{ "schema": "ggii.inverse-certificates-artifact/1", "baseCoreVersion": "0.1.0", "inverseCompilerVersion": "0.1.0-alpha.2", "inverseCompilerSchemaVersion": 2, "scientificScope": { "established": [ "a finite exhaustive Chow--Luo admission for the declared tetrahedral Model CP target", "an exact Chow--Luo subset obstruction with a minimal displayed witness", "an exact Gauss--Bonnet total obstruction", "a degenerate-triangle obstruction in the declared edge-metric model" ], "notEstablished": [ "a general geometry compiler", "extrinsic embedding existence or uniqueness", "continuum convergence", "material or actuator feasibility", "physical validation" ] }, "cases": [ { "id": "tetra-uniform-pass", "request": { "schemaVersion": 2, "id": "tetra-uniform-pass", "seed": "exact-deterministic", "target": { "kind": "intrinsic-curvature", "topology": { "surface": "sphere", "orientable": true, "boundaryComponents": 0 }, "discretization": { "complex": { "nodes": [ { "id": 0, "label": "v0", "x": 0, "y": 0, "z": 0, "radius": 1 }, { "id": 1, "label": "v1", "x": 0, "y": 0, "z": 0, "radius": 1 }, { "id": 2, "label": "v2", "x": 0, "y": 0, "z": 0, "radius": 1 }, { "id": 3, "label": "v3", "x": 0, "y": 0, "z": 0, "radius": 1 } ], "edges": [ { "source": 0, "target": 1, "length": 1, "weight": 1 }, { "source": 0, "target": 2, "length": 1, "weight": 1 }, { "source": 0, "target": 3, "length": 1, "weight": 1 }, { "source": 1, "target": 2, "length": 1, "weight": 1 }, { "source": 1, "target": 3, "length": 1, "weight": 1 }, { "source": 2, "target": 3, "length": 1, "weight": 1 } ], "faces": [ [ 0, 1, 2 ], [ 0, 1, 3 ], [ 0, 2, 3 ], [ 1, 2, 3 ] ] } }, "curvature": { "basis": "symbolic-pi", "piCoefficients": [ { "numerator": 1, "denominator": 1 }, { "numerator": 1, "denominator": 1 }, { "numerator": 1, "denominator": 1 }, { "numerator": 1, "denominator": 1 } ] } }, "models": [ "circle-packing-tangency" ], "constraints": { "radiusBounds": [ 0.5, 2 ], "edgeLengthBounds": [ 1e-9, 10 ], "allowTopologyChange": false }, "verification": { "gbTolerance": 0, "chowLuoGuard": 1e-8, "exactSubsetLimit": 18, "optimizerTolerance": 1e-8, "maxIterations": 0, "refinementLevels": 0, "uncertaintySamples": 0 } }, "report": { "candidates": [ { "caveats": [ "This deterministic reference vector is not an optimizer result.", "A Chow–Luo admission proves existence only under its hypotheses; this reference vector realizes the requested curvature only when the disclosed residual is within tolerance.", "No extrinsic embedding or physical response is established." ], "evidenceClass": "finite-numerical-estimate", "hardBoundsSatisfied": true, "id": "tetra-uniform-pass.cp-reference", "model": "circle-packing-tangency", "objectiveHistory": [ 3.944304526105059e-31 ], "preRuntimeCompilation": { "localityClass": "L2", "operations": [ "closed schema normalization", "exact intrinsic gates", "deterministic in-bound reference-radius construction" ] }, "residuals": { "maximumCurvatureResidual": 4.440892098500626e-16, "minimumTriangleMargin": 2.0000000000000004, "rmsCurvatureResidual": 4.440892098500626e-16 }, "runtimeRule": { "immutableLocalTargets": [ "K_i*", "Phi_ij" ], "localityClass": "L1", "radius": 1, "stateVariables": [ "u_i=log(r_i)" ], "synchronous": true, "updateEquation": "u_i^(n+1)=u_i^n+eta(K_i*-K_i(u^n))" }, "targetUsed": "original", "topology": { "edges": [ { "length": 1, "source": 0, "target": 1, "weight": 1 }, { "length": 1, "source": 0, "target": 2, "weight": 1 }, { "length": 1, "source": 0, "target": 3, "weight": 1 }, { "length": 1, "source": 1, "target": 2, "weight": 1 }, { "length": 1, "source": 1, "target": 3, "weight": 1 }, { "length": 1, "source": 2, "target": 3, "weight": 1 } ], "faces": [ [ 0, 1, 2 ], [ 0, 1, 3 ], [ 0, 2, 3 ], [ 1, 2, 3 ] ], "nodes": [ { "id": 0, "label": "v0", "radius": 1, "x": 0, "y": 0, "z": 0 }, { "id": 1, "label": "v1", "radius": 1, "x": 0, "y": 0, "z": 0 }, { "id": 2, "label": "v2", "radius": 1, "x": 0, "y": 0, "z": 0 }, { "id": 3, "label": "v3", "radius": 1, "x": 0, "y": 0, "z": 0 } ] }, "variables": { "edgeLengths": [ 2.0000000000000004, 2.0000000000000004, 2.0000000000000004, 2.0000000000000004, 2.0000000000000004, 2.0000000000000004 ], "radii": [ 1.0000000000000002, 1.0000000000000002, 1.0000000000000002, 1.0000000000000002 ] } } ], "claims": { "demonstrated": [ "Closed bounded request normalization and ordered finite intrinsic gates were executed deterministically.", "The declared Model CP target satisfies every bounded Chow–Luo condition under the verified theorem hypotheses." ], "notDemonstrated": [ "continuum convergence", "a constructive CP radius solve except when the reference residual passes", "extrinsic embeddability", "mechanical selection", "instrument observability", "physical validation", "fabrication readiness" ] }, "compilerVersion": "0.1.0-alpha.2", "digestAlgorithm": "fnv1a32-ieee754-json-v1", "fabricationPackage": null, "gates": [ { "basis": "combinatorial-identity", "code": "SCHEMA_VALID", "evidenceClass": "exact-finite-identity", "hypotheses": [], "id": "G0", "impossibleInDeclaredModel": false, "name": "schema, units, and finite bounds", "observed": null, "required": "schemaVersion 2 finite bounded request", "signedMargin": null, "stage": "intrinsic", "statement": "The request passed the closed, bounded schema and was normalized deterministically; configuration text was treated only as data.", "status": "PASS", "tolerance": null, "witness": { "limits": { "maxEdges": 1024, "maxExactSubsetLimit": 18, "maxExportedSubsetRows": 1024, "maxFaces": 1024, "maxIdentifierLength": 128, "maxSeedLength": 256, "maxSubsetEvaluations": 262142, "maxVertices": 256 }, "normalizedRequestDigest": "d6acc5c7" } }, { "basis": "combinatorial-identity", "code": "SURFACE_VALID", "evidenceClass": "exact-finite-identity", "hypotheses": [], "id": "G1", "impossibleInDeclaredModel": false, "name": "complex and surface validity", "observed": null, "required": "a finite pure triangular 2-manifold matching the declared topology", "signedMargin": null, "stage": "intrinsic", "statement": "The supplied cells form the declared finite triangular surface. A coherent face orientation exists exactly when the request declares orientability; this is a finite combinatorial statement, not continuum convergence.", "status": "PASS", "tolerance": null, "witness": { "baseSurfaceValidity": { "boundaryComponents": 0, "caveat": "This gate checks a finite pure triangular 2-manifold with nondegenerate intrinsic triangles and path/cycle vertex links; it does not certify orientability or continuum convergence.", "danglingEdges": [], "degenerateFaces": [], "duplicateFaces": [], "errors": [], "identityApplicable": true, "invalidVertexLinks": [], "metric": "circle-packing", "nonManifoldEdges": [], "valid": true }, "bettiGF2": [ 1, 0, 1 ], "boundaryComponents": 0, "errors": [], "eulerCharacteristic": 2, "orientable": true, "orientationContradiction": null, "simplexCounts": { "edges": 6, "faces": 4, "vertices": 4 } } }, { "basis": "theorem-instance", "code": "CURVATURE_LOCAL_BOUNDS_SATISFIED", "evidenceClass": "exact-finite-identity", "hypotheses": [ "nondegenerate Euclidean cone triangles", "interior total angle is positive", "boundary total angle uses the π convention" ], "id": "G2", "impossibleInDeclaredModel": false, "name": "local curvature upper bounds", "observed": 3.141592653589793, "required": "K_i < 2π for interior vertices and K_i < π for boundary vertices", "signedMargin": null, "stage": "intrinsic", "statement": "Every target defect is strictly below its boundary-aware Euclidean cone upper bound. These local inequalities are necessary but not sufficient for realizability.", "status": "PASS", "tolerance": 0, "witness": { "checkedVertices": 4, "symbolicComparison": true } }, { "basis": "theorem-instance", "code": "GAUSS_BONNET_TOTAL_SATISFIED", "evidenceClass": "exact-finite-identity", "hypotheses": [ "valid compact triangular surface", "interior defect 2π minus incident angles", "boundary defect π minus incident angles" ], "id": "G3", "impossibleInDeclaredModel": false, "name": "Gauss–Bonnet total", "observed": 12.566370614359172, "required": "Σ K_i = 2πχ = 12.566370614359172", "signedMargin": 0, "stage": "intrinsic", "statement": "The target total equals 2πχ exactly as a rational multiple of π.", "status": "PASS", "tolerance": 0, "witness": { "comparison": "exact-rational-multiple-of-pi", "eulerCharacteristic": 2, "observedPiCoefficient": { "denominator": 1, "numerator": 4 }, "observedTotal": 12.566370614359172, "requiredPiCoefficient": { "denominator": 1, "numerator": 4 }, "requiredTotal": 12.566370614359172, "unroundedResidual": 0 } }, { "basis": "exhaustive-finite-check", "code": "CHOW_LUO_SUBSETS_SATISFIED", "evidenceClass": "exact-finite-identity", "hypotheses": [ "connected triangulation of a closed surface", "simplicial complex with no loops/double edges/duplicate triples and minimum degree three", "Euclidean weighted circle-packing metric", "Phi:E→[0,π/2]", "Gauss–Bonnet total already passed" ], "id": "G4", "impossibleInDeclaredModel": false, "name": "Chow–Luo subset inequalities", "observed": 3.141592653589793, "required": "Δ_CL(I)>0 for every nonempty proper I⊂V; requested guard 1e-8", "signedMargin": 3.141592653589793, "stage": "model", "statement": "All nonempty proper vertex subsets were exhaustively evaluated. Under the verified closed, connected, simplicial Euclidean Model CP hypotheses, the Chow–Luo theorem admits the target in the normalized declared circle-packing model. This intrinsic statement does not establish an embedding, material response, actuator program, or physical shape.", "status": "PASS", "tolerance": 0, "witness": { "evaluatedSubsetCount": 14, "exactSymbolicTangencyArithmetic": true, "exhaustive": true, "exportedSubsetRowCount": 14, "minimum": { "cardinality": 3, "eulerCharacteristicFullSubcomplex": 1, "infimumCurvatureIncreaseForThisSubset": 0, "internalEdgeCount": 3, "internalFaceCount": 1, "link": [], "linkCount": 0, "linkSum": 0, "requiredLowerBound": 6.283185307179586, "slack": 3.141592653589793, "strictInequalityNote": "The displayed infimum is not sufficient when equality is attained; add a positive guard and rerun every subset.", "subset": [ 0, 1, 2 ], "symbolicSlackPiCoefficient": { "denominator": 1, "numerator": 1 }, "targetCurvatureMass": 9.42477796076938 }, "subsetLedgerDigest": "e4251845", "subsetLedgerDigestAlgorithm": "fnv1a32-ieee754-json-v1", "subsetSlacks": [ { "cardinality": 1, "eulerCharacteristicFullSubcomplex": 1, "internalEdgeCount": 0, "internalFaceCount": 0, "linkCount": 3, "linkSum": 9.42477796076938, "slack": 6.283185307179586, "subset": [ 0 ], "symbolicSlackPiCoefficient": { "denominator": 1, "numerator": 2 }, "targetCurvatureMass": 3.141592653589793 }, { "cardinality": 1, "eulerCharacteristicFullSubcomplex": 1, "internalEdgeCount": 0, "internalFaceCount": 0, "linkCount": 3, "linkSum": 9.42477796076938, "slack": 6.283185307179586, "subset": [ 1 ], "symbolicSlackPiCoefficient": { "denominator": 1, "numerator": 2 }, "targetCurvatureMass": 3.141592653589793 }, { "cardinality": 1, "eulerCharacteristicFullSubcomplex": 1, "internalEdgeCount": 0, "internalFaceCount": 0, "linkCount": 3, "linkSum": 9.42477796076938, "slack": 6.283185307179586, "subset": [ 2 ], "symbolicSlackPiCoefficient": { "denominator": 1, "numerator": 2 }, "targetCurvatureMass": 3.141592653589793 }, { "cardinality": 1, "eulerCharacteristicFullSubcomplex": 1, "internalEdgeCount": 0, "internalFaceCount": 0, "linkCount": 3, "linkSum": 9.42477796076938, "slack": 6.283185307179586, "subset": [ 3 ], "symbolicSlackPiCoefficient": { "denominator": 1, "numerator": 2 }, "targetCurvatureMass": 3.141592653589793 }, { "cardinality": 2, "eulerCharacteristicFullSubcomplex": 1, "internalEdgeCount": 1, "internalFaceCount": 0, "linkCount": 2, "linkSum": 6.283185307179586, "slack": 6.283185307179586, "subset": [ 0, 1 ], "symbolicSlackPiCoefficient": { "denominator": 1, "numerator": 2 }, "targetCurvatureMass": 6.283185307179586 }, { "cardinality": 2, "eulerCharacteristicFullSubcomplex": 1, "internalEdgeCount": 1, "internalFaceCount": 0, "linkCount": 2, "linkSum": 6.283185307179586, "slack": 6.283185307179586, "subset": [ 0, 2 ], "symbolicSlackPiCoefficient": { "denominator": 1, "numerator": 2 }, "targetCurvatureMass": 6.283185307179586 }, { "cardinality": 2, "eulerCharacteristicFullSubcomplex": 1, "internalEdgeCount": 1, "internalFaceCount": 0, "linkCount": 2, "linkSum": 6.283185307179586, "slack": 6.283185307179586, "subset": [ 0, 3 ], "symbolicSlackPiCoefficient": { "denominator": 1, "numerator": 2 }, "targetCurvatureMass": 6.283185307179586 }, { "cardinality": 2, "eulerCharacteristicFullSubcomplex": 1, "internalEdgeCount": 1, "internalFaceCount": 0, "linkCount": 2, "linkSum": 6.283185307179586, "slack": 6.283185307179586, "subset": [ 1, 2 ], "symbolicSlackPiCoefficient": { "denominator": 1, "numerator": 2 }, "targetCurvatureMass": 6.283185307179586 }, { "cardinality": 2, "eulerCharacteristicFullSubcomplex": 1, "internalEdgeCount": 1, "internalFaceCount": 0, "linkCount": 2, "linkSum": 6.283185307179586, "slack": 6.283185307179586, "subset": [ 1, 3 ], "symbolicSlackPiCoefficient": { "denominator": 1, "numerator": 2 }, "targetCurvatureMass": 6.283185307179586 }, { "cardinality": 2, "eulerCharacteristicFullSubcomplex": 1, "internalEdgeCount": 1, "internalFaceCount": 0, "linkCount": 2, "linkSum": 6.283185307179586, "slack": 6.283185307179586, "subset": [ 2, 3 ], "symbolicSlackPiCoefficient": { "denominator": 1, "numerator": 2 }, "targetCurvatureMass": 6.283185307179586 }, { "cardinality": 3, "eulerCharacteristicFullSubcomplex": 1, "internalEdgeCount": 3, "internalFaceCount": 1, "linkCount": 0, "linkSum": 0, "slack": 3.141592653589793, "subset": [ 0, 1, 2 ], "symbolicSlackPiCoefficient": { "denominator": 1, "numerator": 1 }, "targetCurvatureMass": 9.42477796076938 }, { "cardinality": 3, "eulerCharacteristicFullSubcomplex": 1, "internalEdgeCount": 3, "internalFaceCount": 1, "linkCount": 0, "linkSum": 0, "slack": 3.141592653589793, "subset": [ 0, 1, 3 ], "symbolicSlackPiCoefficient": { "denominator": 1, "numerator": 1 }, "targetCurvatureMass": 9.42477796076938 }, { "cardinality": 3, "eulerCharacteristicFullSubcomplex": 1, "internalEdgeCount": 3, "internalFaceCount": 1, "linkCount": 0, "linkSum": 0, "slack": 3.141592653589793, "subset": [ 0, 2, 3 ], "symbolicSlackPiCoefficient": { "denominator": 1, "numerator": 1 }, "targetCurvatureMass": 9.42477796076938 }, { "cardinality": 3, "eulerCharacteristicFullSubcomplex": 1, "internalEdgeCount": 3, "internalFaceCount": 1, "linkCount": 0, "linkSum": 0, "slack": 3.141592653589793, "subset": [ 1, 2, 3 ], "symbolicSlackPiCoefficient": { "denominator": 1, "numerator": 1 }, "targetCurvatureMass": 9.42477796076938 } ], "subsetSlacksComplete": true, "totalProperNonemptySubsetCount": 14 } }, { "basis": "combinatorial-identity", "code": "NOT_APPLICABLE", "evidenceClass": "exact-finite-identity", "hypotheses": [], "id": "G5", "impossibleInDeclaredModel": false, "name": "symmetric disk doubling", "observed": null, "required": "not evaluated", "signedMargin": null, "stage": "model", "statement": "The supplied CP surface is already closed; symmetric disk doubling is unnecessary.", "status": "NOT_APPLICABLE", "tolerance": null, "witness": null }, { "basis": "simulation", "code": "TRIANGLE_MARGINS_SATISFIED", "evidenceClass": "finite-numerical-estimate", "hypotheses": [ "finite declared edge vector", "one length per simplicial edge", "strict Euclidean triangle inequalities" ], "id": "G6", "impossibleInDeclaredModel": false, "name": "metric positivity and triangle margin", "observed": 1, "required": "positive lengths and positive relative triangle margin; guard 1e-8", "signedMargin": 2.0000000000000004, "stage": "intrinsic", "statement": "An analytically intersected global-scale interval yields an in-bound equal-radius CP reference metric. It is a metric-domain witness, not a solved curvature target.", "status": "PASS", "tolerance": 0, "witness": { "checkedFaceCount": 4, "hardBoundFailures": [], "minimumTriangle": { "edgeLengths": [ 2.0000000000000004, 2.0000000000000004, 2.0000000000000004 ], "exactDyadicSign": 1, "face": [ 0, 1, 2 ], "faceIndex": 0, "margin": 2.0000000000000004, "relativeMargin": 1 }, "nonpositiveLengths": [], "referenceOnly": true, "referenceScale": { "feasible": true, "interval": [ 0.5, 2 ], "rejectedInterval": null, "scope": "equal-radius reference ray only; not a solved target radius ratio", "selected": 1.0000000000000002 }, "trianglePredicate": "exact sign of the sum of two supplied binary64 dyadics minus the third" } } ], "normalizedRequest": { "constraints": { "allowTopologyChange": false, "allowedBoundaryValences": [], "allowedInteriorValences": [], "edgeLengthBounds": [ 1e-9, 10 ], "maxStrain": 0.1, "phiByEdge": [ 0, 0, 0, 0, 0, 0 ], "radiusBounds": [ 0.5, 2 ] }, "id": "tetra-uniform-pass", "models": [ "circle-packing-tangency" ], "schemaVersion": 2, "seed": "exact-deterministic", "target": { "curvature": { "basis": "symbolic-pi", "piCoefficients": [ { "denominator": 1, "numerator": 1 }, { "denominator": 1, "numerator": 1 }, { "denominator": 1, "numerator": 1 }, { "denominator": 1, "numerator": 1 } ] }, "discretization": { "complex": { "edges": [ { "length": 1, "source": 0, "target": 1, "weight": 1 }, { "length": 1, "source": 0, "target": 2, "weight": 1 }, { "length": 1, "source": 0, "target": 3, "weight": 1 }, { "length": 1, "source": 1, "target": 2, "weight": 1 }, { "length": 1, "source": 1, "target": 3, "weight": 1 }, { "length": 1, "source": 2, "target": 3, "weight": 1 } ], "faces": [ [ 0, 1, 2 ], [ 0, 1, 3 ], [ 0, 2, 3 ], [ 1, 2, 3 ] ], "nodes": [ { "id": 0, "label": "v0", "radius": 1, "x": 0, "y": 0, "z": 0 }, { "id": 1, "label": "v1", "radius": 1, "x": 0, "y": 0, "z": 0 }, { "id": 2, "label": "v2", "radius": 1, "x": 0, "y": 0, "z": 0 }, { "id": 3, "label": "v3", "radius": 1, "x": 0, "y": 0, "z": 0 } ] } }, "kind": "intrinsic-curvature", "topology": { "boundaryComponents": 0, "expectedBetti": null, "orientable": true, "surface": "sphere" } }, "verification": { "chowLuoGuard": 1e-8, "exactSubsetLimit": 18, "gbTolerance": 0, "maxIterations": 0, "optimizerTolerance": 1e-8, "refinementLevels": 0, "uncertaintySamples": 0 } }, "obstructionCertificates": [], "refinement": null, "repairProposals": [], "replay": { "deterministic": true, "methods": [ "closed-schema-normalization", "release-1-surface-and-GF2-topology", "exact-rational-pi-when-declared", "ordered-local-and-total-curvature-gates", "bounded-Chow-Luo-subset-enumeration", "exhaustive-triangle-margin-audit" ], "platformIndependentInputsDigest": "d6acc5c7", "seed": "exact-deterministic" }, "reportDigest": "321001a1", "requestDigest": "d6acc5c7", "requestId": "tetra-uniform-pass", "robustness": null, "schemaVersion": 2, "selectedCandidateId": "tetra-uniform-pass.cp-reference", "status": "PASS" } }, { "id": "triangle-valid-pass", "request": { "schemaVersion": 2, "id": "triangle-valid-pass", "seed": "exact-deterministic", "target": { "kind": "intrinsic-edge-metric", "topology": { "surface": "disk", "orientable": true, "boundaryComponents": 1 }, "discretization": { "complex": { "nodes": [ { "id": 0, "label": "v0" }, { "id": 1, "label": "v1" }, { "id": 2, "label": "v2" } ], "edges": [ { "source": 0, "target": 1, "length": 1, "weight": 1 }, { "source": 0, "target": 2, "length": 1, "weight": 1 }, { "source": 1, "target": 2, "length": 1, "weight": 1 } ], "faces": [ [ 0, 1, 2 ] ] } }, "edgeLengths": [ 1, 1, 1 ] }, "models": [ "edge-metric" ], "constraints": { "radiusBounds": [ 0.5, 2 ], "edgeLengthBounds": [ 1e-9, 10 ], "allowTopologyChange": false }, "verification": { "gbTolerance": 0, "chowLuoGuard": 1e-8, "exactSubsetLimit": 18, "optimizerTolerance": 1e-8, "maxIterations": 0, "refinementLevels": 0, "uncertaintySamples": 0 } }, "report": { "candidates": [ { "caveats": [ "This is an intrinsic finite edge metric, not an R3 embedding or a physical sheet." ], "evidenceClass": "exact-finite-identity", "hardBoundsSatisfied": true, "id": "triangle-valid-pass.edge-metric", "model": "edge-metric", "objectiveHistory": [], "preRuntimeCompilation": { "localityClass": "L2", "operations": [ "closed schema normalization", "surface validation", "exhaustive face triangle audit" ] }, "residuals": { "minimumTriangleMargin": 1 }, "runtimeRule": { "immutableLocalTargets": [ "declared edge length" ], "localityClass": "L0", "radius": 0, "stateVariables": [ "edge length" ], "synchronous": true, "updateEquation": "ell_e(t+1)=ell_e(t)" }, "targetUsed": "original", "topology": { "edges": [ { "length": 1, "source": 0, "target": 1, "weight": 1 }, { "length": 1, "source": 0, "target": 2, "weight": 1 }, { "length": 1, "source": 1, "target": 2, "weight": 1 } ], "faces": [ [ 0, 1, 2 ] ], "nodes": [ { "id": 0, "label": "v0", "radius": 1, "x": 0, "y": 0, "z": 0 }, { "id": 1, "label": "v1", "radius": 1, "x": 0, "y": 0, "z": 0 }, { "id": 2, "label": "v2", "radius": 1, "x": 0, "y": 0, "z": 0 } ] }, "variables": { "edgeLengths": [ 1, 1, 1 ] } } ], "claims": { "demonstrated": [ "Closed bounded request normalization and ordered finite intrinsic gates were executed deterministically.", "The supplied finite edge vector defines strict Euclidean triangles within the declared bounds." ], "notDemonstrated": [ "continuum convergence", "a constructive CP radius solve except when the reference residual passes", "extrinsic embeddability", "mechanical selection", "instrument observability", "physical validation", "fabrication readiness" ] }, "compilerVersion": "0.1.0-alpha.2", "digestAlgorithm": "fnv1a32-ieee754-json-v1", "fabricationPackage": null, "gates": [ { "basis": "combinatorial-identity", "code": "SCHEMA_VALID", "evidenceClass": "exact-finite-identity", "hypotheses": [], "id": "G0", "impossibleInDeclaredModel": false, "name": "schema, units, and finite bounds", "observed": null, "required": "schemaVersion 2 finite bounded request", "signedMargin": null, "stage": "intrinsic", "statement": "The request passed the closed, bounded schema and was normalized deterministically; configuration text was treated only as data.", "status": "PASS", "tolerance": null, "witness": { "limits": { "maxEdges": 1024, "maxExactSubsetLimit": 18, "maxExportedSubsetRows": 1024, "maxFaces": 1024, "maxIdentifierLength": 128, "maxSeedLength": 256, "maxSubsetEvaluations": 262142, "maxVertices": 256 }, "normalizedRequestDigest": "3a644bd6" } }, { "basis": "combinatorial-identity", "code": "SURFACE_VALID", "evidenceClass": "exact-finite-identity", "hypotheses": [], "id": "G1", "impossibleInDeclaredModel": false, "name": "complex and surface validity", "observed": null, "required": "a finite pure triangular 2-manifold matching the declared topology", "signedMargin": null, "stage": "intrinsic", "statement": "The supplied cells form the declared finite triangular surface. 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These local inequalities are necessary but not sufficient for realizability.", "status": "PASS", "tolerance": 0, "witness": { "checkedVertices": 4, "symbolicComparison": true } }, { "basis": "theorem-instance", "code": "GAUSS_BONNET_TOTAL_SATISFIED", "evidenceClass": "exact-finite-identity", "hypotheses": [ "valid compact triangular surface", "interior defect 2π minus incident angles", "boundary defect π minus incident angles" ], "id": "G3", "impossibleInDeclaredModel": false, "name": "Gauss–Bonnet total", "observed": 12.566370614359174, "required": "Σ K_i = 2πχ = 12.566370614359172", "signedMargin": 0, "stage": "intrinsic", "statement": "The target total equals 2πχ exactly as a rational multiple of π.", "status": "PASS", "tolerance": 0, "witness": { "comparison": "exact-rational-multiple-of-pi", "eulerCharacteristic": 2, "observedPiCoefficient": { "denominator": 1, "numerator": 4 }, "observedTotal": 12.566370614359174, "requiredPiCoefficient": { "denominator": 1, "numerator": 4 }, "requiredTotal": 12.566370614359172, "unroundedResidual": 1.7763568394002505e-15 } }, { "basis": "exhaustive-finite-check", "code": "CHOW_LUO_SUBSET", "evidenceClass": "exact-finite-identity", "hypotheses": [ "connected triangulation of a closed surface", "simplicial complex with no loops/double edges/duplicate triples and minimum degree three", "Euclidean weighted circle-packing metric", "Phi:E→[0,π/2]", "Gauss–Bonnet total already passed" ], "id": "G4", "impossibleInDeclaredModel": true, "name": "Chow–Luo subset inequalities", "observed": -0.6283185307179586, "required": "Δ_CL(I)>0 for every nonempty proper I⊂V; requested guard 1e-8", "signedMargin": -0.6283185307179586, "stage": "model", "statement": "A displayed vertex subset violates the strict Chow–Luo curvature-polytope inequality. Under the verified hypotheses this target is impossible in the declared Model CP, but this says nothing universal about arbitrary edge metrics, embeddings, or physical sheets.", "status": "FAIL", "tolerance": 0, "witness": { "evaluatedSubsetCount": 14, "exactSymbolicTangencyArithmetic": true, "exhaustive": true, "exportedSubsetRowCount": 14, "minimum": { "cardinality": 1, "eulerCharacteristicFullSubcomplex": 1, "infimumCurvatureIncreaseForThisSubset": 0.6283185307179586, "internalEdgeCount": 0, "internalFaceCount": 0, "link": [ { "edge": [ 1, 2 ], "face": 0, "phi": 0, "vertex": 0, "weight": 3.141592653589793 }, { "edge": [ 1, 3 ], "face": 1, "phi": 0, "vertex": 0, "weight": 3.141592653589793 }, { "edge": [ 2, 3 ], "face": 2, "phi": 0, "vertex": 0, "weight": 3.141592653589793 } ], "linkCount": 3, "linkSum": 9.42477796076938, "requiredLowerBound": -3.141592653589793, "slack": -0.6283185307179586, "strictInequalityNote": "The displayed infimum is not sufficient when equality is attained; add a positive guard and rerun every subset.", "subset": [ 0 ], "symbolicSlackPiCoefficient": { "denominator": 5, "numerator": -1 }, "targetCurvatureMass": -3.7699111843077517 }, "subsetLedgerDigest": "62b0c69b", "subsetLedgerDigestAlgorithm": "fnv1a32-ieee754-json-v1", "subsetSlacks": [ { "cardinality": 1, "eulerCharacteristicFullSubcomplex": 1, "internalEdgeCount": 0, "internalFaceCount": 0, "linkCount": 3, "linkSum": 9.42477796076938, "slack": -0.6283185307179586, "subset": [ 0 ], "symbolicSlackPiCoefficient": { "denominator": 5, "numerator": -1 }, "targetCurvatureMass": -3.7699111843077517 }, { "cardinality": 1, "eulerCharacteristicFullSubcomplex": 1, "internalEdgeCount": 0, "internalFaceCount": 0, "linkCount": 3, "linkSum": 9.42477796076938, "slack": 8.587019919812102, "subset": [ 1 ], "symbolicSlackPiCoefficient": { "denominator": 15, "numerator": 41 }, "targetCurvatureMass": 5.445427266222309 }, { "cardinality": 1, "eulerCharacteristicFullSubcomplex": 1, "internalEdgeCount": 0, "internalFaceCount": 0, "linkCount": 3, "linkSum": 9.42477796076938, "slack": 8.587019919812102, "subset": [ 2 ], "symbolicSlackPiCoefficient": { "denominator": 15, "numerator": 41 }, "targetCurvatureMass": 5.445427266222309 }, { "cardinality": 1, "eulerCharacteristicFullSubcomplex": 1, "internalEdgeCount": 0, "internalFaceCount": 0, "linkCount": 3, "linkSum": 9.42477796076938, "slack": 8.587019919812102, "subset": [ 3 ], "symbolicSlackPiCoefficient": { "denominator": 15, "numerator": 41 }, "targetCurvatureMass": 5.445427266222309 }, { "cardinality": 2, "eulerCharacteristicFullSubcomplex": 1, "internalEdgeCount": 1, "internalFaceCount": 0, "linkCount": 2, "linkSum": 6.283185307179586, "slack": 1.675516081914557, "subset": [ 0, 1 ], "symbolicSlackPiCoefficient": { "denominator": 15, "numerator": 8 }, "targetCurvatureMass": 1.675516081914557 }, { "cardinality": 2, "eulerCharacteristicFullSubcomplex": 1, "internalEdgeCount": 1, "internalFaceCount": 0, "linkCount": 2, "linkSum": 6.283185307179586, "slack": 1.675516081914557, "subset": [ 0, 2 ], "symbolicSlackPiCoefficient": { "denominator": 15, "numerator": 8 }, "targetCurvatureMass": 1.675516081914557 }, { "cardinality": 2, "eulerCharacteristicFullSubcomplex": 1, "internalEdgeCount": 1, "internalFaceCount": 0, "linkCount": 2, "linkSum": 6.283185307179586, "slack": 1.675516081914557, "subset": [ 0, 3 ], "symbolicSlackPiCoefficient": { "denominator": 15, "numerator": 8 }, "targetCurvatureMass": 1.675516081914557 }, { "cardinality": 2, "eulerCharacteristicFullSubcomplex": 1, "internalEdgeCount": 1, "internalFaceCount": 0, "linkCount": 2, "linkSum": 6.283185307179586, "slack": 10.890854532444617, "subset": [ 1, 2 ], "symbolicSlackPiCoefficient": { "denominator": 15, "numerator": 52 }, "targetCurvatureMass": 10.890854532444617 }, { "cardinality": 2, "eulerCharacteristicFullSubcomplex": 1, "internalEdgeCount": 1, "internalFaceCount": 0, "linkCount": 2, "linkSum": 6.283185307179586, "slack": 10.890854532444617, "subset": [ 1, 3 ], "symbolicSlackPiCoefficient": { "denominator": 15, "numerator": 52 }, "targetCurvatureMass": 10.890854532444617 }, { "cardinality": 2, "eulerCharacteristicFullSubcomplex": 1, "internalEdgeCount": 1, "internalFaceCount": 0, "linkCount": 2, "linkSum": 6.283185307179586, "slack": 10.890854532444617, "subset": [ 2, 3 ], "symbolicSlackPiCoefficient": { "denominator": 15, "numerator": 52 }, "targetCurvatureMass": 10.890854532444617 }, { "cardinality": 3, "eulerCharacteristicFullSubcomplex": 1, "internalEdgeCount": 3, "internalFaceCount": 1, "linkCount": 0, "linkSum": 0, "slack": 0.8377580409572793, "subset": [ 0, 1, 2 ], "symbolicSlackPiCoefficient": { "denominator": 15, "numerator": 4 }, "targetCurvatureMass": 7.120943348136866 }, { "cardinality": 3, "eulerCharacteristicFullSubcomplex": 1, "internalEdgeCount": 3, "internalFaceCount": 1, "linkCount": 0, "linkSum": 0, "slack": 0.8377580409572793, "subset": [ 0, 1, 3 ], "symbolicSlackPiCoefficient": { "denominator": 15, "numerator": 4 }, "targetCurvatureMass": 7.120943348136866 }, { "cardinality": 3, "eulerCharacteristicFullSubcomplex": 1, "internalEdgeCount": 3, "internalFaceCount": 1, "linkCount": 0, "linkSum": 0, "slack": 0.8377580409572793, "subset": [ 0, 2, 3 ], "symbolicSlackPiCoefficient": { "denominator": 15, "numerator": 4 }, "targetCurvatureMass": 7.120943348136866 }, { "cardinality": 3, "eulerCharacteristicFullSubcomplex": 1, "internalEdgeCount": 3, "internalFaceCount": 1, "linkCount": 0, "linkSum": 0, "slack": 10.053096491487338, "subset": [ 1, 2, 3 ], "symbolicSlackPiCoefficient": { "denominator": 5, "numerator": 16 }, "targetCurvatureMass": 16.336281798666924 } ], "subsetSlacksComplete": true, "totalProperNonemptySubsetCount": 14 } }, { "basis": "combinatorial-identity", "code": "NOT_APPLICABLE", "evidenceClass": "exact-finite-identity", "hypotheses": [], "id": "G5", "impossibleInDeclaredModel": false, "name": "symmetric disk doubling", "observed": null, "required": "not evaluated", "signedMargin": null, "stage": "model", "statement": "The supplied CP surface is already closed; symmetric disk doubling is unnecessary.", "status": "NOT_APPLICABLE", "tolerance": null, "witness": null }, { "basis": "simulation", "code": "TRIANGLE_MARGINS_SATISFIED", "evidenceClass": "finite-numerical-estimate", "hypotheses": [ "finite declared edge vector", "one length per simplicial edge", "strict Euclidean triangle inequalities" ], "id": "G6", "impossibleInDeclaredModel": false, "name": "metric positivity and triangle margin", "observed": 1, "required": "positive lengths and positive relative triangle margin; guard 1e-8", "signedMargin": 2.0000000000000004, "stage": "intrinsic", "statement": "An analytically intersected global-scale interval yields an in-bound equal-radius CP reference metric. It is a metric-domain witness, not a solved curvature target.", "status": "PASS", "tolerance": 0, "witness": { "checkedFaceCount": 4, "hardBoundFailures": [], "minimumTriangle": { "edgeLengths": [ 2.0000000000000004, 2.0000000000000004, 2.0000000000000004 ], "exactDyadicSign": 1, "face": [ 0, 1, 2 ], "faceIndex": 0, "margin": 2.0000000000000004, "relativeMargin": 1 }, "nonpositiveLengths": [], "referenceOnly": true, "referenceScale": { "feasible": true, "interval": [ 0.5, 2 ], "rejectedInterval": null, "scope": "equal-radius reference ray only; not a solved target radius ratio", "selected": 1.0000000000000002 }, "trianglePredicate": "exact sign of the sum of two supplied binary64 dyadics minus the third" } } ], "normalizedRequest": { "constraints": { "allowTopologyChange": false, "allowedBoundaryValences": [], "allowedInteriorValences": [], "edgeLengthBounds": [ 1e-9, 10 ], "maxStrain": 0.1, "phiByEdge": [ 0, 0, 0, 0, 0, 0 ], "radiusBounds": [ 0.5, 2 ] }, "id": "tetra-subset-reject", "models": [ "circle-packing-tangency" ], "schemaVersion": 2, "seed": "exact-deterministic", "target": { "curvature": { "basis": "symbolic-pi", "piCoefficients": [ { "denominator": 5, "numerator": -6 }, { "denominator": 15, "numerator": 26 }, { "denominator": 15, "numerator": 26 }, { "denominator": 15, "numerator": 26 } ] }, "discretization": { "complex": { "edges": [ { "length": 1, "source": 0, "target": 1, "weight": 1 }, { "length": 1, "source": 0, "target": 2, "weight": 1 }, { "length": 1, "source": 0, "target": 3, "weight": 1 }, { "length": 1, "source": 1, "target": 2, "weight": 1 }, { "length": 1, "source": 1, "target": 3, "weight": 1 }, { "length": 1, "source": 2, "target": 3, "weight": 1 } ], "faces": [ [ 0, 1, 2 ], [ 0, 1, 3 ], [ 0, 2, 3 ], [ 1, 2, 3 ] ], "nodes": [ { "id": 0, "label": "v0", "radius": 1, "x": 0, "y": 0, "z": 0 }, { "id": 1, "label": "v1", "radius": 1, "x": 0, "y": 0, "z": 0 }, { "id": 2, "label": "v2", "radius": 1, "x": 0, "y": 0, "z": 0 }, { "id": 3, "label": "v3", "radius": 1, "x": 0, "y": 0, "z": 0 } ] } }, "kind": "intrinsic-curvature", "topology": { "boundaryComponents": 0, "expectedBetti": null, "orientable": true, "surface": "sphere" } }, "verification": { "chowLuoGuard": 1e-8, "exactSubsetLimit": 18, "gbTolerance": 0, "maxIterations": 0, "optimizerTolerance": 1e-8, "refinementLevels": 0, "uncertaintySamples": 0 } }, "obstructionCertificates": [ { "basis": "exhaustive-finite-check", "code": "CHOW_LUO_SUBSET", "evidenceClass": "exact-finite-identity", "hypotheses": [ "connected triangulation of a closed surface", "simplicial complex with no loops/double edges/duplicate triples and minimum degree three", "Euclidean weighted circle-packing metric", "Phi:E→[0,π/2]", "Gauss–Bonnet total already passed" ], "impossibleInDeclaredModel": true, "observed": -0.6283185307179586, "repairHints": [ "Redistribute curvature mass away from the violating subset while preserving the total, then rerun every subset inequality." ], "required": "Δ_CL(I)>0 for every nonempty proper I⊂V; requested guard 1e-8", "signedMargin": -0.6283185307179586, "stage": "model", "statement": "A displayed vertex subset violates the strict Chow–Luo curvature-polytope inequality. Under the verified hypotheses this target is impossible in the declared Model CP, but this says nothing universal about arbitrary edge metrics, embeddings, or physical sheets.", "status": "FAIL", "tolerance": 0, "witness": { "evaluatedSubsetCount": 14, "exactSymbolicTangencyArithmetic": true, "exhaustive": true, "exportedSubsetRowCount": 14, "minimum": { "cardinality": 1, "eulerCharacteristicFullSubcomplex": 1, "infimumCurvatureIncreaseForThisSubset": 0.6283185307179586, "internalEdgeCount": 0, "internalFaceCount": 0, "link": [ { "edge": [ 1, 2 ], "face": 0, "phi": 0, "vertex": 0, "weight": 3.141592653589793 }, { "edge": [ 1, 3 ], "face": 1, "phi": 0, "vertex": 0, "weight": 3.141592653589793 }, { "edge": [ 2, 3 ], "face": 2, "phi": 0, "vertex": 0, "weight": 3.141592653589793 } ], "linkCount": 3, "linkSum": 9.42477796076938, "requiredLowerBound": -3.141592653589793, "slack": -0.6283185307179586, "strictInequalityNote": "The displayed infimum is not sufficient when equality is attained; add a positive guard and rerun every subset.", "subset": [ 0 ], "symbolicSlackPiCoefficient": { "denominator": 5, "numerator": -1 }, "targetCurvatureMass": -3.7699111843077517 }, "subsetLedgerDigest": "62b0c69b", "subsetLedgerDigestAlgorithm": "fnv1a32-ieee754-json-v1", "subsetSlacks": [ { "cardinality": 1, "eulerCharacteristicFullSubcomplex": 1, "internalEdgeCount": 0, "internalFaceCount": 0, "linkCount": 3, "linkSum": 9.42477796076938, "slack": -0.6283185307179586, "subset": [ 0 ], "symbolicSlackPiCoefficient": { "denominator": 5, "numerator": -1 }, "targetCurvatureMass": -3.7699111843077517 }, { "cardinality": 1, "eulerCharacteristicFullSubcomplex": 1, "internalEdgeCount": 0, "internalFaceCount": 0, "linkCount": 3, "linkSum": 9.42477796076938, "slack": 8.587019919812102, "subset": [ 1 ], "symbolicSlackPiCoefficient": { "denominator": 15, "numerator": 41 }, "targetCurvatureMass": 5.445427266222309 }, { "cardinality": 1, "eulerCharacteristicFullSubcomplex": 1, "internalEdgeCount": 0, "internalFaceCount": 0, "linkCount": 3, "linkSum": 9.42477796076938, "slack": 8.587019919812102, "subset": [ 2 ], "symbolicSlackPiCoefficient": { "denominator": 15, "numerator": 41 }, "targetCurvatureMass": 5.445427266222309 }, { "cardinality": 1, "eulerCharacteristicFullSubcomplex": 1, "internalEdgeCount": 0, "internalFaceCount": 0, "linkCount": 3, "linkSum": 9.42477796076938, "slack": 8.587019919812102, "subset": [ 3 ], "symbolicSlackPiCoefficient": { "denominator": 15, "numerator": 41 }, "targetCurvatureMass": 5.445427266222309 }, { "cardinality": 2, "eulerCharacteristicFullSubcomplex": 1, "internalEdgeCount": 1, "internalFaceCount": 0, "linkCount": 2, "linkSum": 6.283185307179586, "slack": 1.675516081914557, "subset": [ 0, 1 ], "symbolicSlackPiCoefficient": { "denominator": 15, "numerator": 8 }, "targetCurvatureMass": 1.675516081914557 }, { "cardinality": 2, "eulerCharacteristicFullSubcomplex": 1, "internalEdgeCount": 1, "internalFaceCount": 0, "linkCount": 2, "linkSum": 6.283185307179586, "slack": 1.675516081914557, "subset": [ 0, 2 ], "symbolicSlackPiCoefficient": { "denominator": 15, "numerator": 8 }, "targetCurvatureMass": 1.675516081914557 }, { "cardinality": 2, "eulerCharacteristicFullSubcomplex": 1, "internalEdgeCount": 1, "internalFaceCount": 0, "linkCount": 2, "linkSum": 6.283185307179586, "slack": 1.675516081914557, "subset": [ 0, 3 ], "symbolicSlackPiCoefficient": { "denominator": 15, "numerator": 8 }, "targetCurvatureMass": 1.675516081914557 }, { "cardinality": 2, "eulerCharacteristicFullSubcomplex": 1, "internalEdgeCount": 1, "internalFaceCount": 0, "linkCount": 2, "linkSum": 6.283185307179586, "slack": 10.890854532444617, "subset": [ 1, 2 ], "symbolicSlackPiCoefficient": { "denominator": 15, "numerator": 52 }, "targetCurvatureMass": 10.890854532444617 }, { "cardinality": 2, "eulerCharacteristicFullSubcomplex": 1, "internalEdgeCount": 1, "internalFaceCount": 0, "linkCount": 2, "linkSum": 6.283185307179586, "slack": 10.890854532444617, "subset": [ 1, 3 ], "symbolicSlackPiCoefficient": { "denominator": 15, "numerator": 52 }, "targetCurvatureMass": 10.890854532444617 }, { "cardinality": 2, "eulerCharacteristicFullSubcomplex": 1, "internalEdgeCount": 1, "internalFaceCount": 0, "linkCount": 2, "linkSum": 6.283185307179586, "slack": 10.890854532444617, "subset": [ 2, 3 ], "symbolicSlackPiCoefficient": { "denominator": 15, "numerator": 52 }, "targetCurvatureMass": 10.890854532444617 }, { "cardinality": 3, "eulerCharacteristicFullSubcomplex": 1, "internalEdgeCount": 3, "internalFaceCount": 1, "linkCount": 0, "linkSum": 0, "slack": 0.8377580409572793, "subset": [ 0, 1, 2 ], "symbolicSlackPiCoefficient": { "denominator": 15, "numerator": 4 }, "targetCurvatureMass": 7.120943348136866 }, { "cardinality": 3, "eulerCharacteristicFullSubcomplex": 1, "internalEdgeCount": 3, "internalFaceCount": 1, "linkCount": 0, "linkSum": 0, "slack": 0.8377580409572793, "subset": [ 0, 1, 3 ], "symbolicSlackPiCoefficient": { "denominator": 15, "numerator": 4 }, "targetCurvatureMass": 7.120943348136866 }, { "cardinality": 3, "eulerCharacteristicFullSubcomplex": 1, "internalEdgeCount": 3, "internalFaceCount": 1, "linkCount": 0, "linkSum": 0, "slack": 0.8377580409572793, "subset": [ 0, 2, 3 ], "symbolicSlackPiCoefficient": { "denominator": 15, "numerator": 4 }, "targetCurvatureMass": 7.120943348136866 }, { "cardinality": 3, "eulerCharacteristicFullSubcomplex": 1, "internalEdgeCount": 3, "internalFaceCount": 1, "linkCount": 0, "linkSum": 0, "slack": 10.053096491487338, "subset": [ 1, 2, 3 ], "symbolicSlackPiCoefficient": { "denominator": 5, "numerator": 16 }, "targetCurvatureMass": 16.336281798666924 } ], "subsetSlacksComplete": true, "totalProperNonemptySubsetCount": 14 } } ], "refinement": null, "repairProposals": [], "replay": { "deterministic": true, "methods": [ "closed-schema-normalization", "release-1-surface-and-GF2-topology", "exact-rational-pi-when-declared", "ordered-local-and-total-curvature-gates", "bounded-Chow-Luo-subset-enumeration", "exhaustive-triangle-margin-audit" ], "platformIndependentInputsDigest": "7f4ed74d", "seed": "exact-deterministic" }, "reportDigest": "740d3a4a", "requestDigest": "7f4ed74d", "requestId": "tetra-subset-reject", "robustness": null, "schemaVersion": 2, "selectedCandidateId": null, "status": "FAIL" } }, { "id": "sphere-zero-reject", "request": { "schemaVersion": 2, "id": "sphere-zero-reject", "seed": "exact-deterministic", "target": { "kind": "intrinsic-curvature", "topology": { "surface": "sphere", "orientable": true, "boundaryComponents": 0 }, "discretization": { "complex": { "nodes": [ { "id": 0, "label": "v0", "x": 0, "y": 0, "z": 0, "radius": 1 }, { "id": 1, "label": "v1", "x": 0, "y": 0, "z": 0, "radius": 1 }, { "id": 2, "label": "v2", "x": 0, "y": 0, "z": 0, "radius": 1 }, { "id": 3, "label": "v3", "x": 0, "y": 0, "z": 0, "radius": 1 } ], "edges": [ { "source": 0, "target": 1, "length": 1, "weight": 1 }, { "source": 0, "target": 2, "length": 1, "weight": 1 }, { "source": 0, "target": 3, "length": 1, "weight": 1 }, { "source": 1, "target": 2, "length": 1, "weight": 1 }, { "source": 1, "target": 3, "length": 1, "weight": 1 }, { "source": 2, "target": 3, "length": 1, "weight": 1 } ], "faces": [ [ 0, 1, 2 ], [ 0, 1, 3 ], [ 0, 2, 3 ], [ 1, 2, 3 ] ] } }, "curvature": { "basis": "symbolic-pi", "piCoefficients": [ { "numerator": 0, "denominator": 1 }, { "numerator": 0, "denominator": 1 }, { "numerator": 0, "denominator": 1 }, { "numerator": 0, "denominator": 1 } ] } }, "models": [ "circle-packing-tangency" ], "constraints": { "radiusBounds": [ 0.5, 2 ], "edgeLengthBounds": [ 1e-9, 10 ], "allowTopologyChange": false }, "verification": { "gbTolerance": 0, "chowLuoGuard": 1e-8, "exactSubsetLimit": 18, "optimizerTolerance": 1e-8, "maxIterations": 0, "refinementLevels": 0, "uncertaintySamples": 0 } }, "report": { "candidates": [], "claims": { "demonstrated": [ "Closed bounded request normalization and ordered finite intrinsic gates were executed deterministically." ], "notDemonstrated": [ "continuum convergence", "a constructive CP radius solve except when the reference residual passes", "extrinsic embeddability", "mechanical selection", "instrument observability", "physical validation", "fabrication readiness" ] }, "compilerVersion": "0.1.0-alpha.2", "digestAlgorithm": "fnv1a32-ieee754-json-v1", "fabricationPackage": null, "gates": [ { "basis": "combinatorial-identity", "code": "SCHEMA_VALID", "evidenceClass": "exact-finite-identity", "hypotheses": [], "id": "G0", "impossibleInDeclaredModel": false, "name": "schema, units, and finite bounds", "observed": null, "required": "schemaVersion 2 finite bounded request", "signedMargin": null, "stage": "intrinsic", "statement": "The request passed the closed, bounded schema and was normalized deterministically; configuration text was treated only as data.", "status": "PASS", "tolerance": null, "witness": { "limits": { "maxEdges": 1024, "maxExactSubsetLimit": 18, "maxExportedSubsetRows": 1024, "maxFaces": 1024, "maxIdentifierLength": 128, "maxSeedLength": 256, "maxSubsetEvaluations": 262142, "maxVertices": 256 }, "normalizedRequestDigest": "99549406" } }, { "basis": "combinatorial-identity", "code": "SURFACE_VALID", "evidenceClass": "exact-finite-identity", "hypotheses": [], "id": "G1", "impossibleInDeclaredModel": false, "name": "complex and surface validity", "observed": null, "required": "a finite pure triangular 2-manifold matching the declared topology", "signedMargin": null, "stage": "intrinsic", "statement": "The supplied cells form the declared finite triangular surface. A coherent face orientation exists exactly when the request declares orientability; this is a finite combinatorial statement, not continuum convergence.", "status": "PASS", "tolerance": null, "witness": { "baseSurfaceValidity": { "boundaryComponents": 0, "caveat": "This gate checks a finite pure triangular 2-manifold with nondegenerate intrinsic triangles and path/cycle vertex links; it does not certify orientability or continuum convergence.", "danglingEdges": [], "degenerateFaces": [], "duplicateFaces": [], "errors": [], "identityApplicable": true, "invalidVertexLinks": [], "metric": "circle-packing", "nonManifoldEdges": [], "valid": true }, "bettiGF2": [ 1, 0, 1 ], "boundaryComponents": 0, "errors": [], "eulerCharacteristic": 2, "orientable": true, "orientationContradiction": null, "simplexCounts": { "edges": 6, "faces": 4, "vertices": 4 } } }, { "basis": "theorem-instance", "code": "CURVATURE_LOCAL_BOUNDS_SATISFIED", "evidenceClass": "exact-finite-identity", "hypotheses": [ "nondegenerate Euclidean cone triangles", "interior total angle is positive", "boundary total angle uses the π convention" ], "id": "G2", "impossibleInDeclaredModel": false, "name": "local curvature upper bounds", "observed": 0, "required": "K_i < 2π for interior vertices and K_i < π for boundary vertices", "signedMargin": null, "stage": "intrinsic", "statement": "Every target defect is strictly below its boundary-aware Euclidean cone upper bound. These local inequalities are necessary but not sufficient for realizability.", "status": "PASS", "tolerance": 0, "witness": { "checkedVertices": 4, "symbolicComparison": true } }, { "basis": "theorem-instance", "code": "GAUSS_BONNET_TOTAL", "evidenceClass": "exact-finite-identity", "hypotheses": [ "valid compact triangular surface", "interior defect 2π minus incident angles", "boundary defect π minus incident angles" ], "id": "G3", "impossibleInDeclaredModel": true, "name": "Gauss–Bonnet total", "observed": 0, "required": "Σ K_i = 2πχ = 12.566370614359172", "signedMargin": -12.566370614359172, "stage": "intrinsic", "statement": "The symbolic target violates the finite Gauss–Bonnet total exactly. It was not silently projected or repaired.", "status": "FAIL", "tolerance": 0, "witness": { "comparison": "exact-rational-multiple-of-pi", "eulerCharacteristic": 2, "observedPiCoefficient": { "denominator": 1, "numerator": 0 }, "observedTotal": 0, "requiredPiCoefficient": { "denominator": 1, "numerator": 4 }, "requiredTotal": 12.566370614359172, "unroundedResidual": -12.566370614359172 } }, { "basis": "general-inverse-design", "code": "NOT_RUN", "evidenceClass": "research-target", "hypotheses": [], "id": "G4", "impossibleInDeclaredModel": false, "name": "Chow–Luo subset inequalities", "observed": null, "required": "not evaluated", "signedMargin": null, "stage": "model", "statement": "Stopped by the exact-first compiler because G3 did not establish exact total-curvature membership.", "status": "NOT_RUN", "tolerance": null, "witness": null }, { "basis": "general-inverse-design", "code": "NOT_RUN", "evidenceClass": "research-target", "hypotheses": [], "id": "G5", "impossibleInDeclaredModel": false, "name": "symmetric disk doubling", "observed": null, "required": "not evaluated", "signedMargin": null, "stage": "model", "statement": "Stopped by the exact-first compiler because G3 did not establish exact total-curvature membership.", "status": "NOT_RUN", "tolerance": null, "witness": null }, { "basis": "general-inverse-design", "code": "NOT_RUN", "evidenceClass": "research-target", "hypotheses": [], "id": "G6", "impossibleInDeclaredModel": false, "name": "metric positivity and triangle margin", "observed": null, "required": "not evaluated", "signedMargin": null, "stage": "intrinsic", "statement": "Stopped by the exact-first compiler because G3 did not establish exact total-curvature membership.", "status": "NOT_RUN", "tolerance": null, "witness": null } ], "normalizedRequest": { "constraints": { "allowTopologyChange": false, "allowedBoundaryValences": [], "allowedInteriorValences": [], "edgeLengthBounds": [ 1e-9, 10 ], "maxStrain": 0.1, "phiByEdge": [ 0, 0, 0, 0, 0, 0 ], "radiusBounds": [ 0.5, 2 ] }, "id": "sphere-zero-reject", "models": [ "circle-packing-tangency" ], "schemaVersion": 2, "seed": "exact-deterministic", "target": { "curvature": { "basis": "symbolic-pi", "piCoefficients": [ { "denominator": 1, "numerator": 0 }, { "denominator": 1, "numerator": 0 }, { "denominator": 1, "numerator": 0 }, { "denominator": 1, "numerator": 0 } ] }, "discretization": { "complex": { "edges": [ { "length": 1, "source": 0, "target": 1, "weight": 1 }, { "length": 1, "source": 0, "target": 2, "weight": 1 }, { "length": 1, "source": 0, "target": 3, "weight": 1 }, { "length": 1, "source": 1, "target": 2, "weight": 1 }, { "length": 1, "source": 1, "target": 3, "weight": 1 }, { "length": 1, "source": 2, "target": 3, "weight": 1 } ], "faces": [ [ 0, 1, 2 ], [ 0, 1, 3 ], [ 0, 2, 3 ], [ 1, 2, 3 ] ], "nodes": [ { "id": 0, "label": "v0", "radius": 1, "x": 0, "y": 0, "z": 0 }, { "id": 1, "label": "v1", "radius": 1, "x": 0, "y": 0, "z": 0 }, { "id": 2, "label": "v2", "radius": 1, "x": 0, "y": 0, "z": 0 }, { "id": 3, "label": "v3", "radius": 1, "x": 0, "y": 0, "z": 0 } ] } }, "kind": "intrinsic-curvature", "topology": { "boundaryComponents": 0, "expectedBetti": null, "orientable": true, "surface": "sphere" } }, "verification": { "chowLuoGuard": 1e-8, "exactSubsetLimit": 18, "gbTolerance": 0, "maxIterations": 0, "optimizerTolerance": 1e-8, "refinementLevels": 0, "uncertaintySamples": 0 } }, "obstructionCertificates": [ { "basis": "theorem-instance", "code": "GAUSS_BONNET_TOTAL", "evidenceClass": "exact-finite-identity", "hypotheses": [ "valid compact triangular surface", "interior defect 2π minus incident angles", "boundary defect π minus incident angles" ], "impossibleInDeclaredModel": true, "observed": 0, "repairHints": [ "Create a disclosed Gauss–Bonnet projection candidate; do not overwrite the original target." ], "required": "Σ K_i = 2πχ = 12.566370614359172", "signedMargin": -12.566370614359172, "stage": "intrinsic", "statement": "The symbolic target violates the finite Gauss–Bonnet total exactly. It was not silently projected or repaired.", "status": "FAIL", "tolerance": 0, "witness": { "comparison": "exact-rational-multiple-of-pi", "eulerCharacteristic": 2, "observedPiCoefficient": { "denominator": 1, "numerator": 0 }, "observedTotal": 0, "requiredPiCoefficient": { "denominator": 1, "numerator": 4 }, "requiredTotal": 12.566370614359172, "unroundedResidual": -12.566370614359172 } }, { "basis": "general-inverse-design", "code": "NOT_RUN", "evidenceClass": "research-target", "hypotheses": [], "impossibleInDeclaredModel": false, "observed": null, "repairHints": [], "required": "not evaluated", "signedMargin": null, "stage": "model", "statement": "Stopped by the exact-first compiler because G3 did not establish exact total-curvature membership.", "status": "NOT_RUN", "tolerance": null, "witness": {} } ], "refinement": null, "repairProposals": [ { "id": "sphere-zero-reject.gb-projection", "operation": "Khat_i=K_i-w_i/sum(w)*(sum(K)-2πχ)", "preservesOriginalTarget": true, "status": "PROPOSED_NOT_APPLIED" } ], "replay": { "deterministic": true, "methods": [ "closed-schema-normalization", "release-1-surface-and-GF2-topology", "exact-rational-pi-when-declared", "ordered-local-and-total-curvature-gates", "bounded-Chow-Luo-subset-enumeration", "exhaustive-triangle-margin-audit" ], "platformIndependentInputsDigest": "99549406", "seed": "exact-deterministic" }, "reportDigest": "a6cd5b0b", "requestDigest": "99549406", "requestId": "sphere-zero-reject", "robustness": null, "schemaVersion": 2, "selectedCandidateId": null, "status": "FAIL" } }, { "id": "triangle-equality-reject", "request": { "schemaVersion": 2, "id": "triangle-equality-reject", "seed": "exact-deterministic", "target": { "kind": "intrinsic-edge-metric", "topology": { "surface": "disk", "orientable": true, "boundaryComponents": 1 }, "discretization": { "complex": { "nodes": [ { "id": 0 }, { "id": 1 }, { "id": 2 } ], "edges": [ { "source": 0, "target": 1 }, { "source": 0, "target": 2 }, { "source": 1, "target": 2 } ], "faces": [ [ 0, 1, 2 ] ] } }, "edgeLengths": [ 1, 1, 2 ] }, "models": [ "edge-metric" ], "constraints": { "radiusBounds": [ 0.5, 2 ], "edgeLengthBounds": [ 1e-9, 10 ], "allowTopologyChange": false }, "verification": { "gbTolerance": 0, "chowLuoGuard": 1e-8, "exactSubsetLimit": 18, "optimizerTolerance": 1e-8, "maxIterations": 0, "refinementLevels": 0, "uncertaintySamples": 0 } }, "report": { "candidates": [], "claims": { "demonstrated": [ "Closed bounded request normalization and ordered finite intrinsic gates were executed deterministically." ], "notDemonstrated": [ "continuum convergence", "a constructive CP radius solve except when the reference residual passes", "extrinsic embeddability", "mechanical selection", "instrument observability", "physical validation", "fabrication readiness" ] }, "compilerVersion": "0.1.0-alpha.2", "digestAlgorithm": "fnv1a32-ieee754-json-v1", "fabricationPackage": null, "gates": [ { "basis": "combinatorial-identity", "code": "SCHEMA_VALID", "evidenceClass": "exact-finite-identity", "hypotheses": [], "id": "G0", "impossibleInDeclaredModel": false, "name": "schema, units, and finite bounds", "observed": null, "required": "schemaVersion 2 finite bounded request", "signedMargin": null, "stage": "intrinsic", "statement": "The request passed the closed, bounded schema and was normalized deterministically; configuration text was treated only as data.", "status": "PASS", "tolerance": null, "witness": { "limits": { "maxEdges": 1024, "maxExactSubsetLimit": 18, "maxExportedSubsetRows": 1024, "maxFaces": 1024, "maxIdentifierLength": 128, "maxSeedLength": 256, "maxSubsetEvaluations": 262142, "maxVertices": 256 }, "normalizedRequestDigest": "71aaaea5" } }, { "basis": "combinatorial-identity", "code": "SURFACE_VALID", "evidenceClass": "exact-finite-identity", "hypotheses": [], "id": "G1", "impossibleInDeclaredModel": false, "name": "complex and surface validity", "observed": null, "required": "a finite pure triangular 2-manifold matching the declared topology", "signedMargin": null, "stage": "intrinsic", "statement": "The supplied cells form the declared finite triangular surface. 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