{
  "schemaVersion": "1.6",
  "slug": "four-point-inductive-cubical-excision",
  "title": "A four-point failure of (J+, boxdot)-cubical excision",
  "shortTitle": "Four-point cubical-excision failure",
  "url": "https://evidencepress.org/releases/four-point-inductive-cubical-excision/",
  "oneLine": "A directed four-point closure space sends a nonzero integral relative one-cycle to zero, disproving interior-cover excision for normalized (J+, boxdot)-cubical homology.",
  "abstract": "This anonymous, unrefereed candidate gives a four-point interior cover in the inductive square of the directed interval J+ for which the integral relative cubical-homology map is not injective. In an explicit basis, the source has d1=[-1 -1] and d2=0, so H1(B,L) is Z; in the target, the identity square supplies the primitive column (1,-1), so H1(X,A)=0. The result is limited to normalized (J+, boxdot)-cubical singular homology and does not extend to categorical products, other intervals, symmetric graph closures, or other coefficient conventions. A written proof, complete Python and JavaScript stable-schema concordance, six semantic mutations, 15 tests, internal review, and hash-bound release gates support the candidate. Historical priority, independent reproduction, formal verification, specialist review, and peer review remain unestablished.",
  "datePublished": "2026-08-30",
  "dateModified": "2026-08-30",
  "version": "0.2.0-candidate",
  "doi": "10.5281/zenodo.22171896",
  "doiUrl": "https://doi.org/10.5281/zenodo.22171896",
  "conceptDoi": "10.5281/zenodo.22171895",
  "pdfUrl": "https://github.com/ipitchford/four-point-inductive-cubical-excision/releases/download/v0.2.0-candidate/four-point-inductive-cubical-excision-v0.2.0-candidate.pdf",
  "altPdfUrl": "https://zenodo.org/records/22171896/files/four-point-inductive-cubical-excision-v0.2.0-candidate.pdf?download=1",
  "altPdfLabel": null,
  "zenodoUrl": "https://zenodo.org/records/22171896",
  "repoUrl": "https://github.com/ipitchford/four-point-inductive-cubical-excision",
  "releaseUrl": "https://github.com/ipitchford/four-point-inductive-cubical-excision/releases/tag/v0.2.0-candidate",
  "markdownUrl": "https://evidencepress.org/releases/four-point-inductive-cubical-excision/index.md",
  "bibtexUrl": "https://evidencepress.org/releases/four-point-inductive-cubical-excision/cite.bib",
  "audioUrl": "https://evidencepress.org/assets/audio/four-point-inductive-cubical-excision.mp3?v=e1e7b68a78",
  "imageUrl": "https://evidencepress.org/assets/og/four-point-inductive-cubical-excision.png?v=41d83a5f0c",
  "coverArtUrl": "https://evidencepress.org/assets/art/four-point-inductive-cubical-excision.svg?v=fbd7b27eaf",
  "media": [
    {
      "type": "audio",
      "url": "https://evidencepress.org/assets/audio/four-point-inductive-cubical-excision.mp3",
      "name": "Audio briefing — four-point cubical-excision failure",
      "description": "Plain-English synthetic-voice account of the four-point cover, the map from Z to zero, the exact theory boundary, and assurance limits; a communication aid, not additional evidence.",
      "transcriptUrl": "https://evidencepress.org/assets/audio/four-point-inductive-cubical-excision.txt"
    }
  ],
  "authors": [
    "Anonymous"
  ],
  "license": "CC0-1.0",
  "status": "unrefereed-candidate",
  "verification": {
    "peerReviewed": false,
    "independentlyReproduced": false,
    "formallyVerified": false,
    "internallyReplayed": true,
    "detail": "Anonymous, unrefereed algebraic-topology candidate at internal PASS_WITH_NOTES. The finite theorem has a direct chain-level proof and same-producer exact replay. The candidate coarsens Bubenik and Milićević Example 5.9 but makes no first or priority claim because related work remains listed in preparation. The result is only for integral normalized (J+, boxdot)-cubical homology."
  },
  "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/22171896",
      "note": "The tagged GitHub prerelease and published Zenodo record expose the DOI-bearing PDF, complete ZIP, checksum ledger, claims, manifest and replay receipt; public downloads are checked against local bytes."
    },
    {
      "dimension": "internalReplay",
      "label": "Internal replay",
      "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?",
      "state": "passed",
      "note": "Normal and optimized Python agree, JavaScript matches the complete stable schema, all 15 tests pass, six semantic corruptions are rejected, the PDF compiles semantically, and complete source inventories remain unchanged across the four-job public Linux matrix.",
      "evidenceUrl": "https://github.com/ipitchford/four-point-inductive-cubical-excision/actions/runs/33305262433"
    },
    {
      "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 rerun is reported."
    },
    {
      "dimension": "independentReimplementation",
      "label": "Independent reimplementation",
      "question": "Has someone else reached the result from an independent implementation?",
      "state": "not-assessed",
      "note": "Python and JavaScript were produced within one coordinated workflow and are not independent implementations in the assurance sense."
    },
    {
      "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 the closure calculations, normalized chain complexes, or induced map."
    },
    {
      "dimension": "specialistReview",
      "label": "Specialist review",
      "question": "Has a domain specialist assessed the argument?",
      "state": "not-assessed",
      "note": "The supplied review, five internal roles and confirmation are not authenticated unaffiliated closure-space specialists."
    },
    {
      "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/four-point-inductive-cubical-excision/blob/v0.2.0-candidate/ENVIRONMENT.txt",
      "note": "Commands, tested versions, exact certificates, manifests and Linux CI are public, but no bit-reproducible system image or unaffiliated reconstruction is reported."
    }
  ],
  "provenance": {
    "aiGenerated": true,
    "aiAssisted": true,
    "generatedBy": [
      "AI systems under human direction",
      "OpenAI Codex",
      "Python 3 standard library",
      "Node.js standard library"
    ],
    "humanRole": "Research direction, candidate 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": "AI systems contributed mathematical reconstruction, exact computation, proof drafting, source and novelty auditing, role-separated internal review, review repair, editing, replay, and release engineering within one producer-coordinated workflow. Cross-language agreement and internal review are not unaffiliated reconstruction, specialist review, or editorial peer review."
  },
  "attributionBoundary": null,
  "problem": {
    "name": "AIM Problem 4.2: cubical homology and excision",
    "url": "http://aimpl.org/combhomotop/4/"
  },
  "orderedProductAverage": null,
  "corrections": [],
  "publicCorrections": [],
  "publicCorrectionIndexes": null,
  "pageStructureVersion": "2.0",
  "pageStructureVariant": "reader-first-reference",
  "pageStructureWaivers": [],
  "recordMaturity": "current",
  "metadataProvenance": "Authored prospectively against the reviewed theorem boundary, then reconciled after exact canonical Evidence Press readback on 30 August 2026. The public HTML, paper.json, content-versioned media, GitHub release and Zenodo DOI were checked without changing the theorem or assurance ceiling.",
  "grandfatheredAtSchemaVersion": null,
  "keywords": [
    "closure spaces",
    "cubical homology",
    "excision",
    "relative homology",
    "counterexample",
    "finite certificate",
    "unrefereed candidate"
  ],
  "keyResults": [
    "The sets A={10,01,11} and B={00,10,01} form an interior cover of X=J+ boxdot J+.",
    "With integral coefficients, H1(B,L) is Z, generated by the difference of the two coordinate edges.",
    "The identity square has relative boundary equal to that generator, so H1(X,A)=0.",
    "The inclusion-induced map Z to 0 is not injective, and interior-cover excision fails for this named cubical theory.",
    "The matrix certificate labels all four source and eight target degree-two columns; six semantic mutations are rejected in both encodings."
  ],
  "reviews": [],
  "evidencePackage": "A seven-page canonical manuscript and aligned Markdown; explicit singleton closures and integer matrices; a complete stable JSON certificate; dependency-free Python and JavaScript encodings; 15 unit/audit tests; six semantic mutations; complete pre/post payload inventories; a 53-entry manifest over a 56-file package; bundled fast-gate and terminal receipts; source, novelty, licence, review and review-response records; a five-role producer-organised substantive review; one bounded confirmation that found a gate-plan portability P1, followed by explicit-bytecode deterministic repair and reseal; a clean public GitHub history; and exact GitHub/Zenodo asset readback. These establish public availability, integrity, producer replay and the stated finite computation—not historical priority, unaffiliated reproduction, formal verification, external specialist review, peer review, or impact.",
  "openProblems": [
    "Determine excision for the inductive theories based on J1 and the ordinary interval I.",
    "Determine whether an analogous failure exists under symmetric graph-closure restrictions.",
    "Independently reconstruct the four-point chain complexes in a materially separate stack.",
    "Formalize the finite closure, continuity, normalization and relative-homology calculation in a proof assistant.",
    "Obtain an unaffiliated specialist source and priority assessment, including the related work listed in preparation."
  ],
  "mathObjects": [],
  "relatedWorks": [
    {
      "citation": "Bubenik, P., & Milićević, N. (2021). Eilenberg–Steenrod homology and cohomology theories for Čech's closure spaces. arXiv:2112.13421v1.",
      "url": "https://arxiv.org/abs/2112.13421v1",
      "doi": "10.48550/arXiv.2112.13421"
    },
    {
      "citation": "Bubenik, P., & Milićević, N. (2024). Homotopy, homology, and persistent homology using closure spaces. Journal of Applied and Computational Topology 8, 579–641.",
      "url": "https://doi.org/10.1007/s41468-024-00183-8",
      "doi": "10.1007/s41468-024-00183-8"
    },
    {
      "citation": "Milićević, N. (2025). Singular homology of roots of unity. Topology and its Applications 366, 109291.",
      "url": "https://doi.org/10.1016/j.topol.2025.109291",
      "doi": "10.1016/j.topol.2025.109291"
    },
    {
      "citation": "Jamil, S. S., Staecker, P. C., & Ali, D. (2022). Computability of digital cubical singular homology of c1-digital images. arXiv:2205.07457.",
      "url": "https://arxiv.org/abs/2205.07457",
      "doi": "10.48550/arXiv.2205.07457"
    },
    {
      "citation": "Bubenik, P. (2025). Relative cell complexes in closure spaces. Canadian Mathematical Bulletin 68, 1338–1346.",
      "url": "https://doi.org/10.4153/S0008439525100738",
      "doi": "10.4153/S0008439525100738"
    }
  ],
  "claimAssurance": null,
  "operatingModel": {
    "version": "1.0",
    "workId": "ep-work:four-point-inductive-cubical-excision",
    "attemptIds": [
      "ep-attempt:four-point-inductive-cubical-excision-assurance-publication"
    ],
    "aims": [
      "science"
    ],
    "artifactRoles": [
      "research-output",
      "evidence-assessment",
      "method-demonstration",
      "communication"
    ],
    "lineageId": null,
    "accelerationPrimitives": [
      "counterexample-proxy-first",
      "certificate-first",
      "structural-compression",
      "adversarial-controls",
      "assurance-vector",
      "agent-readable-research-object"
    ],
    "decisionObject": {
      "type": "counterexample",
      "description": "An exact four-point interior cover whose integral normalized (J+, boxdot)-cubical relative-homology map is Z to 0.",
      "scope": "The directed interval J+ with the inductive box product, normalized cubical chains, integral coefficients and degree one; not other intervals, products, graph restrictions, coefficient systems, novelty or external validation."
    },
    "bottleneckTargeted": [
      "discovery",
      "assurance",
      "publication"
    ],
    "semanticBridge": {
      "state": "explicit",
      "description": "The package derives the four-point closure from the cited J+ inductive-product definitions, states the exact relative excision map, and labels every basis column needed for H1.",
      "remainingRisks": [
        "The live AIM statement does not select one of the six cubical theories.",
        "Related in-preparation work is unavailable for collision testing.",
        "All executable and internal-review evidence remains producer coordinated."
      ]
    },
    "humanJudgmentGates": [
      "Keep the result limited to integral normalized (J+, boxdot)-cubical homology.",
      "Check the written relative-chain proof rather than treating replay as a proof oracle.",
      "Keep producer concordance separate from independent reproduction.",
      "Keep bounded novelty search separate from first or priority language."
    ],
    "parentLinks": [
      {
        "relation": "depends-on-claim",
        "workId": null,
        "legacyReleaseSlug": null,
        "externalUrl": "https://arxiv.org/abs/2112.13421",
        "inheritedClaim": "Bubenik and Milićević define the interval-and-product cubical theories, prove categorical-product excision and pose the inductive-product excision cases as open.",
        "inheritedAssuranceCeiling": "The source fixes the definitions and question but does not validate this candidate's computation, proof, novelty or priority."
      }
    ],
    "assuranceTarget": {
      "dimensions": [
        "independentRerun",
        "independentReimplementation",
        "formalVerification",
        "specialistReview",
        "dataEnvironmentReproducibility",
        "semanticValidation",
        "noveltyAssessment",
        "priorityAssessment"
      ],
      "nextAction": "Obtain an authenticated unaffiliated reconstruction and closure-space specialist assessment, then investigate the remaining inductive interval theories.",
      "claimCeiling": "An anonymous unrefereed four-point counterexample for integral normalized (J+, boxdot)-cubical homology with public immutable assets and producer-side replay; not historical priority, unaffiliated validation, formal verification, external specialist review, or editorial peer review."
    },
    "impactClaims": [
      {
        "id": "four-point-cubical-excision-science-no-impact",
        "aim": "science",
        "outcome": "Faster or more reliable resolution and assurance of topology open problems",
        "setting": "AI-assisted finite counterexample reconstruction, proof compression, source audit, review repair and guarded candidate publication",
        "status": "NO_IMPACT_EVIDENCE",
        "designClass": "none",
        "comparator": "No matched conventional workflow was registered.",
        "estimand": "No effect on discovery time, human effort, compute, correction rate, proof quality, 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:four-point-inductive-cubical-excision-assurance-publication",
        "status": "published",
        "resultClass": "positive",
        "metrics": {
          "schemaVersion": "1.0",
          "measurementScope": "assurance-through-publication",
          "scopeBoundary": "From prospective registration before substantive action on the supplied Major Revision report through terminal public readback or a preserved stop. The earlier problem selection, theorem discovery, proof construction, prepare-only review and package build are excluded and are not reconstructed.",
          "forecast": {
            "frozenAt": "2026-08-30T08:43:51Z",
            "procedureClass": "major-review-replay-redesign-through-full-mathematics-candidate-publication",
            "targetOutcome": "Action every required review item, redesign replay so a clean cross-environment run is successful and non-mutating, obtain frozen internal editorial closure, publish synchronized immutable GitHub and Zenodo assets, and complete both guarded Evidence Press release cycles without exceeding the exact (J_+, inductive-product, integral H_1) claim boundary.",
            "expectedActiveMinutes": 140,
            "plausibleLowMinutes": 90,
            "plausibleHighMinutes": 220,
            "expectedUnattendedWaitMinutes": 60,
            "referenceClass": {
              "label": "Reviewed exact mathematics candidates taken through full Evidence Press publication",
              "sampleSize": 2,
              "basis": "The two latest prospectively measured assurance-through-publication releases completed in 95 and 55 active agent minutes. This candidate has a smaller theorem but a material replay-contract redesign and a fresh five-role review."
            },
            "fermiComponents": [
              {
                "component": "non-mutating replay redesign, schema consistency and clean-environment validation",
                "count": 1,
                "lowMinutesPerUnit": 30,
                "centralMinutesPerUnit": 45,
                "highMinutesPerUnit": 70,
                "basis": "One compact Python and JavaScript package, but the manifest, stable and volatile receipts, PDF build and clean-tree test must be separated coherently."
              },
              {
                "component": "manuscript, source relationship, literature and frozen editorial closure",
                "count": 1,
                "lowMinutesPerUnit": 20,
                "centralMinutesPerUnit": 30,
                "highMinutesPerUnit": 50,
                "basis": "One six-page note, one supplied Major Revision, four primary related-work records, one five-role round and at most one confirmation."
              },
              {
                "component": "public research repository, CI, immutable release and Zenodo identity",
                "count": 1,
                "lowMinutesPerUnit": 15,
                "centralMinutesPerUnit": 25,
                "highMinutesPerUnit": 40,
                "basis": "One dependency-light anonymous candidate with standard licences, one tagged CI run and three principal archive assets."
              },
              {
                "component": "Evidence Press page, media, two composite seals, deployments and readback",
                "count": 1,
                "lowMinutesPerUnit": 25,
                "centralMinutesPerUnit": 40,
                "highMinutesPerUnit": 60,
                "basis": "One new reader-first release using established art, audio, thumbnail, A/B then C/D protocol seals and zero-cost guarded deployment tooling."
              }
            ],
            "probabilityPositiveSignal": 0.96,
            "probabilityTargetClosure": 0.82,
            "probabilityHorizonMinutes": 220,
            "assumptions": [
              "The supplied review's chain calculation remains consistent with the foundational source conventions after the notation and matrix repairs.",
              "A non-mutating build-directory design passes both local and public Linux clean-checkout validation.",
              "GitHub, Zenodo, OpenAI fable TTS and zero-cost Cloudflare authentication remain available without weakening a gate or exposing a secret.",
              "The current Evidence Press operating-model and preservation contracts admit the new bounded topology counterexample record."
            ],
            "reforecastTriggers": [
              "The exact stable-result receipt disagrees between Python and JavaScript or a semantic mutation survives.",
              "A primary source or frozen internal review identifies a load-bearing source-convention or originality error.",
              "GitHub, Zenodo, CI, audio, Cloudflare or canonical readback remains unavailable after bounded recovery.",
              "Origin main advances with overlapping operating-model, ledger or release changes requiring reconciliation."
            ],
            "stopRule": "Stop on a fatal proof, semantic, novelty, licence, authentication, preservation, CI, accessibility or exact-readback failure. At 140 active agent minutes consolidate to load-bearing gates and report variance; at 220 active minutes pause rather than weaken a gate unless the user explicitly authorises continuation after seeing the exact blocker."
          },
          "outcome": {
            "measuredAt": "2026-08-30T10:16:05Z",
            "calendarElapsedMinutes": 92,
            "activeAgentMinutes": 84,
            "activeHumanMinutes": null,
            "computeMinutes": 2,
            "unattendedWaitMinutes": 8,
            "blockedMinutes": 0,
            "reworkMinutes": 6,
            "agentRuns": 1,
            "maxParallelAgents": 1,
            "modelTurns": 1,
            "deduplicatedModelTokens": 1543001,
            "uncachedInputTokens": null,
            "researchCycles": {
              "positive": 0,
              "negative": 0,
              "inconclusive": 0
            },
            "falsificationGatesRun": 6,
            "candidateArchitecturesTested": 1,
            "candidateArchitecturesRejected": 0,
            "substantiveReviewRounds": 2,
            "p0Findings": 0,
            "p1Findings": 4,
            "prepublicationClaimCorrections": 18,
            "resultState": "target-closed",
            "resultSummary": "The supplied Major Revision was fully actioned, including the manifest-mutating replay redesign, frozen-record reconciliation, precise source relationship, adjacent-theory comparison, exact matrix certificate and bounded terminology and presentation repairs. The final non-mutating package passed 15 tests, six semantic corruptions, Python optimization and JavaScript concordance, a pristine-archive gate, a four-environment public Linux matrix and a frozen internal review plus confirmation. Byte-identical immutable GitHub and Zenodo assets then reached canonical Evidence Press publication with provenance-bound media, composite CI, accessibility, 41-release and 31-artifact preservation, protocol integrity and desktop/mobile browser readback. The result remains an anonymous unrefereed candidate with unresolved priority and no external-assurance or impact promotion.",
            "positiveSignalObserved": true,
            "targetReached": true,
            "forecastErrorMinutes": -56,
            "forecastRatio": 0.6,
            "withinForecastInterval": false,
            "varianceReason": "The theorem was mathematically sound at intake, the replay redesign admitted one compact build-directory architecture, and established exact-package, GitHub, Zenodo, media, composite-seal and guarded-deployment tooling allowed immutable-identity and publication work to close below the 90-minute lower active-time bound despite a concurrent-main reconciliation and one bounded confirmation enhancement.",
            "missingFields": [
              {
                "field": "activeHumanMinutes",
                "reason": "No instrument captured human direction or review time at the prospective attempt boundary."
              },
              {
                "field": "uncachedInputTokens",
                "reason": "The runtime does not expose an uncached-input counter."
              }
            ]
          }
        },
        "missingnessReason": null
      }
    ]
  }
}