{
  "schemaVersion": "1.6",
  "slug": "factorial-spikes-fi-noetherianity",
  "title": "Factorial spikes separate componentwise Noetherianity from FI-Noetherianity",
  "shortTitle": "Factorial spikes and FI-Noetherianity",
  "url": "https://evidencepress.org/releases/factorial-spikes-fi-noetherianity/",
  "oneLine": "A characteristic-free sub-FI-algebra has affine Noetherian components at every finite width but a strict infinite chain of FI-ideals across widths.",
  "abstract": "Let V(S)=k[x_s | s in S] be the standard width-one polynomial FI-algebra over an arbitrary field. This anonymous, unrefereed theorem candidate defines a sub-FI-algebra A generated by every FI-translate of g_r=x_1...x_(r-1)x_r^(r!) for r>=2. For each finite S, only widths r<=|S| occur, so A(S) is an affine, finitely presented Noetherian k-algebra. Globally, however, the FI-ideals J_R generated through width R form a strict infinite ascending chain. The exact obstruction is elementary: a native width-R spike could factor through lower widths only if the light-variable counts and peak exponents satisfied two simultaneous partition equations, while R!>(R-1)(R-1)! rules this out. The proof is characteristic-free. The regular A-module is finite free of rank one and contains a non-finitely-generated FI-submodule, which is a formal consequence over A rather than a non-free example or a module over V. The construction is not finitely generated as an FI-algebra; stronger finite-generation variants remain open. Producer-side exact replay, four hostile controls, a supplied release review and a five-role internal editorial gate support the candidate without establishing independent reconstruction, authenticated specialist review, formal verification, peer review, novelty or priority.",
  "datePublished": "2026-08-31",
  "dateModified": "2026-08-31",
  "version": "0.1.0-candidate",
  "doi": "10.5281/zenodo.22206978",
  "doiUrl": "https://doi.org/10.5281/zenodo.22206978",
  "conceptDoi": "10.5281/zenodo.22206977",
  "pdfUrl": "https://github.com/ipitchford/factorial-spikes-fi-noetherianity/releases/download/v0.1.0-candidate/paper.pdf",
  "altPdfUrl": "https://zenodo.org/records/22206978/files/paper.pdf?download=1",
  "altPdfLabel": null,
  "zenodoUrl": "https://zenodo.org/records/22206978",
  "repoUrl": "https://github.com/ipitchford/factorial-spikes-fi-noetherianity",
  "releaseUrl": "https://github.com/ipitchford/factorial-spikes-fi-noetherianity/releases/tag/v0.1.0-candidate",
  "markdownUrl": "https://evidencepress.org/releases/factorial-spikes-fi-noetherianity/index.md",
  "bibtexUrl": "https://evidencepress.org/releases/factorial-spikes-fi-noetherianity/cite.bib",
  "audioUrl": "https://evidencepress.org/assets/audio/factorial-spikes-fi-noetherianity.mp3?v=0ec3b27f7a",
  "imageUrl": "https://evidencepress.org/assets/og/factorial-spikes-fi-noetherianity.png?v=6deec5873d",
  "coverArtUrl": "https://evidencepress.org/assets/art/factorial-spikes-fi-noetherianity.svg?v=4e32f262c8",
  "media": [
    {
      "type": "audio",
      "url": "https://evidencepress.org/assets/audio/factorial-spikes-fi-noetherianity.mp3",
      "name": "Audio briefing — factorial spikes and FI-Noetherianity",
      "description": "Plain-English synthetic-voice account of the fixed-width versus FI-width distinction, factorial obstruction, module scope and assurance limits; a communication aid, not additional evidence.",
      "transcriptUrl": "https://evidencepress.org/assets/audio/factorial-spikes-fi-noetherianity.txt"
    }
  ],
  "authors": [
    "Anonymous"
  ],
  "license": "CC0-1.0",
  "status": "unrefereed-candidate",
  "verification": {
    "peerReviewed": false,
    "independentlyReproduced": false,
    "formallyVerified": false,
    "internallyReplayed": true,
    "detail": "Anonymous, AI-assisted, unrefereed FI-algebra theorem candidate after actioning a supplied full review, completing a five-role producer-coordinated editorial gate and passing one bounded confirmation on an exact frozen archive. Reviewer identity, specialist credentials and unaffiliated status were not authenticated. The literal unrestricted subalgebra and characteristic clauses receive a complete candidate proof; the regular-module corollary is deliberately scoped over A. The construction is not FI-finitely generated, and stronger finite-generation variants remain open."
  },
  "assurance": [
    {
      "dimension": "availability",
      "label": "Availability and archiving",
      "question": "Is the evidence package publicly retrievable from an archive under a persistent identifier?",
      "state": "passed",
      "evidenceUrl": "https://zenodo.org/records/22206978",
      "note": "The tagged GitHub prerelease and published Zenodo record expose the DOI-bearing PDF, ZIP and manifest; fresh unauthenticated downloads from both services matched all local SHA-256 values."
    },
    {
      "dimension": "internalReplay",
      "label": "Internal replay",
      "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?",
      "state": "passed",
      "note": "Public Linux CI checks the manifest, runs the dependency-free exact verifier normally and with optimization, compares the replay receipt, checks release metadata and rejects raw TeX in the PDF.",
      "evidenceUrl": "https://github.com/ipitchford/factorial-spikes-fi-noetherianity/actions/runs/33388925405"
    },
    {
      "dimension": "independentRerun",
      "label": "Independent rerun",
      "question": "Has someone else run the supplied implementation and obtained the stated result?",
      "state": "not-assessed",
      "note": "No authenticated unaffiliated clean-environment execution receipt is reported."
    },
    {
      "dimension": "independentReimplementation",
      "label": "Independent reimplementation",
      "question": "Has someone else reached the result from an independent implementation?",
      "state": "not-assessed",
      "note": "The structural and ambient search routes are algorithmically distinct but producer-authored."
    },
    {
      "dimension": "formalVerification",
      "label": "Formal verification",
      "question": "Is a formalised statement machine-checked, and over which trusted base?",
      "state": "not-assessed",
      "note": "No proof assistant formalizes FI-functoriality, the monomial criterion, the strict chain or the module corollary."
    },
    {
      "dimension": "specialistReview",
      "label": "Specialist review",
      "question": "Has a domain specialist assessed the argument?",
      "state": "not-assessed",
      "note": "The supplied review and internal role reports are not authenticated unaffiliated review by an FI-algebra specialist."
    },
    {
      "dimension": "editorialPeerReview",
      "label": "Editorial peer review",
      "question": "Has a journal or venue run peer review to a decision?",
      "state": "not-assessed",
      "note": "No journal or comparable external venue has conducted editorial peer review."
    },
    {
      "dimension": "dataEnvironmentReproducibility",
      "label": "Data and environment reproducibility",
      "question": "Are data and computational environment pinned well enough to rebuild?",
      "state": "partial",
      "evidenceUrl": "https://github.com/ipitchford/factorial-spikes-fi-noetherianity/blob/v0.1.0-candidate/ENVIRONMENT.txt",
      "note": "The Python replay is dependency-free and public Linux CI passes, but PDF rebuilding depends on host Pandoc, Poppler and TeX distributions and no bit-reproducible system image is supplied."
    }
  ],
  "provenance": {
    "aiGenerated": true,
    "aiAssisted": true,
    "generatedBy": [
      "AI systems under human direction",
      "OpenAI Codex",
      "Python 3 standard library",
      "Node.js standard library"
    ],
    "humanRole": "Research direction, problem selection, review provision, claim-boundary decisions, publication authorization and institutional responsibility by the human creator; repository maintenance and Evidence Press publication are not scholarly authorship, and scholarly attribution remains Anonymous.",
    "disclosure": "The current UnsolvedMath record already contained a machine-generated factorial-spike attempt. AI systems reconstructed and tightened that argument, implemented replay and controls, actioned the supplied review, coordinated internal editorial roles and performed release engineering. Same-producer checks and internal review do not establish unaffiliated reconstruction, authenticated specialist review or editorial peer review."
  },
  "attributionBoundary": null,
  "problem": {
    "name": "AIM representation stability problem list, section 1, item 1.2",
    "url": "https://aimpl.org/repnstability/1/"
  },
  "orderedProductAverage": null,
  "corrections": [],
  "publicCorrections": [],
  "publicCorrectionIndexes": null,
  "pageStructureVersion": "2.0",
  "pageStructureVariant": "canonical-theorem",
  "pageStructureWaivers": [],
  "recordMaturity": "current",
  "metadataProvenance": "Authored prospectively against the DOI-bearing tagged v0.1.0-candidate, supplied-review response, five-role internal editorial record, exact confirmation, two successful public hosted replay runs and exact GitHub and Zenodo three-asset readback on 31 August 2026. Canonical Evidence Press deployment and terminal publication metrics are recorded separately and do not promote the mathematical assurance ceiling.",
  "grandfatheredAtSchemaVersion": null,
  "keywords": [
    "FI-algebra",
    "Noetherianity",
    "monomial subalgebra",
    "representation stability",
    "factorial exponent",
    "unrefereed candidate"
  ],
  "keyResults": [
    "The displayed generators define a sub-FI-algebra A of V(S)=k[x_s | s in S] over every field.",
    "Every component A(S) is finitely generated, finitely presented and Noetherian; for |S|=n there are n(2^(n-1)-1) distinct displayed generator monomials.",
    "The FI-ideals generated by widths at most R form a strict chain J_2 strictly contained in J_3 strictly contained in J_4 and so on, so A is not FI-Noetherian.",
    "The exact lower-width factorisation criterion separates light-variable accounting from peak-exponent accounting, and the factorial inequality rules out every lower-width factorisation.",
    "The regular A-module is finite free of rank one and contains a non-finitely-generated FI-submodule; this does not give a non-free example or a module over the ambient polynomial FI-algebra V."
  ],
  "reviews": [],
  "evidencePackage": "An eight-page DOI-bearing manuscript and aligned Markdown; a universal monomial proof; an exact native-width factorisation lemma and monomial-ideal membership lemma; dependency-free Python replay in normal and optimized modes; a structurally distinct ambient exponent-vector search through width seven; four hostile controls; an exact replay receipt; semantic metadata and raw-TeX checks; all-page visual inspection; source, citation and bounded novelty audits; the supplied release review; five differentiated internal editorial reports; one exact frozen-target confirmation; a 41-member ZIP containing 40 manifested payloads plus its root manifest; two successful public GitHub Actions runs; and exact unauthenticated GitHub and Zenodo asset readback. These establish availability, integrity, producer replay and the stated unrefereed candidate. They do not establish the universal theorem independently, authenticated specialist review, formal verification, editorial peer review, novelty, priority or impact.",
  "openProblems": [
    "Find a non-Noetherian sub-FI-algebra of the standard width-one polynomial FI-algebra that is finitely generated, or prove that none exists under a natural finite-presentation hypothesis.",
    "Construct a non-free finitely presented module over a finitely presented non-Noetherian sub-FI-algebra, if the intended module clause requires more than the regular module.",
    "Independently reconstruct the exact lower-width factorisation criterion and strict FI-ideal chain in a materially separate stack.",
    "Formalize the FI-algebra construction, monomial-ideal lemma, factorial obstruction and regular-module consequence in a proof assistant.",
    "Obtain authenticated FI-algebra specialist review and a broader MathSciNet or zbMATH novelty and priority assessment."
  ],
  "mathObjects": [],
  "relatedWorks": [
    {
      "citation": "American Institute of Mathematics. Representation stability, section 1, item 1.2, sub-FI-algebra problem.",
      "url": "https://aimpl.org/repnstability/1/",
      "doi": null
    },
    {
      "citation": "Nagel, U., & Römer, T. (2019). FI- and OI-modules with varying coefficients. Journal of Algebra 535, 286-322.",
      "url": "https://doi.org/10.1016/j.jalgebra.2019.06.029",
      "doi": "10.1016/j.jalgebra.2019.06.029"
    },
    {
      "citation": "Draisma, J., Eggermont, R. H., & Farooq, A. (2022). Components of symmetric wide-matrix varieties. Journal für die reine und angewandte Mathematik 793, 143-184.",
      "url": "https://doi.org/10.1515/crelle-2022-0064",
      "doi": "10.1515/crelle-2022-0064"
    },
    {
      "citation": "Draisma, J., Eggermont, R. H., Farooq, A., & Meier, L. (2025). Image closures of symmetric wide-matrix varieties. Journal of Algebra 668, 190-207.",
      "url": "https://doi.org/10.1016/j.jalgebra.2024.12.028",
      "doi": "10.1016/j.jalgebra.2024.12.028"
    },
    {
      "citation": "Maraj, A., & Nagel, U. (2023). Shift invariant algebras, Segre products and regular languages. Journal of Algebra 631, 236-266.",
      "url": "https://doi.org/10.1016/j.jalgebra.2023.04.016",
      "doi": "10.1016/j.jalgebra.2023.04.016"
    },
    {
      "citation": "Morrow, M., & Nagel, U. (2025). Equivariant free resolutions of sequences of symmetric modules. arXiv:2507.11650.",
      "url": "https://arxiv.org/abs/2507.11650",
      "doi": "10.48550/arXiv.2507.11650"
    }
  ],
  "claimAssurance": null,
  "operatingModel": {
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    "workId": "ep-work:factorial-spikes-fi-noetherianity",
    "attemptIds": [
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    "aims": [
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    "artifactRoles": [
      "research-output",
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      "communication"
    ],
    "lineageId": null,
    "accelerationPrimitives": [
      "certificate-first",
      "structural-compression",
      "adversarial-controls",
      "assurance-vector",
      "agent-readable-research-object"
    ],
    "decisionObject": {
      "type": "counterexample",
      "description": "An explicit characteristic-free sub-FI-algebra with affine Noetherian fixed components and a strict infinite ascending chain of FI-ideals.",
      "scope": "The unrestricted sub-FI-algebra A generated by factorial spikes and the regular A-module consequence; not an FI-finitely-generated algebra, a non-free module, a module over V, a classification theorem, a novelty finding or external validation."
    },
    "bottleneckTargeted": [
      "assurance",
      "publication",
      "translation"
    ],
    "semanticBridge": {
      "state": "explicit",
      "description": "The package records the recovered AIM wording, FI convention, factorial-spike generators, exact native-width factorisation equations, monomial-ideal membership step, strict FI-ideal witnesses, component count, module interpretation and replay predicates. This exposes every transition from the source request to the separation claim.",
      "remainingRisks": [
        "The live AIMPL item returned HTTP 502 during the audit, so exact current wording was recovered through the UnsolvedMath v1.6.0 record rather than direct readback.",
        "Both finite checker routes are producer-authored and do not independently establish the universal proof.",
        "The literal module interpretation may be weaker than the source author's intended challenge.",
        "The bounded novelty search cannot establish historical priority."
      ]
    },
    "humanJudgmentGates": [
      "Keep componentwise Noetherianity separate from Noetherianity of the FI-system.",
      "State prominently that A is not finitely generated as an FI-algebra.",
      "Scope the finitely presented module consequence over A and do not call it non-free or a module over V.",
      "Treat finite replay and hostile controls as regression evidence for the encoding, not as proof of the universal theorem.",
      "Keep the supplied review, internal editorial confirmation, authenticated specialist review, formal verification and peer review distinct.",
      "Make no first-discovery or historical-priority claim from a bounded search."
    ],
    "parentLinks": [
      {
        "relation": "depends-on-claim",
        "workId": null,
        "legacyReleaseSlug": null,
        "externalUrl": "https://aimpl.org/repnstability/1/",
        "inheritedClaim": "The AIM problem asks for a concrete non-Noetherian sub-FI-algebra of the standard polynomial FI-algebra, asks about finitely presented modules and asks whether characteristic matters.",
        "inheritedAssuranceCeiling": "The source request fixes the target wording but does not establish this construction, its correctness, intended module scope, novelty or priority."
      },
      {
        "relation": "extends-result",
        "workId": null,
        "legacyReleaseSlug": null,
        "externalUrl": "https://doi.org/10.1016/j.jalgebra.2019.06.029",
        "inheritedClaim": "Nagel and Römer provide the FI-algebra and FI-module conventions, positive Noetherianity for finitely generated modules over the standard polynomial FI-algebra and adjacent monomial-subalgebra results.",
        "inheritedAssuranceCeiling": "Their theorems supply framework and nearby positive regimes, not the present unrestricted non-finitely-generated subalgebra or a novelty finding for it."
      }
    ],
    "assuranceTarget": {
      "dimensions": [
        "independentRerun",
        "independentReimplementation",
        "formalVerification",
        "specialistReview",
        "dataEnvironmentReproducibility",
        "semanticValidation",
        "noveltyAssessment",
        "priorityAssessment"
      ],
      "nextAction": "Obtain an authenticated unaffiliated FI-algebra reconstruction of the strict-chain proof and clarify whether a finitely generated or non-free module variant is the intended stronger target.",
      "claimCeiling": "An anonymous, AI-assisted, unrefereed theorem candidate separating componentwise Noetherianity from FI-Noetherianity by a characteristic-free factorial-spike construction, with a carefully delimited regular-module consequence, complete written proof, public immutable assets, producer replay and internal review; not an FI-finitely-generated counterexample, a non-free module, historical-priority finding, unaffiliated validation, formal verification, authenticated external specialist review or editorial peer review."
    },
    "impactClaims": [
      {
        "id": "factorial-spikes-fi-noetherianity-science-no-impact",
        "aim": "science",
        "outcome": "Faster or more reliable construction and assurance of FI-algebra examples addressing open problem-list requests",
        "setting": "AI-assisted proof reconstruction, source correspondence, adversarial replay, internal review and guarded candidate publication",
        "status": "NO_IMPACT_EVIDENCE",
        "designClass": "none",
        "comparator": "No matched conventional research or publication workflow was registered.",
        "estimand": "No effect on discovery time, active human effort, compute, correction rate, proof quality, independent-assurance time, uptake, citation or field outcomes was estimated.",
        "evidenceRefs": [],
        "registeredDesignRef": null,
        "independentAssessment": null
      }
    ]
  },
  "researchMetrics": {
    "schemaVersion": "1.0",
    "policyUrl": "https://evidencepress.org/api/research-metrics-policy.json",
    "workLedgerUrl": "https://evidencepress.org/api/work-ledger.json",
    "claimCeiling": "These records describe scoped workflow inputs and outcomes. They support calibration and reference-class construction, but do not by themselves establish research acceleration, theorem correctness, novelty, independent assurance or impact.",
    "attempts": [
      {
        "attemptId": "ep-attempt:factorial-spikes-fi-noetherianity-publication",
        "status": "published",
        "resultClass": "positive",
        "metrics": {
          "schemaVersion": "1.0",
          "measurementScope": "publication-only",
          "scopeBoundary": "From prospective registration before Evidence Press page authoring through terminal canonical site readback or a preserved stop. Earlier theorem construction, source recovery, review repair, editorial confirmation, package freezing, GitHub identity and Zenodo publication are excluded and not reconstructed.",
          "forecast": {
            "frozenAt": "2026-08-31T11:46:57Z",
            "procedureClass": "reviewed-candidate-canonical-site-publication",
            "targetOutcome": "Publish a reader-first Evidence Press release with provenance-bound media, append-only operating records, two composite seals, hosted CI, guarded zero-cost deployment and exact canonical readback without broadening the literal unrestricted-algebra result or its assurance status.",
            "expectedActiveMinutes": 110,
            "plausibleLowMinutes": 65,
            "plausibleHighMinutes": 165,
            "expectedUnattendedWaitMinutes": 45,
            "referenceClass": {
              "label": "Recent reviewed exact-mathematics canonical-site publication closures",
              "sampleSize": 5,
              "basis": "Recent Evidence Press candidates used the same reader-first authoring, provenance-bound media, append-only operating records, composite sealing, hosted CI, guarded deployment and canonical-readback architecture."
            },
            "fermiComponents": [
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                "basis": "One eight-page FI-algebra candidate with a delicate componentwise-versus-FI and module-scope boundary."
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              {
                "component": "Deterministic art, byte-bound audio, Open Graph image and thumbnail",
                "count": 1,
                "lowMinutesPerUnit": 15,
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                "highMinutesPerUnit": 35,
                "basis": "One release using established generators but requiring a new factorial-spike visual that must not imply FI-finite generation."
              },
              {
                "component": "Operating records and composite A/B and C/D seals",
                "count": 1,
                "lowMinutesPerUnit": 15,
                "centralMinutesPerUnit": 25,
                "highMinutesPerUnit": 40,
                "basis": "One new prospective work link and method cluster with append-only ledger reconciliation."
              },
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                "component": "Hosted CI, guarded deployment and exact readback",
                "count": 1,
                "lowMinutesPerUnit": 20,
                "centralMinutesPerUnit": 35,
                "highMinutesPerUnit": 55,
                "basis": "One canonical URL requiring merged-main verification, zero-cost deployment and bounded custom-domain convergence checks."
              }
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            "probabilityPositiveSignal": 0.97,
            "probabilityTargetClosure": 0.84,
            "probabilityHorizonMinutes": 165,
            "assumptions": [
              "The published Zenodo bytes, OpenAI fable TTS, hosted CI and zero-cost Cloudflare credentials remain available without weakening a gate or exposing a secret.",
              "GitHub workflow-scope authorization completes without changing the already-frozen candidate commit or tag.",
              "Origin main does not acquire an irreconcilable overlapping release or operating-ledger change.",
              "The site generators accept the new candidate without a release-contract migration."
            ],
            "reforecastTriggers": [
              "GitHub, TTS, CI, Cloudflare or canonical readback remains unavailable after bounded recovery.",
              "A site, citation, licence, accessibility or preservation gate finds a claim-boundary conflict.",
              "Origin main advances with overlapping release, method-registry or work-ledger edits requiring reconciliation.",
              "Media inspection reveals a persistent representation or provenance defect."
            ],
            "stopRule": "Stop on a fatal claim, source, licence, authentication, preservation, CI, accessibility or exact-readback failure. At 110 active minutes prioritize load-bearing release gates; at 165 active minutes preserve the exact blocker rather than weaken a gate."
          },
          "outcome": {
            "measuredAt": "2026-08-31T12:26:46Z",
            "calendarElapsedMinutes": 40,
            "activeAgentMinutes": 33,
            "activeHumanMinutes": null,
            "computeMinutes": 2,
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            "agentRuns": 1,
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            "deduplicatedModelTokens": null,
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            "researchCycles": {
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            "candidateArchitecturesTested": 0,
            "candidateArchitecturesRejected": 0,
            "substantiveReviewRounds": 0,
            "p0Findings": 0,
            "p1Findings": 0,
            "prepublicationClaimCorrections": 0,
            "resultState": "target-closed",
            "resultSummary": "The review-repaired candidate reached a reader-first canonical Evidence Press release with byte-bound fable audio and transcript, deterministic factorial-spike art, an inspected Open Graph card and thumbnail, append-only operating records, protocol revision 194, green Node 18, 20 and 22 plus accessibility CI, guarded zero-cost deployment, one bounded five-second custom-domain protocol convergence retry, exact 45-release and 31-institutional-artifact readback, full publication preservation and accepted IndexNow submission. CI exposed one reader-function heading omission; the heading and protocol seal were repaired without changing any mathematical claim, source archive or assurance state. The result remains an anonymous unrefereed theorem candidate separating componentwise Noetherianity from FI-Noetherianity over the constructed non-FI-finitely-generated subalgebra, with only the carefully scoped regular-module consequence and no claim of a non-free module, a module counterexample over the ambient polynomial FI-algebra, exhaustive novelty, historical priority, independent reconstruction, formal verification, external specialist review, editorial peer review or impact.",
            "positiveSignalObserved": true,
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            "forecastRatio": 0.3,
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            "varianceReason": "Established page, media, composite-seal, hosted-CI and guarded-deployment tooling closed the canonical-site route below the 65-minute lower active-time bound; immutable GitHub and Zenodo identity were already complete at registration, and the single reader-heading repair was bounded and presentation-only.",
            "missingFields": [
              {
                "field": "activeHumanMinutes",
                "reason": "No instrument captured human direction or review time at the prospective publication-only boundary."
              },
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                "field": "deduplicatedModelTokens",
                "reason": "No active fork-aware goal counter exposed exact task-local model-token usage at or after the 11:46:57Z publication-ledger boundary, so tokens are not reconstructed from conversation context."
              },
              {
                "field": "uncachedInputTokens",
                "reason": "The runtime does not expose an uncached-input token counter."
              }
            ]
          }
        },
        "missingnessReason": null
      }
    ]
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}