{
  "schema": "catenation-nivat-history/v1",
  "sourceGraph": {
    "schema": "nivat-historical-proof-graph/v1",
    "sha256": "f9aade3b539ced0d035a5f7d1a64e93b4e01d8425a24d54e4a31323f3f74a1b7",
    "historicalCampaignDate": "2026-09-08",
    "importDate": "2026-09-21",
    "totalClaimNodes": 46,
    "dependencyEdges": 25,
    "displayedConnectedNodes": 20,
    "edgeOrientation": "dependent -> prerequisite",
    "nativeStatus": "UNVERIFIED_IMPORT",
    "scope": "Dependency map extracted from the actual campaign’s original manuscript. Archival import did not perform a fresh proof audit or audit the later 21-page manuscript."
  },
  "manuscript": {
    "title": "A proof of Nivat’s conjecture",
    "authors": [
      "Pranava Kumar"
    ],
    "date": "2026-09-14",
    "pages": 21,
    "sha256": "6b3b5efdeb7aa549fdd0ca06cfda61b4be60241b369eb790d49b1f8fbf6b16e0",
    "scope": "For every finite alphabet and every configuration on the integer plane, at most k × n distinct patterns on some positive k × n rectangle imply a nonzero global period. Equality is included; no recurrence assumption is added.",
    "status": "Full-scope proof manuscript. External peer review and full formal verification are not established in the inspected record.",
    "originalAuditsTransferred": false
  },
  "counts": {
    "originalAgentHistories": 7,
    "approachFamilies": 11,
    "historicalAuditRecords": 10,
    "historicalRefutations": 8,
    "nativeRecertificationObligations": 6,
    "archivedArtifactNodes": 176,
    "historicalComputationGroups": 5
  },
  "routeStory": [
    {
      "title": "Freeze the target",
      "description": "Keep arbitrary finite alphabets, all positive rectangle sizes, the equality case, and a global period of the original configuration in scope."
    },
    {
      "title": "Explore eleven approach families",
      "description": "Algebra, boundary counting, dynamical systems, geometry, and exact falsification develop different routes against the same target."
    },
    {
      "title": "Keep the failed shortcuts",
      "description": "The history retains counterexamples to mixed-increment sign rules, apex-placement counting, unique-marker assumptions, and rare-pattern bounds."
    },
    {
      "title": "Assemble the closing route",
      "description": "The manuscript combines nonperiodic wedge selection with quadratic observables and pointed-cone support to recover an affine dimension deficit on the original rectangle."
    },
    {
      "title": "Retain the negative review",
      "description": "An assembled draft received FAIL AS WRITTEN for a missing residual step. The corrected original source received two internal final PASS audits, each tied to its exact version."
    }
  ],
  "nodes": [
    {
      "id": "T0",
      "title": "Full Nivat statement",
      "statement": "Full classical Nivat target",
      "historicalStatus": "Proved/audited (local agent audits)",
      "nativeStatus": "UNVERIFIED_IMPORT"
    },
    {
      "id": "W1",
      "title": "At least kn + 1 patterns",
      "statement": "A nonperiodic finitary configuration with a distinct-direction product annihilator and one fully periodic rational wedge has P(k,n)>=kn+1 for every positive k,n",
      "historicalStatus": "Proved/audited",
      "nativeStatus": "UNVERIFIED_IMPORT"
    },
    {
      "id": "W2",
      "title": "Bound the affine deficit",
      "statement": "Constant-convolution relations on a k by n rectangle have dimension at most (k-r)_+(n-s)_+, where r,s are widths of the common directional divisor",
      "historicalStatus": "Proved/audited",
      "nativeStatus": "UNVERIFIED_IMPORT"
    },
    {
      "id": "W3",
      "title": "Recover independent translates",
      "statement": "One quadratic observable supplies that many fitting translates independent modulo affine functions of the same rectangle",
      "historicalStatus": "Proved/audited",
      "nativeStatus": "UNVERIFIED_IMPORT"
    },
    {
      "id": "S3",
      "title": "Nonperiodic wedge selection",
      "statement": "A nonperiodic low-rectangle configuration has a nonperiodic orbit point agreeing with a fully periodic background on a pointed rational wedge",
      "historicalStatus": "Proved/audited (source reconstruction)",
      "nativeStatus": "UNVERIFIED_IMPORT"
    },
    {
      "id": "T1",
      "title": "Contradiction on the same rectangle",
      "statement": "S3 plus W1 contradict the low bound at the original rectangle",
      "historicalStatus": "Proved/audited",
      "nativeStatus": "UNVERIFIED_IMPORT"
    },
    {
      "id": "I-ENC",
      "title": "Finite alphabet and one-cell cases",
      "statement": "Injective finite integer encoding and one-cell cases",
      "historicalStatus": "reconstructed subclaim in historically audited manuscript",
      "nativeStatus": "UNVERIFIED_IMPORT"
    },
    {
      "id": "I-KS-PRODUCT",
      "title": "Directional annihilator",
      "statement": "Low rectangle gives distinct-direction product annihilator",
      "historicalStatus": "reconstructed subclaim in historically audited manuscript",
      "nativeStatus": "UNVERIFIED_IMPORT"
    },
    {
      "id": "I-KS-DECOMP",
      "title": "Prescribed-period decomposition",
      "statement": "Prescribed-period decomposition theorem",
      "historicalStatus": "reconstructed subclaim in historically audited manuscript",
      "nativeStatus": "UNVERIFIED_IMPORT"
    },
    {
      "id": "I-COLLE",
      "title": "Regional selection inputs",
      "statement": "Generating-set, nonexpansive, and regional continuation inputs",
      "historicalStatus": "reconstructed subclaim in historically audited manuscript",
      "nativeStatus": "UNVERIFIED_IMPORT"
    },
    {
      "id": "SEL-MIN",
      "title": "Minimum decomposition order",
      "statement": "Minimum integer decomposition order among nonperiodic orbit points",
      "historicalStatus": "reconstructed subclaim in historically audited manuscript",
      "nativeStatus": "UNVERIFIED_IMPORT"
    },
    {
      "id": "SEL-CASES",
      "title": "Corrected selection cases",
      "statement": "Corrected half-strip dichotomy, nested envelopes, signed layers, fixed seeds",
      "historicalStatus": "reconstructed subclaim in historically audited manuscript",
      "nativeStatus": "UNVERIFIED_IMPORT"
    },
    {
      "id": "SEL-NONPER",
      "title": "Keep the point nonperiodic",
      "statement": "Selected point remains nonperiodic by two distinct singly periodic limit directions",
      "historicalStatus": "reconstructed subclaim in historically audited manuscript",
      "nativeStatus": "UNVERIFIED_IMPORT"
    },
    {
      "id": "SEL-BACKGROUND",
      "title": "A fixed periodic background",
      "statement": "Two regional periods extend to a fixed global periodic background on a wedge",
      "historicalStatus": "reconstructed subclaim in historically audited manuscript",
      "nativeStatus": "UNVERIFIED_IMPORT"
    },
    {
      "id": "ALG-GCD",
      "title": "Common directional divisor",
      "statement": "Common directional divisor and fully periodic residual via coprime annihilators",
      "historicalStatus": "reconstructed subclaim in historically audited manuscript",
      "nativeStatus": "UNVERIFIED_IMPORT"
    },
    {
      "id": "ALG-FILTER",
      "title": "Small directional filters",
      "statement": "At least two essential directions; small filters singly periodic and nonfully periodic",
      "historicalStatus": "reconstructed subclaim in historically audited manuscript",
      "nativeStatus": "UNVERIFIED_IMPORT"
    },
    {
      "id": "GEO-HALFPLANE",
      "title": "Half-plane defects",
      "statement": "Fixed wedge agreement propagates to half-plane tails under tangent period",
      "historicalStatus": "reconstructed subclaim in historically audited manuscript",
      "nativeStatus": "UNVERIFIED_IMPORT"
    },
    {
      "id": "GEO-ALIGN",
      "title": "Align the defects",
      "statement": "Nonzero defect product aligned inside factor boxes by recurrence endpoints and floor",
      "historicalStatus": "reconstructed subclaim in historically audited manuscript",
      "nativeStatus": "UNVERIFIED_IMPORT"
    },
    {
      "id": "CONE-NOANN",
      "title": "Pointed-cone support",
      "statement": "Nonzero pointed-cone-supported array admits no Laurent annihilator",
      "historicalStatus": "reconstructed subclaim in historically audited manuscript",
      "nativeStatus": "UNVERIFIED_IMPORT"
    },
    {
      "id": "OBS-QUADRATIC",
      "title": "Quadratic local observable",
      "statement": "Quadratic local observable has nonzero pointed-cone mixed difference",
      "historicalStatus": "reconstructed subclaim in historically audited manuscript",
      "nativeStatus": "UNVERIFIED_IMPORT"
    }
  ],
  "edges": [
    {
      "source": "I-KS-PRODUCT",
      "target": "I-ENC",
      "relation": "DEPENDS_ON",
      "reconstructed": true
    },
    {
      "source": "SEL-MIN",
      "target": "I-KS-DECOMP",
      "relation": "DEPENDS_ON",
      "reconstructed": true
    },
    {
      "source": "SEL-CASES",
      "target": "I-COLLE",
      "relation": "DEPENDS_ON",
      "reconstructed": true
    },
    {
      "source": "SEL-CASES",
      "target": "SEL-MIN",
      "relation": "DEPENDS_ON",
      "reconstructed": true
    },
    {
      "source": "SEL-NONPER",
      "target": "SEL-CASES",
      "relation": "DEPENDS_ON",
      "reconstructed": true
    },
    {
      "source": "SEL-BACKGROUND",
      "target": "SEL-CASES",
      "relation": "DEPENDS_ON",
      "reconstructed": true
    },
    {
      "source": "S3",
      "target": "SEL-NONPER",
      "relation": "DEPENDS_ON",
      "reconstructed": true
    },
    {
      "source": "S3",
      "target": "SEL-BACKGROUND",
      "relation": "DEPENDS_ON",
      "reconstructed": true
    },
    {
      "source": "S3",
      "target": "I-KS-PRODUCT",
      "relation": "DEPENDS_ON",
      "reconstructed": true
    },
    {
      "source": "ALG-GCD",
      "target": "I-KS-PRODUCT",
      "relation": "DEPENDS_ON",
      "reconstructed": true
    },
    {
      "source": "ALG-FILTER",
      "target": "ALG-GCD",
      "relation": "DEPENDS_ON",
      "reconstructed": true
    },
    {
      "source": "GEO-HALFPLANE",
      "target": "ALG-FILTER",
      "relation": "DEPENDS_ON",
      "reconstructed": true
    },
    {
      "source": "GEO-HALFPLANE",
      "target": "SEL-BACKGROUND",
      "relation": "DEPENDS_ON",
      "reconstructed": true
    },
    {
      "source": "GEO-ALIGN",
      "target": "GEO-HALFPLANE",
      "relation": "DEPENDS_ON",
      "reconstructed": true
    },
    {
      "source": "GEO-ALIGN",
      "target": "ALG-FILTER",
      "relation": "DEPENDS_ON",
      "reconstructed": true
    },
    {
      "source": "OBS-QUADRATIC",
      "target": "GEO-ALIGN",
      "relation": "DEPENDS_ON",
      "reconstructed": true
    },
    {
      "source": "OBS-QUADRATIC",
      "target": "CONE-NOANN",
      "relation": "DEPENDS_ON",
      "reconstructed": true
    },
    {
      "source": "W2",
      "target": "ALG-GCD",
      "relation": "DEPENDS_ON",
      "reconstructed": true
    },
    {
      "source": "W3",
      "target": "OBS-QUADRATIC",
      "relation": "DEPENDS_ON",
      "reconstructed": true
    },
    {
      "source": "W1",
      "target": "W2",
      "relation": "DEPENDS_ON",
      "reconstructed": true
    },
    {
      "source": "W1",
      "target": "W3",
      "relation": "DEPENDS_ON",
      "reconstructed": true
    },
    {
      "source": "T1",
      "target": "S3",
      "relation": "DEPENDS_ON",
      "reconstructed": true
    },
    {
      "source": "T1",
      "target": "W1",
      "relation": "DEPENDS_ON",
      "reconstructed": true
    },
    {
      "source": "T0",
      "target": "T1",
      "relation": "DEPENDS_ON",
      "reconstructed": true
    },
    {
      "source": "T0",
      "target": "I-ENC",
      "relation": "DEPENDS_ON",
      "reconstructed": true
    }
  ],
  "historicalRefutations": [
    {
      "claimId": "F3",
      "statement": "Mixed complexity increment has a fixed sign under P<=area, even with minimal area",
      "historicalStatus": "False",
      "evidence": "Explicit diagonal-periodic counterexamples in boundary_wave1"
    },
    {
      "claimId": "F4",
      "statement": "Binary values plus integer annihilator plus finitely many straight star interfaces imply periodicity",
      "historicalStatus": "False",
      "evidence": "H(i)+H(j)-H(i+j), three independent derivations"
    },
    {
      "claimId": "F5",
      "statement": "Every low-complexity orbit closure is countable or has finite Cantor-Bendixson height",
      "historicalStatus": "False",
      "evidence": "All vertically constant binary configurations: four allowed 2x2 blocks"
    },
    {
      "claimId": "C2",
      "statement": "Every rational three-wall fan gives kn distinct patterns by placing its apex at each rectangle site",
      "historicalStatus": "False",
      "evidence": "H(y)+H(3x+y)-H(3x+2y), two apex placements give same all-one 2x2 pattern"
    },
    {
      "claimId": "F6",
      "statement": "An annihilated star has a unique block inside its minimal annihilator box",
      "historicalStatus": "False",
      "evidence": "Separated identical directional pulses; dynamics marker wave"
    },
    {
      "claimId": "F7",
      "statement": "Number of rare R-patterns is at least the local annihilator dimension",
      "historicalStatus": "False",
      "evidence": "Two de Bruijn-periodic tails with finite defects: R2x2 has M_R=0 and dimAnn_R=1"
    },
    {
      "claimId": "F1",
      "statement": "A periodic point in orbit closure forces original periodicity",
      "historicalStatus": "False",
      "evidence": "Single defect"
    },
    {
      "claimId": "F2",
      "statement": "Finite tori establish periodicity of arbitrary configurations",
      "historicalStatus": "False as inference",
      "evidence": "Periodic boundary conditions impose periods"
    }
  ],
  "historicalReviews": [
    {
      "id": "conditional-finite-algebra",
      "verdict": "PASS conditional finite-tail theorem; later cone upgrade also recorded",
      "subjectClaims": [
        "ALG-GCD",
        "ALG-FILTER",
        "GEO-ALIGN",
        "W2",
        "W3"
      ],
      "reviewedSourceSha256": null,
      "nativeStatus": "UNVERIFIED_IMPORT"
    },
    {
      "id": "conditional-boundary",
      "verdict": "PASS conditional on ideal/filter and periodic-region inputs",
      "subjectClaims": [
        "GEO-ALIGN",
        "W2",
        "W3"
      ],
      "reviewedSourceSha256": null,
      "nativeStatus": "UNVERIFIED_IMPORT"
    },
    {
      "id": "cone-region",
      "verdict": "conditional cone-region audit",
      "subjectClaims": [
        "SEL-BACKGROUND",
        "GEO-HALFPLANE",
        "CONE-NOANN",
        "OBS-QUADRATIC"
      ],
      "reviewedSourceSha256": null,
      "nativeStatus": "UNVERIFIED_IMPORT"
    },
    {
      "id": "fresh-wedge",
      "verdict": "PASS conditional theorem, source-selection excluded",
      "subjectClaims": [
        "W1"
      ],
      "reviewedSourceSha256": null,
      "nativeStatus": "UNVERIFIED_IMPORT"
    },
    {
      "id": "selection-boundary",
      "verdict": "selection survives with recorded corrections",
      "subjectClaims": [
        "S3"
      ],
      "reviewedSourceSha256": null,
      "nativeStatus": "UNVERIFIED_IMPORT"
    },
    {
      "id": "selection-redteam",
      "verdict": "PASS corrected selection only",
      "subjectClaims": [
        "S3"
      ],
      "reviewedSourceSha256": null,
      "nativeStatus": "UNVERIFIED_IMPORT"
    },
    {
      "id": "source-application-failed",
      "verdict": "FAIL AS WRITTEN: missing residual step",
      "subjectClaims": [
        "S3",
        "T1"
      ],
      "reviewedSourceSha256": "26C8F844A6DD68CB5EFA6E3EFF1AF4FF1D961CEBE3286E18475B34BEA3A202C9",
      "nativeStatus": "UNVERIFIED_IMPORT"
    },
    {
      "id": "source-application-recheck",
      "verdict": "PASS corrected source applications and closing implication",
      "subjectClaims": [
        "S3",
        "T1"
      ],
      "reviewedSourceSha256": "EFA80C217988B3C1D96B8DB236DADFF552C511C9141FC054187D94CD6F42C2FB",
      "nativeStatus": "UNVERIFIED_IMPORT"
    },
    {
      "id": "end-to-end",
      "verdict": "PASS complete mathematical implication at audited snapshot",
      "subjectClaims": [
        "T0"
      ],
      "reviewedSourceSha256": "EFA80C217988B3C1D96B8DB236DADFF552C511C9141FC054187D94CD6F42C2FB",
      "nativeStatus": "UNVERIFIED_IMPORT"
    },
    {
      "id": "release-identity",
      "verdict": "PASS mathematical body byte-identical; release identity only",
      "subjectClaims": [
        "T0"
      ],
      "reviewedSourceSha256": "9C74E3BCE55AC6476F919C3509BA9D8CAC3672858B7279883A2105295FC9A613",
      "nativeStatus": "UNVERIFIED_IMPORT"
    }
  ],
  "statusNotes": [
    "All 46 native claims remain UNVERIFIED_IMPORT. Historical labels are preserved source assertions, not new native certification.",
    "The 25 arrows preserve the manuscript-derived DEPENDS_ON relationships, directed from dependent claim to prerequisite. Display labels are shortened for readability; extraction provenance remains in each edge record.",
    "The 20 displayed nodes are exactly the endpoints of those 25 edges. Other archived claims and rejected mechanisms are not connected by invented arrows.",
    "The two final internal agent audits apply to the original 8 September source. The supplied 21-page revision does not inherit those audits.",
    "Preserved histories and document-navigation views are archival records, not newly executing agents."
  ]
}