The dense implementation reports numerical linear residuals and conditioning, but these are not certified relative energy-error bounds and do not implement the conditional nearly-linear randomized solver."
      },
      "coverage_gap": "No nearly-linear solver integration, scaling benchmark, failure-probability accounting, finite-precision stability proof, or geometric-error certification. Dense numerical traces are not evidence for the stated arithmetic-work scale.",
      "source_evidence": [
        {
          "source": "SpielmanTengSolvers",
          "url": "https://arxiv.org/pdf/cs/0607105v5",
          "pin": "Theorem 5.5 and Section 6",
          "access": "Primary relative energy-norm guarantee and separate precision scope checked."
        }
      ]
    },
    {
      "id": "s9.rippa_energy",
      "labels": [
        "subsec:ipolyFlips",
        "fig:mono-flip"
      ],
      "status": "external_theorem_statement_checked",
      "hypotheses": [
        "Common fixed planar data sites and nodal values; triangulations of their convex hull.",
        "Quadratic-form comparison on that common nodal vector space."
      ],
      "review_evidence": [
        "docs/execution/section9-chapter-2026-10-31/companion-audit.md"
      ],
      "computation": {
        "status": "not_implemented",
        "code_paths": [],
        "evidence_paths": [],
        "description": "The figure illustrates a fixed Euclidean diagonal exchange; no numerical Rippa comparison is supplied by the Section 9 driver."
      },
      "coverage_gap": "No inference about volume monotonicity under arbitrary topological flips.",
      "source_evidence": [
        {
          "source": "RippaEnergy",
          "url": "https://doi.org/10.1016/0167-8396(90)90011-F",
          "pin": "Primary publisher abstract",
          "access": "The stated minimization theorem is supported by the abstract; full original proof was not read (403/timeout)."
        }
      ]
    },
    {
      "id": "s9.flip_graph",
      "labels": [
        "subsec:ipolyFlips"
      ],
      "status": "external_theorem_statement_checked",
      "hypotheses": [
        "Connected orientable finite-type triangulable marked surface, with a marked point on every boundary component.",
        "Orientation-preserving mapping classes and isotopies fixing marked points individually; retain the cited marking conventions."
      ],
      "review_evidence": [
        "docs/execution/section9-chapter-2026-10-31/companion-audit.md"
      ],
      "computation": {
        "status": "not_implemented",
        "code_paths": [],
        "evidence_paths": [],
        "description": "The proper cocompact action and orbit quasi-isometry are external inputs."
      },
      "coverage_gap": "No marked-surface triangulation enumerator or mapping-class word metric implementation.",
      "source_evidence": [
        {
          "source": "DisarloParlier",
          "url": "https://arxiv.org/pdf/1411.4285v1",
          "pin": "Section 2.1, Lemma 2.3",
          "access": "Primary theorem statement and relevant scope paragraphs checked; not a complete re-refereeing of the paper."
        }
      ]
    },
    {
      "id": "s9.brock_comparisons",
      "labels": [
        "subsec:ipolyFlips"
      ],
      "status": "external_theorem_statement_checked",
      "hypotheses": [
        "Closed oriented surface of genus at least two.",
        "Convex-core comparison uses quasi-Fuchsian conformal ends X,Y and multiplicative/additive topology-dependent constants.",
        "Mapping-torus comparison uses pseudo-Anosov monodromy and topology-dependent multiplicative constants."
      ],
      "review_evidence": [
        "docs/execution/section9-chapter-2026-10-31/companion-audit.md"
      ],
      "computation": {
        "status": "not_implemented",
        "code_paths": [],
        "evidence_paths": [],
        "description": "Pants graph, WP metric and mapping-torus volume comparisons are cited external inputs."
      },
      "coverage_gap": "No WP metric/distance evaluator; no unverified extension to punctured or boundary surfaces.",
      "source_evidence": [
        {
          "source": "BrockWPVolumes",
          "url": "https://arxiv.org/pdf/math/0109048v2",
          "pin": "Introduction, Theorems 1.1-1.2",
          "access": "Primary statement and closed-surface scope checked."
        },
        {
          "source": "BrockMappingTori",
          "url": "https://arxiv.org/pdf/math/0109050v1",
          "pin": "Theorem 1.1",
          "access": "Primary statement and closed pseudo-Anosov scope checked."
        }
      ]
    },
    {
      "id": "s9.layered_volume_bound",
      "labels": [
        "subsec:ipolyFlips"
      ],
      "status": "proved_in_chapter",
      "hypotheses": [
        "The flip sequence actually supplies an ideal triangulation of the specified finite-volume hyperbolic mapping torus representing its fundamental class.",
        "N is its tetrahedron count; straightened signed volumes are bounded in absolute value by the regular ideal tetrahedron volume."
      ],
      "review_evidence": [
        "docs/execution/section9-chapter-2026-10-31/companion-audit.md"
      ],
      "computation": {
        "status": "not_implemented",
        "code_paths": [],
        "evidence_paths": [],
        "description": "Straightening and the ideal-tetrahedron maximum give Vol(M)<=N*v3, independently of positivity of the chosen shapes."
      },
      "coverage_gap": "No proof that an arbitrary flip word yields the required triangulation; no WP comparison transferred to a punctured fiber."
    },
    {
      "id": "s9.three_dimensional_hessian",
      "labels": [
        "eq:ipolyThreeDimensionalHessian"
      ],
      "status": "proved_in_chapter",
      "hypotheses": [
        "Positive ideal tetrahedron angles; leading-trailing edge variations with p_alpha the sum of the opposite-edge variables.",
        "A full angle-space coordinate system also requires peripheral directions and removal of dependencies."
      ],
      "review_evidence": [
        "docs/execution/section9-chapter-2026-10-31/math-audit.md",
        "docs/execution/section9-chapter-2026-10-31/companion-audit.md"
      ],
      "computation": {
        "status": "exact_local_counterexample_only",
        "code_paths": [
          "docs/execution/section9-chapter-2026-10-31/math-audit-checks.py"
        ],
        "evidence_paths": [
          "docs/execution/section9-chapter-2026-10-31/math-audit-checks.json"
        ],
        "description": "The regular-tetrahedron edge-variable matrix is an exact rank-two Gram matrix with positive off-diagonal entries and no diagonal dominance."
      },
      "coverage_gap": "No complete peripheral tangent basis, 3-manifold Hessian solver, or general SDD reduction. The local formula does not establish one."
    },
    {
      "id": "s9.casson_rivin",
      "labels": [
        "subsec:ipolyThreeManifolds"
      ],
      "status": "external_theorem_statement_checked",
      "hypotheses": [
        "Ideal triangulation of an orientable compact 3-manifold with nonempty boundary a union of tori.",
        "Positive angle structures with per-tetrahedron sums pi and global edge sums 2pi.",
        "Criticality on the full angle-space tangent, including peripheral directions."
      ],
      "review_evidence": [
        "docs/execution/section9-chapter-2026-10-31/math-audit.md",
        "docs/execution/section9-chapter-2026-10-31/companion-audit.md"
      ],
      "computation": {
        "status": "not_implemented",
        "code_paths": [],
        "evidence_paths": [],
        "description": "The complete-metric implication is an external theorem; edge-only stationary equations do not impose completeness."
      },
      "coverage_gap": "No full angle-space peripheral-holonomy solver or complete hyperbolic metric certification.",
      "source_evidence": [
        {
          "source": "FuterGueritaud",
          "url": "https://arxiv.org/pdf/1004.0440v2",
          "pin": "Theorem 1.2; Proposition 2.5; Section 5, Proposition 5.4",
          "access": "Primary statements and scope/proof paragraphs checked; complete proofs not independently re-audited."
        }
      ]
    },
    {
      "id": "s9.fixture_exact",
      "labels": [
        "subsec:ipolyComputation",
        "eq:ipolyForestDeterminant",
        "eq:ipolySchur"
      ],
      "status": "exact_finite_check",
      "hypotheses": [
        "The unchanged public 3 by 2 integer-grid fixture, twelve triangles, two interior vertices.",
        "Independent rational finite-element assembly and exhaustive rooted-forest enumeration; electrical voltage x/3."
      ],
      "review_evidence": [
        "docs/execution/section9-chapter-2026-10-31/math-audit.md",
        "docs/execution/2026-10-31/section9-review.md",
        "docs/execution/2026-10-31/section9-rereview.md"
      ],
      "computation": {
        "status": "implemented_exact_fixture_oracle",
        "code_paths": [
          "experiments/section9/run.py",
          "packages/volume-toolkit/tests/test_variational.py"
        ],
        "evidence_paths": [
          "experiments/section9/fixture.json",
          "artifacts/section9/result.json"
        ],
        "description": "L_II=[[4,-1],[-1,4]], K=2*L_II, determinant 60; fifteen unit-weight forests, Schur coefficient 15/2, harmonic interior values 1/3,2/3 and energy 2/3."
      },
      "coverage_gap": "Exact arithmetic is limited to the named finite identities; it is not a general interval implementation."
    },
    {
      "id": "s9.fixture_numerical",
      "labels": [
        "subsec:ipolyComputation",
        "eq:ipolyHessian"
      ],
      "status": "numerical_check",
      "hypotheses": [
        "The fixed fixture and the specific off-origin angle perturbation recorded in the result.",
        "Binary64 maintained operators and independent 50-digit numerical differentiation; explicit finite residual tolerances."
      ],
      "review_evidence": [
        "docs/execution/2026-10-31/section9-rereview.md",
        "docs/execution/numerical-correctness-2026-10-31/closeout.md"
      ],
      "computation": {
        "status": "implemented_floating_point_checks",
        "code_paths": [
          "packages/volume-toolkit/ideal_poly_volume_toolkit/variational.py",
          "experiments/section9/run.py",
          "packages/volume-toolkit/tests/test_variational.py"
        ],
        "evidence_paths": [
          "experiments/section9/fixture.json",
          "artifacts/section9/result.json",
          "site/section9/downloads/section9-reproduction.zip"
        ],
        "description": "The result compares the volume with 12*Catalan and off-origin derivatives with an independent numerical reference; it records actual source/input hashes and run provenance."
      },
      "coverage_gap": "No interval enclosure, general geometric realization, strict convex-polyhedron certificate, or asymptotic benchmark. A regenerated result must carry its own provenance."
    },
    {
      "id": "s9.wp_continuum_limit",
      "labels": [
        "subsec:ipolyThreeManifolds"
      ],
      "status": "open",
      "hypotheses": [
        "A proposed limit must specify the geometric convergence, normalization, and control of second variations."
      ],
      "review_evidence": [
        "docs/execution/section9-chapter-2026-10-31/math-audit.md",
        "docs/execution/section9-chapter-2026-10-31/companion-audit.md"
      ],
      "computation": {
        "status": "not_implemented",
        "code_paths": [],
        "evidence_paths": [],
        "description": "No continuum-limit theorem or experiment is supplied."
      },
      "coverage_gap": "The real symmetric FEM Hessian is not identified with the alternating WP form."
    },
    {
      "id": "s9.quantitative_angle_bound",
      "labels": [
        "subsec:ipolyThreeManifolds"
      ],
      "status": "open",
      "hypotheses": [
        "Seek a positive interior maximum and a quantitative feasible margin; boundary maxima can have zero angles."
      ],
      "review_evidence": [
        "docs/execution/section9-chapter-2026-10-31/math-audit.md",
        "docs/execution/section9-chapter-2026-10-31/companion-audit.md"
      ],
      "computation": {
        "status": "not_implemented",
        "code_paths": [],
        "evidence_paths": [],
        "description": "No effective smallest-angle bound in terms of size and feasible margin is established."
      },
      "coverage_gap": "A volume-gap certificate does not imply angle error, geometric embedding, or combinatorial-type certification."
    }
  ],
  "computational_revision": {
    "baseline_commit": "014d1c4e70dfdfc852fa542f3a9c469f08dbd618",
    "scope": "Dense binary64 affine-angle barrier implementation and computational addendum; original proof mathematics and numbering retained.",
    "review_evidence": [
      "docs/execution/section9-barrier-2026-10-31/barrier-review.md",
      "docs/execution/section9-barrier-2026-10-31/barrier-publication-review.md"
    ],
    "evidence_boundary": "Review dispositions and exact source identities are recorded in those reports; computations are not formal or interval certification."
  }
}