{ "schemaVersion": "1.6", "slug": "ap4-ramsey-diameter-36", "title": "No 26-point four-term-AP Ramsey set has primitive diameter at most 36", "shortTitle": "Four-term-AP Ramsey diameter 36", "url": "https://evidencepress.org/releases/ap4-ramsey-diameter-36/", "oneLine": "Checked SAT certificates establish that 27 is the minimum inside [1,37]; the unrestricted sparse value and the AIM asymptotic questions remain open.", "abstract": "This anonymous, AI-assisted, unrefereed computer-assisted finite theorem candidate addresses the bounded core of AIM-COMBINATORICS-0088. A 1,579-variable, 5,761-clause CNF with 2,012 audited colouring cuts and a checked DRAT refutation excludes every 26-point four-term-arithmetic-progression Ramsey subset of [37]. A separate 27-variable, 196-clause CNF and DRAT refutation verifies Graham's displayed 27-point witness, giving v_{2,4}([37]) = 27. The paper proves the source-to-encoding bridge, exact-cardinality and edge-indicator lemmas, colouring-cut soundness, monotonicity and primitive-affine-diameter corollary. Public replay reconstructs both formulas, checks both proofs, validates 28 positive colourings, exhausts a 512-subset oracle, recomputes a separate 21,329-candidate structural neighbourhood and requires fifteen hostile controls to fail. The result does not establish W*(4) = 27 or answer either asymptotic question in AIM Problem 1.9. Independent reconstruction, formal verification, external specialist review, editorial peer review, exhaustive novelty and absolute priority remain open or unassessed.", "datePublished": "2026-09-03", "dateModified": "2026-09-03", "version": "0.1.0-candidate", "doi": "10.5281/zenodo.22286322", "doiUrl": "https://doi.org/10.5281/zenodo.22286322", "conceptDoi": "10.5281/zenodo.22286321", "pdfUrl": "https://github.com/ipitchford/ap4-ramsey-diameter-36/releases/download/v0.1.0-candidate/ap4-ramsey-diameter-36-paper-v0.1.0-candidate.pdf", "altPdfUrl": "https://zenodo.org/records/22286322/files/ap4-ramsey-diameter-36-paper-v0.1.0-candidate.pdf?download=1", "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/22286322", "repoUrl": "https://github.com/ipitchford/ap4-ramsey-diameter-36", "releaseUrl": "https://github.com/ipitchford/ap4-ramsey-diameter-36/releases/tag/v0.1.0-candidate", "markdownUrl": "https://evidencepress.org/releases/ap4-ramsey-diameter-36/index.md", "bibtexUrl": "https://evidencepress.org/releases/ap4-ramsey-diameter-36/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/ap4-ramsey-diameter-36.mp3?v=53d0255af4", "imageUrl": "https://evidencepress.org/assets/og/ap4-ramsey-diameter-36.png?v=26feca8d32", "coverArtUrl": "https://evidencepress.org/assets/art/ap4-ramsey-diameter-36.svg?v=38bfa36a7b", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/ap4-ramsey-diameter-36.mp3", "name": "Audio briefing — four-term-AP Ramsey diameter 36", "description": "Plain-English OpenAI API synthetic-voice account of the exact bounded equality, its two checked certificates and the open unrestricted boundary; a communication aid, not additional evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/ap4-ramsey-diameter-36.txt" } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Anonymous, AI-assisted, unrefereed computer-assisted finite theorem candidate at internal PASS_WITH_NOTES with no residual or new P0/P1 findings in the final exact-byte confirmation. The finite theorem is a written source-to-encoding argument supported by separately checked lower and upper DRAT certificates. The 21,329-candidate replacement neighbourhood is a non-load-bearing local structural calculation. The unrestricted sparse value W*(4), every primitive diameter above 36, both asymptotic questions in AIM Problem 1.9, independent reconstruction, proof-assistant formalization, external specialist review, journal peer review, exhaustive novelty assessment and priority adjudication remain open or unassessed." }, "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/22286322", "note": "The tagged GitHub prerelease and published Zenodo record expose the same archive, PDF and checksum sidecar; independently downloaded SHA-256 values match the frozen local assets." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "A clean public runner reconstructs both exact formulas, checks both DRAT proofs, validates 28 positive colourings, exhausts a small oracle, recomputes the structural neighbourhood and requires fifteen hostile mutations to fail.", "evidenceUrl": "https://github.com/ipitchford/ap4-ramsey-diameter-36/actions/runs/33792599171" }, { "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 party has rerun the complete immutable package and published a separate environment and result receipt." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "not-assessed", "note": "The direct oracle and separately structured checks remain producer-coordinated and do not constitute an unaffiliated reimplementation." }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed", "note": "DRAT proves unsatisfiability of the exact CNFs, but no proof assistant formalizes the source-to-CNF bridge, monotonicity or primitive-affine normalization argument." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "The supplied review, five-role reports and bounded confirmation are internal workflow records, not authenticated unaffiliated specialist review." }, { "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 to a decision." }, { "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/ap4-ramsey-diameter-36/blob/v0.1.0-candidate/ENVIRONMENT.txt", "note": "Exact solver commits, build commands, environment details and clean hosted Linux replay are public, but no authenticated unaffiliated clean-environment reconstruction is reported and the environment is not bit-identical." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "AI systems under human direction", "OpenAI Codex", "Python 3.14.6", "Kissat 4.0.4", "CaDiCaL 3.0.1", "DRAT-trim" ], "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": "AI systems contributed finite search, semantic encoding, verification code, literature and GitHub search, drafting, action of the supplied review, internal role-separated review, replay and release engineering within one producer-coordinated workflow. Exact proof checking, implementation-diverse internal checks, matching public hashes and publication are not unaffiliated reconstruction, authenticated specialist review or editorial peer review." }, "attributionBoundary": null, "problem": { "name": "Sparse van der Waerden sets (AIM-COMBINATORICS-0088, Problem 1.9)", "url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf" }, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "certificate-backed-computation", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "Authored prospectively against the DOI-bearing tagged v0.1.0-candidate, exact GitHub release, green public replay, published Zenodo record, supplied-review response, five-role internal editorial record and bounded exact-byte confirmation on 3 September 2026. Canonical Evidence Press deployment and terminal publication metrics are recorded separately and do not promote the mathematical assurance ceiling.", "grandfatheredAtSchemaVersion": null, "keywords": [ "Ramsey theory", "four-term arithmetic progression", "sparse van der Waerden set", "Property B", "SAT certificate", "DRAT", "computer-assisted proof", "unrefereed candidate" ], "keyResults": [ "The minimum cardinality of a four-term-AP Ramsey subset of [1,37] is exactly 27.", "Equivalently, no 26-point four-term-AP Ramsey set has primitive affine diameter at most 36.", "A semantically reconstructed 1,579-variable, 5,761-clause lower CNF with 2,012 audited colouring cuts has a checked DRAT refutation.", "Graham's displayed 27-point set is verified by a separate 27-variable, 196-clause colourability CNF and DRAT refutation.", "A distinct 21,329-candidate replacement neighbourhood contains no witness, but this local structural result is not used in the theorem and does not settle W*(4)." ], "reviews": [], "evidencePackage": "A seven-page theorem manuscript and accessible Markdown; a 1,579-variable, 5,761-clause lower-bound CNF, 2,012 audited colouring cuts and DRAT refutation; a separate 27-variable, 196-clause upper-witness CNF and DRAT refutation; exact source-to-clause reconstruction; 28 explicit satisfying colourings; a dependency-free 512-subset small oracle; ordinary and optimized Python tests; complete 21,329-set replacement-neighbourhood recomputation; fifteen fail-closed semantic, target, witness, proof, hash and inventory corruptions; source-pinned solver bootstrap; timestamped replay receipts; a supplied Major Revision review; producer-coordinated five-role internal review, response matrix and bounded exact-byte confirmation; green public GitHub replay; and byte-identical GitHub and Zenodo archive, PDF and checksum assets. These establish public availability, package integrity, producer replay and the exact bounded certificate—not W*(4) = 27, the AIM asymptotic questions, independent reconstruction, formal verification, external specialist review, peer review, exhaustive novelty or priority.", "openProblems": [ "Find a certified 26-point four-term-AP Ramsey set of primitive diameter greater than 36, or prove that none exists at any diameter.", "Develop a global structural reduction that covers infinite families of primitive 26-point sets rather than one declared replacement neighbourhood.", "Formally verify the source-to-CNF bridge and connect it to machine-checked DRAT or LRAT proof validation.", "Independently reimplement the normalized set search, colouring-cut generation and both certificate checks in a materially separate stack.", "Determine what, if anything, exact small-k sparse results imply for W(k)-W*(k) or W*(k)/W(k) as k grows.", "Obtain an unaffiliated specialist assessment of correctness, closest prior art and historical priority." ], "mathObjects": [], "relatedWorks": [ { "citation": "American Institute of Mathematics. Open Problems in Additive Combinatorics, Problem 1.9.", "url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "doi": null }, { "citation": "Graham, R. L. (1983). Recent developments in Ramsey theory. Proceedings of the International Congress of Mathematicians, 1555-1567.", "url": "https://www.math.ucsd.edu/~ronspubs/83_01_recent_ramsey.pdf", "doi": null }, { "citation": "Graham, R. L., and Nesetril, J. (1986). Large Minimal Sets Which Force Long Arithmetic Progressions. Journal of Combinatorial Theory, Series A 42(2), 270-276.", "url": "https://doi.org/10.1016/0097-3165(86)90097-X", "doi": "10.1016/0097-3165(86)90097-X" }, { "citation": "Hindman, N., and Jordan, J. (2013). On the Size of Minimal Hales-Jewett Sets. Scientiae Mathematicae Japonicae 76(2), 195-207.", "url": "https://doi.org/10.32219/isms.76.2_195", "doi": "10.32219/isms.76.2_195" }, { "citation": "Dransfield, M. R., Liu, L., Marek, V. W., and Truszczynski, M. (2004). Satisfiability and computing van der Waerden numbers. Electronic Journal of Combinatorics 11.", "url": "https://doi.org/10.37236/1794", "doi": "10.37236/1794" }, { "citation": "Sinz, C. (2005). Towards an optimal CNF encoding of Boolean cardinality constraints. CP 2005, 827-831.", "url": "https://doi.org/10.1007/11564751_73", "doi": "10.1007/11564751_73" }, { "citation": "Wetzler, N., Heule, M. J. H., and Hunt, W. A. Jr. (2014). DRAT-trim: Efficient checking and trimming using expressive clausal proofs. SAT 2014, 422-429.", "url": "https://doi.org/10.1007/978-3-319-09284-3_31", "doi": "10.1007/978-3-319-09284-3_31" } ], "claimAssurance": null, "operatingModel": { "version": "1.0", "workId": "ep-work:ap4-ramsey-diameter-36", "attemptIds": [ "ep-attempt:ap4-ramsey-diameter-36-publication-completion" ], "aims": [ "science" ], "artifactRoles": [ "research-output", "evidence-assessment", "method-demonstration", "communication" ], "lineageId": null, "accelerationPrimitives": [ "certificate-first", "structural-compression", "adversarial-controls", "assurance-vector", "agent-readable-research-object" ], "decisionObject": { "type": "certificate", "description": "A semantically reconstructed SAT and DRAT certificate pair for the exact bounded four-term-AP Ramsey value v_{2,4}([37]) = 27.", "scope": "Finite subsets of [1,37], equivalently 26-point sets of primitive affine diameter at most 36; not arbitrary primitive diameter, W*(4), the AIM asymptotic questions, independent validation, novelty or priority." }, "bottleneckTargeted": [ "assurance", "publication", "translation" ], "semanticBridge": { "state": "explicit", "description": "The paper maps selected-point variables, exact-cardinality counters and progression-indicator equivalences to the finite set problem; proves every full-colouring clause necessary for a Ramsey set; and joins the lower refutation to a separately certified 27-point upper witness, monotonicity and primitive affine normalization.", "remainingRisks": [ "The source-to-CNF bridge and both certificate pipelines have not been reconstructed by an unaffiliated specialist.", "DRAT checking certifies CNF unsatisfiability but does not formally verify the mathematical translation into that CNF.", "The 21,329-candidate structural neighbourhood is local and does not control arbitrary larger primitive diameters.", "The targeted literature and GitHub search may miss differently phrased, unindexed or unpublished prior work." ] }, "humanJudgmentGates": [ "Check the exact-cardinality, edge-indicator and colouring-cut soundness lemmas as mathematics rather than inferring the theorem from a green SAT proof.", "Keep v_{2,4}([37]) = 27 separate from W*(4) = 27 and from both asymptotic questions in AIM Problem 1.9.", "Keep the 21,329-candidate replacement neighbourhood separate from the load-bearing lower certificate and from global structural classification.", "Treat public proof checking, internal review and archive availability as distinct from unaffiliated validation, formal verification, peer review, novelty and priority." ], "parentLinks": [ { "relation": "depends-on-claim", "workId": null, "legacyReleaseSlug": null, "externalUrl": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "inheritedClaim": "AIM Problem 1.9 asks asymptotic questions about the gap and ratio between ordinary and sparse two-colour van der Waerden parameters.", "inheritedAssuranceCeiling": "The problem source fixes the broader target but does not state or establish this exact bounded equality, its certificate, correctness, novelty or priority." }, { "relation": "extends-result", "workId": null, "legacyReleaseSlug": null, "externalUrl": "https://www.math.ucsd.edu/~ronspubs/83_01_recent_ramsey.pdf", "inheritedClaim": "Graham displays the 27-point four-term-AP Ramsey set that supplies this candidate's upper bound.", "inheritedAssuranceCeiling": "The published construction supplies the witness, not the complete exclusion of every 26-point subset of [37]." } ], "assuranceTarget": { "dimensions": [ "independentRerun", "independentReimplementation", "formalVerification", "specialistReview", "dataEnvironmentReproducibility", "semanticValidation", "noveltyAssessment", "priorityAssessment" ], "nextAction": "Obtain an authenticated unaffiliated reconstruction of the source-to-CNF bridge and both certificates, then pursue either a certified 26-point witness at larger primitive diameter or a global structural exclusion.", "claimCeiling": "An anonymous, AI-assisted, unrefereed computer-assisted finite theorem candidate for v_{2,4}([37]) = 27 with a written soundness bridge, two checked DRAT certificates, hostile controls, public producer replay and internal editorial closure; not W*(4) = 27, an answer to AIM Problem 1.9, independent validation, formal verification, external specialist or editorial peer review, historical priority or demonstrated workflow impact." }, "impactClaims": [ { "id": "ap4-ramsey-diameter-36-science-no-impact", "aim": "science", "outcome": "Faster or more reliable resolution and assurance of sparse van der Waerden problems", "setting": "AI-assisted finite search, semantic SAT encoding, adversarial replay, internal review and guarded candidate publication", "status": "NO_IMPACT_EVIDENCE", "designClass": "none", "comparator": "No matched conventional research, theorem-review 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:ap4-ramsey-diameter-36-publication-completion", "status": "published", "resultClass": "partial", "metrics": { "schemaVersion": "1.0", "measurementScope": "publication-only", "scopeBoundary": "From registration after reviewed GitHub and Zenodo identity through reader-first Evidence Press authoring, media, reciprocal operating records, composite sealing, hosted site CI, merge, guarded zero-cost deployment and exact canonical readback or a preserved stop. All research, review and immutable-identity work is excluded and not reconstructed.", "forecast": { "frozenAt": "2026-09-03T18:55:39.000Z", "procedureClass": "canonical-site-authoring-media-seal-deploy-and-readback", "targetOutcome": "Publish a reader-first Evidence Press release with provenance-bound fable audio, deterministic art and thumbnail, reciprocal method and work ledgers, green composite CI, guarded zero-cost deployment and exact canonical HTML, paper.json and media readback.", "expectedActiveMinutes": 70, "plausibleLowMinutes": 35, "plausibleHighMinutes": 120, "expectedUnattendedWaitMinutes": 25, "referenceClass": { "label": "Recent Evidence Press publication-completion releases", "sampleSize": 6, "basis": "Recent mathematics releases used the same reader-first authoring, media generators, composite protocol seal, four-job hosted CI, guarded deployment and canonical readback architecture." }, "fermiComponents": [ { "component": "Reader-first page, metadata and reciprocal operating records", "count": 1, "lowMinutesPerUnit": 10, "centralMinutesPerUnit": 20, "highMinutesPerUnit": 35, "basis": "One exact finite result with an unusually important global-versus-finite claim boundary." }, { "component": "Claim-disciplined art, Open Graph image, thumbnail and fable audio", "count": 1, "lowMinutesPerUnit": 10, "centralMinutesPerUnit": 15, "highMinutesPerUnit": 25, "basis": "One new mathematical slug using established deterministic renderers and the house TTS profile." }, { "component": "Composite build, protocol seals and hosted CI", "count": 1, "lowMinutesPerUnit": 10, "centralMinutesPerUnit": 20, "highMinutesPerUnit": 35, "basis": "One reader-first release on the current schema with reciprocal work and method records." }, { "component": "Merge, guarded deployment and canonical readback", "count": 1, "lowMinutesPerUnit": 5, "centralMinutesPerUnit": 15, "highMinutesPerUnit": 25, "basis": "One zero-cost Cloudflare deployment with exact release, media, protocol and preservation checks." } ], "probabilityPositiveSignal": 0.99, "probabilityTargetClosure": 0.92, "probabilityHorizonMinutes": 120, "assumptions": [ "The already public GitHub and Zenodo identities remain reachable and byte-stable.", "OpenAI fable TTS, hosted CI and zero-cost Cloudflare credentials remain available without weakening a gate.", "Origin main does not acquire an irreconcilable overlapping release or append-only ledger change.", "The current reader-first and protocol schemas accept the new finite Ramsey record without migration." ], "reforecastTriggers": [ "A media, accessibility, registry, manifest or preservation gate finds a release-blocking conflict.", "Hosted CI, TTS, Cloudflare or canonical readback remains unavailable after bounded recovery.", "Origin main advances with overlapping release or append-only ledger edits requiring reconciliation." ], "stopRule": "Stop on a fatal claim, licence, authentication, preservation, CI, accessibility or exact-readback failure. At 70 active minutes prioritize load-bearing gates; at 120 active minutes preserve the exact blocker rather than weaken a gate." }, "outcome": { "measuredAt": "2026-09-03T19:23:24.000Z", "calendarElapsedMinutes": 28, "activeAgentMinutes": 20, "activeHumanMinutes": null, "computeMinutes": 0, "unattendedWaitMinutes": 8, "blockedMinutes": 0, "reworkMinutes": 6, "agentRuns": 1, "maxParallelAgents": 1, "modelTurns": 1, "deduplicatedModelTokens": null, "uncachedInputTokens": null, "researchCycles": { "positive": 0, "negative": 0, "inconclusive": 0 }, "falsificationGatesRun": 0, "candidateArchitecturesTested": 0, "candidateArchitecturesRejected": 0, "substantiveReviewRounds": 0, "p0Findings": 0, "p1Findings": 0, "prepublicationClaimCorrections": 0, "resultState": "target-closed", "resultSummary": "The honestly left-bounded publication-completion attempt closed with a reader-first Evidence Press page, provenance-bound fable narration, deterministic art and thumbnail, reciprocal method and work records, exact protocol seals, green Node 18, 20 and 22 plus accessibility CI, guarded zero-cost Cloudflare deployment, byte-identical canonical HTML, paper.json and protocol readback, full 50-release and 31-institutional-artifact preservation, and accepted IndexNow submission. A stale derived audio index and an in-place expansion of five published method representative lists were corrected through additive, reviewed commits without changing the theorem, research package or media. The result remains an anonymous, AI-assisted, unrefereed finite computer-assisted theorem candidate for v_{2,4}([37]) = 27; W*(4), primitive diameter above 36, the AIM asymptotic questions, independent reconstruction, formal verification, external specialist review, peer review, novelty and priority remain open or unassessed.", "positiveSignalObserved": true, "targetReached": true, "forecastErrorMinutes": -50, "forecastRatio": 0.29, "withinForecastInterval": false, "varianceReason": "The attempt began only after the research package, supplied-review repairs, five-role review, GitHub prerelease, research CI and Zenodo record were complete; established authoring, media, seal and deployment tooling then closed the remaining route below the 35-minute lower active-time bound despite two bounded integration repairs.", "missingFields": [ { "field": "activeHumanMinutes", "reason": "No instrument captured human direction or review time inside the prospective publication-only boundary." }, { "field": "deduplicatedModelTokens", "reason": "No active fork-aware task counter exposed exact task-local model-token usage at or after the 18:55:39Z publication boundary, so tokens are not reconstructed from conversation context." }, { "field": "uncachedInputTokens", "reason": "The runtime does not expose an uncached-input token counter." } ] } }, "missingnessReason": null } ] } }