{ "schemaVersion": "1.6", "slug": "odd-order-one-third-popular-differences", "title": "A Three-Layer Obstruction for Uniformly Popular Differences", "shortTitle": "Three-layer popular-difference obstruction", "url": "https://evidencepress.org/releases/odd-order-one-third-popular-differences/", "oneLine": "An eight-point subset of Z/21Z refutes the strict one-third AIM question, while a three-layer family obstructs every fixed threshold below 2/5 in odd cyclic groups.", "abstract": "AIM Problem 1.18 asks whether a finite nonempty subset A of an abelian group must have A-A equal to a subgroup or a union of at most three cosets whenever every represented difference has more than |A|/3 ordered representations. This anonymous, unrefereed candidate gives a negative answer in Z/21Z using the eight-point set {0,1,2,7,9,14,15,16}. A direct three-layer construction further shows that for every real theta below 2/5 there is an odd cyclic counterexample with r_A(d)>theta|A| throughout A-A. Exact Python and JavaScript replay, four mutation controls, fresh-extraction checks, and producer-coordinated internal review pass. The release does not claim that 2/5 is sufficient or sharp, that the C21 witness is minimal, that novelty or priority is established, or that independent, formal, specialist, or editorial peer review has occurred.", "datePublished": "2026-08-29", "dateModified": "2026-08-29", "version": "0.1.0-candidate", "doi": "10.5281/zenodo.22161245", "doiUrl": "https://doi.org/10.5281/zenodo.22161245", "conceptDoi": "10.5281/zenodo.22161244", "pdfUrl": "https://github.com/ipitchford/odd-order-one-third-popular-differences/releases/download/v0.1.0-candidate/odd-order-one-third-popular-differences-v0.1.0-candidate.pdf", "altPdfUrl": "https://zenodo.org/records/22161245/files/odd-order-one-third-popular-differences-v0.1.0-candidate.pdf?download=1", "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/22161245", "repoUrl": "https://github.com/ipitchford/odd-order-one-third-popular-differences", "releaseUrl": "https://github.com/ipitchford/odd-order-one-third-popular-differences/releases/tag/v0.1.0-candidate", "markdownUrl": "https://evidencepress.org/releases/odd-order-one-third-popular-differences/index.md", "bibtexUrl": "https://evidencepress.org/releases/odd-order-one-third-popular-differences/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/odd-order-one-third-popular-differences.mp3?v=c162fcead5", "imageUrl": "https://evidencepress.org/assets/og/odd-order-one-third-popular-differences.png?v=e1b443920a", "coverArtUrl": "https://evidencepress.org/assets/art/odd-order-one-third-popular-differences.svg?v=f710c19dd4", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/odd-order-one-third-popular-differences.mp3", "name": "Audio briefing — three-layer popular-difference obstruction", "description": "Plain-English OpenAI API synthetic-voice account of the C21 witness, three-layer family, below-2/5 obstruction, and open assurance boundary; a communication aid, not additional evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/odd-order-one-third-popular-differences.txt" }, { "type": "video", "url": "https://youtu.be/Ma7gtKIwQ7Y", "name": "A Three-Layer Obstruction for Uniformly Popular Differences", "description": "Evidence Press video overview of this release. This is a communication aid; it does not extend the release's claims or assurance status and is not additional research evidence." } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Anonymous, unrefereed counterexample and elementary-family candidate at internal PASS_WITH_NOTES. The C21 computation and bounded family checks replay across Python and JavaScript, and the argument is supplied in full for specialist inspection. The source attribution was corrected to AIM Problem 1.18 after audit. A targeted novelty search found no direct prior statement, but novelty and priority remain unresolved. No unaffiliated rerun, independent reimplementation, proof-assistant formalization, external specialist review, or editorial peer review has occurred." }, "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/22161245", "note": "The tagged GitHub prerelease and published Zenodo record expose the PDF, deterministic archive, checksum sidecar, claims, sources, review boundary, and replay files; independently downloaded GitHub and Zenodo assets match 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": "Producer-side Python and JavaScript verifiers agree, all four mutations fail as intended, fresh extraction passes 12 of 12 publication-baseline gates, and hosted checks pass on Node 18, 20, and 22.", "evidenceUrl": "https://github.com/ipitchford/odd-order-one-third-popular-differences/actions/runs/33251769656" }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed", "note": "No unaffiliated party has reported rerunning the immutable package in a separately provisioned environment." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "not-assessed", "note": "The Python and JavaScript implementations were produced within the same coordinated research process and are not independent reconstructions." }, { "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 finite witness, three-layer counting proposition, or asymptotic odd-cyclic corollary." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "The five-role review and confirmation were internal AI editorial quality control, not unaffiliated additive-combinatorics 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 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/odd-order-one-third-popular-differences/blob/v0.1.0-candidate/ENVIRONMENT.txt", "note": "The checks use only Python 3.11 or later and Node 18 or later standard libraries, with no network, randomness, solver, or third-party package; commands and hashes are public, but no declarative system image or unaffiliated clean-environment 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, source selection, 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 search, exact computation, proof drafting, source auditing, internal role-separated review, editing, replay, and release engineering within one producer-coordinated workflow. Cross-language agreement and internal review are not independent reconstruction, specialist review, or editorial peer review." }, "attributionBoundary": null, "problem": { "name": "AIM Problem 1.18: strict one-third popular differences", "url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf" }, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "reader-first-reference", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "Authored prospectively for reader-first schema 2.0 against immutable GitHub and Zenodo identities, the final source-to-claim audit, fresh-extraction replay, and frozen internal PASS_WITH_NOTES review record.", "grandfatheredAtSchemaVersion": null, "keywords": [ "additive combinatorics", "popular differences", "difference sets", "abelian groups", "cyclic groups", "counterexample", "three-layer construction", "unrefereed candidate" ], "keyResults": [ "In Z/21Z, A={0,1,2,7,9,14,15,16} has minimum difference multiplicity 3>8/3, while A-A has size 15 and is neither a subgroup nor a union of at most three cosets of a common subgroup.", "For finite abelian H of order h, nonempty proper Y of size m, and q at least 7, the three-layer set has A-A=H times {0, plus or minus 1, plus or minus 2} and minimum difference multiplicity min(h,2m).", "The three-layer family satisfies the strict one-third condition exactly when 2h/5theta|A| for every represented difference.", "The result leaves the threshold 2/5 and above open and makes no minimality, novelty, or priority claim." ], "reviews": [], "evidencePackage": "A two-page candidate manuscript and accessible Markdown; an indexed three-claim and four-nonclaim registry; direct finite enumeration in independent Python and JavaScript encodings; a 264-case bounded regression for the general counting formula; four mutation controls that reject corrupted predicates; a frozen 13/13 producer confirmation receipt; a later 12/12 fresh-extraction publication-baseline receipt; source-to-claim and targeted novelty audits; and a five-role producer-coordinated internal review with bounded confirmation. Public GitHub Actions passed on Node 18, 20, and 22, and the archive, PDF, and checksum sidecar were byte-matched across local, GitHub, and Zenodo copies. These establish availability, integrity, internal replay, and the declared finite computations—not independent mathematical validation, exhaustive novelty, or peer review.", "openProblems": [ "Determine whether 2/5 is sufficient or sharp for the subgroup-or-three-coset conclusion, including the strict and non-strict endpoint formulations.", "Classify near-extremal sets whose minimum represented-difference multiplicity approaches 2|A|/5.", "Determine the smallest odd-order or cyclic counterexample; the C21 witness is not claimed minimal.", "Obtain an unaffiliated additive-combinatorics audit of the source mapping, proof, and exact relationship to prior literature.", "Independently reimplement the construction and formalize the elementary counting and coset-size arguments in a proof assistant." ], "mathObjects": [], "relatedWorks": [ { "citation": "American Institute of Mathematics. (2004). Palo Alto Problems, Problem 1.18 (V. Lev).", "url": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "doi": null }, { "citation": "Lev, V. F. Popular differences problem page.", "url": "https://math.haifa.ac.il/seva/Problems/difset.html", "doi": null }, { "citation": "Croot, E., & Lev, V. F. (2007). Open problems in additive combinatorics. In Additive Combinatorics, CRM Proceedings and Lecture Notes 43, 207–233.", "url": "https://math.haifa.ac.il/seva/Papers/Montpr.pdf", "doi": null }, { "citation": "Hou, J., Li, W., & Yang, K. (2026). Popular Differences and the Croot–Lev Half-Threshold Problem. arXiv:2606.29297v1.", "url": "https://arxiv.org/abs/2606.29297", "doi": "10.48550/arXiv.2606.29297" } ], "claimAssurance": null, "operatingModel": { "version": "1.0", "workId": "ep-work:odd-order-one-third-popular-differences", "attemptIds": [ "ep-attempt:odd-order-one-third-popular-differences-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 explicit eight-point C21 witness and an elementary three-layer family that refute the strict one-third implication and obstruct every fixed threshold below 2/5.", "scope": "The exact C21 witness, the H times C_q three-layer counting identity, and the odd-cyclic below-2/5 corollary; not the endpoint 2/5, a sharp-threshold theorem, witness minimality, exhaustive novelty, or priority." }, "bottleneckTargeted": [ "discovery", "assurance", "publication" ], "semanticBridge": { "state": "explicit", "description": "The package maps AIM Problem 1.18's pointwise representation hypothesis directly to a quotient-layer construction, derives every multiplicity by elementary counting, and separates the finite counterexample, asymptotic lower obstruction, and unresolved endpoint.", "remainingRisks": [ "The source interpretation and proof have not been reconstructed by an unaffiliated additive-combinatorics specialist.", "The targeted novelty search may miss differently phrased, unindexed, or informal prior work.", "The construction approaches 2/5 only from below and supplies no information proving sufficiency at the endpoint or above.", "Finite enumeration and two producer-authored encodings do not establish independent reproduction." ] }, "humanJudgmentGates": [ "Check the AIM source wording and keep the one-third question distinct from the published Croot–Lev half-threshold problem.", "Check the three quotient-layer multiplicity count and the coset-cardinality obstruction as mathematics rather than inferring truth from replay.", "Keep a lower obstruction below 2/5 separate from sufficiency or sharpness at 2/5.", "Keep a found C21 witness separate from a minimality claim.", "Require external specialist search and assessment before any first, novelty, priority, solved-problem, or field-acceptance language." ], "parentLinks": [ { "relation": "depends-on-claim", "workId": null, "legacyReleaseSlug": null, "externalUrl": "https://aimath.org/WWN/additivecomb/additivecomb.pdf", "inheritedClaim": "AIM Problem 1.18 asks whether the strict one-third pointwise popularity hypothesis forces the difference set to be a subgroup or a union of three cosets.", "inheritedAssuranceCeiling": "The external problem statement fixes the target and attribution; it does not validate this candidate's construction, proof, novelty, or priority." } ], "assuranceTarget": { "dimensions": [ "independentRerun", "independentReimplementation", "formalVerification", "specialistReview", "dataEnvironmentReproducibility", "semanticValidation", "noveltyAssessment", "priorityAssessment" ], "nextAction": "Obtain an unaffiliated additive-combinatorics reconstruction and literature assessment, then investigate the endpoint 2/5 and the smallest odd cyclic counterexample without promoting the present claim ceiling.", "claimCeiling": "An anonymous unrefereed counterexample and elementary-family candidate with exact producer-side replay, adversarial controls, public immutable assets, and internal PASS_WITH_NOTES; not a proof of 2/5 sufficiency or sharpness, C21 minimality, independent validation, formal verification, external specialist or editorial peer review, exhaustive novelty, absolute priority, or demonstrated workflow impact." }, "impactClaims": [ { "id": "odd-order-popular-differences-science-no-impact", "aim": "science", "outcome": "Faster or more reliable resolution and assurance of additive-combinatorics open problems", "setting": "AI-assisted counterexample search, exact replay, proof compression, source audit, 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:odd-order-one-third-popular-differences-publication", "status": "published", "resultClass": "positive", "metrics": { "schemaVersion": "1.0", "measurementScope": "publication-only", "scopeBoundary": "From prospective registration immediately before release-specific Evidence Press page and media authoring through the final canonical HTML, Markdown, paper.json, work-ledger and protocol readback. Research, review, public-repository creation, GitHub CI and Zenodo publication before registration are excluded.", "forecast": { "frozenAt": "2026-08-29T12:18:49Z", "procedureClass": "existing-public-candidate-two-cycle-evidence-press-publication", "targetOutcome": "Publish the immutable candidate on Evidence Press with complete reader-first metadata and communication assets, passed composite CI and accessibility, exact two-cycle protocol sealing, canonical public readback, and terminal linked metrics without changing the research claim ceiling.", "expectedActiveMinutes": 90, "plausibleLowMinutes": 60, "plausibleHighMinutes": 150, "expectedUnattendedWaitMinutes": 45, "referenceClass": { "label": "Existing-public-candidate Evidence Press two-cycle releases", "sampleSize": 0, "basis": "No fully instrumented post-policy attempt exists in this exact class; the four-part Fermi decomposition is the prior, with earlier closeout notes used only as qualitative route anchors." }, "fermiComponents": [ { "component": "release page, operating metadata and source binding", "count": 1, "lowMinutesPerUnit": 15, "centralMinutesPerUnit": 25, "highMinutesPerUnit": 45, "basis": "One short exact mathematical candidate with established public identifiers and a new prospective operating-model record." }, { "component": "art, Open Graph, audio and thumbnail generation plus inspection", "count": 1, "lowMinutesPerUnit": 15, "centralMinutesPerUnit": 25, "highMinutesPerUnit": 40, "basis": "One standard deterministic motif, one two-minute narration and one reproducible thumbnail." }, { "component": "first composite seal, CI and guarded deployment", "count": 1, "lowMinutesPerUnit": 15, "centralMinutesPerUnit": 20, "highMinutesPerUnit": 35, "basis": "One A/B content and protocol pair followed by one PR, required CI and a guarded deployment." }, { "component": "live-slug record, second seal, redeploy and canonical readback", "count": 1, "lowMinutesPerUnit": 15, "centralMinutesPerUnit": 20, "highMinutesPerUnit": 30, "basis": "One C/D ledger pair and one final exact browser and machine-surface readback." } ], "probabilityPositiveSignal": 0.95, "probabilityTargetClosure": 0.85, "probabilityHorizonMinutes": 150, "assumptions": [ "The current Evidence Press main branch remains safely reconcilable with this isolated release branch.", "OpenAI fable TTS, GitHub CI and zero-cost Cloudflare Pages authentication remain available.", "The current renderer and operating-model schema accept the new release without requiring a scientific claim change.", "The immutable GitHub and Zenodo research assets remain unchanged and publicly retrievable." ], "reforecastTriggers": [ "The site build exposes a schema or preservation conflict not attributable to the new release metadata.", "OpenAI TTS is unavailable or the generated briefing fails full-file intelligibility review.", "Required CI, accessibility, guarded deployment or canonical byte readback fails.", "Origin main advances with overlapping release or protocol changes that require reconciliation." ], "stopRule": "At 90 active agent minutes, consolidate to the load-bearing gates and report variance. Stop the positive publication route on any preservation, identity, media provenance, CI, accessibility or exact-readback failure; at 150 active minutes, pause rather than weakening a gate." }, "outcome": { "measuredAt": "2026-08-29T12:50:12Z", "calendarElapsedMinutes": 31, "activeAgentMinutes": 27, "activeHumanMinutes": null, "computeMinutes": 0, "unattendedWaitMinutes": 4, "blockedMinutes": 0, "reworkMinutes": 0, "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 existing immutable PASS_WITH_NOTES candidate reached canonical Evidence Press publication with reader-first media, complete composite checks, a merged two-commit content/protocol seal, exact preservation and first-deployment live readback. The publication attempt performed no new mathematical research and did not raise the research assurance ceiling.", "positiveSignalObserved": true, "targetReached": true, "forecastErrorMinutes": -63, "forecastRatio": 0.3, "withinForecastInterval": false, "varianceReason": "The short candidate reused established deterministic Evidence Press tooling and already immutable GitHub and Zenodo identities; the site build, media checks, CI and guarded deployment all passed without a scientific repair. The first-of-class 60--150 active-minute interval was therefore conservative.", "missingFields": [ { "field": "activeHumanMinutes", "reason": "No instrument captured human direction or review time at the publication-attempt boundary." }, { "field": "deduplicatedModelTokens", "reason": "No task-local model-token counter was active for the publication-only attempt, so a scoped value cannot be separated without reconstruction. The earlier research goal and auxiliary transcription-QA counters remain separately reported." }, { "field": "uncachedInputTokens", "reason": "The runtime does not expose an uncached-input counter." } ] } }, "missingnessReason": null } ] } }