{ "schemaVersion": "1.6", "slug": "extremes-do-not-always-maximise", "title": "Extremes do not always maximise: a minimal monotone reversible counterexample", "shortTitle": "Extremes do not always maximise", "url": "https://evidencepress.org/releases/extremes-do-not-always-maximise/", "oneLine": "A four-state monotone reversible heat-bath chain is furthest apart at two incomparable states—not its minimum and maximum—at every positive time.", "abstract": "For a finite monotone reversible Markov chain with a unique minimum and maximum state, must those two extremes maximise the total-variation distance between time-t laws? This anonymous, unrefereed candidate answers the general question negatively with a cardinality-minimal four-state example. On the Boolean diamond, random-scan single-site heat-bath dynamics with Gibbs weights (1,1/2,2,1) is irreducible, aperiodic, reversible and monotone. For every integer t at least one, the incomparable pair has row total variation (2/3)(1/2)^(t-1), while the extremal and all adjacent pairs have distance (1/2)^t. The candidate also classifies the endpoint-symmetric family (1,b,c,1) under bc at most one and proves that no poset with unique bottom and top and at most three states can furnish such a failure. Exact producer replay, negative controls, a separate producer encoding, public multi-version CI, fresh-extraction checks and an internal five-role review pass. The separate zero-field Ising question on every graph remains open, and novelty, priority, independent reconstruction, formal verification, external specialist review and peer review are not established.", "datePublished": "2026-08-29", "dateModified": "2026-08-29", "version": "0.2.0-candidate", "doi": "10.5281/zenodo.22163995", "doiUrl": "https://doi.org/10.5281/zenodo.22163995", "conceptDoi": "10.5281/zenodo.22163994", "pdfUrl": "https://github.com/ipitchford/extremes-do-not-always-maximise/releases/download/v0.2.0-candidate/extremes-do-not-always-maximise-v0.2.0-candidate.pdf", "altPdfUrl": "https://zenodo.org/records/22163995/files/extremes-do-not-always-maximise-v0.2.0-candidate.pdf?download=1", "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/22163995", "repoUrl": "https://github.com/ipitchford/extremes-do-not-always-maximise", "releaseUrl": "https://github.com/ipitchford/extremes-do-not-always-maximise/releases/tag/v0.2.0-candidate", "markdownUrl": "https://evidencepress.org/releases/extremes-do-not-always-maximise/index.md", "bibtexUrl": "https://evidencepress.org/releases/extremes-do-not-always-maximise/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/extremes-do-not-always-maximise.mp3?v=c32dc0db2a", "imageUrl": "https://evidencepress.org/assets/og/extremes-do-not-always-maximise.png?v=aa0541d2e4", "coverArtUrl": "https://evidencepress.org/assets/art/extremes-do-not-always-maximise.svg?v=b3bdc20d4b", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/extremes-do-not-always-maximise.mp3", "name": "Audio briefing — extremes do not always maximise", "description": "Plain-English OpenAI API synthetic-voice account of the four-state counterexample, all-time gap, minimality and unresolved zero-field boundary; a communication aid, not additional mathematical evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/extremes-do-not-always-maximise.txt" } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Anonymous, unrefereed theorem candidate at internal PASS_WITH_NOTES. The universal counterexample, family classification and cardinality-minimality claims rest on the written proofs; the programs replay exact formulas and stated finite predicates. The closest 1992 coefficient-of-ergodicity paper could not be inspected in full, so the bounded prior-art search does not establish novelty or priority. No unaffiliated rerun, independent reimplementation, proof-assistant formalisation, external specialist review or editorial peer review has occurred. The zero-field Ising question on every graph and AIM diagnostics 1.6(2) and 1.6(3) 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/22163995", "note": "The tagged GitHub prerelease and published Zenodo record expose the six-page PDF, complete 45-file replay ZIP and checksum ledger; public downloads match the local reviewed assets byte for byte." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "The DOI-bearing tagged commit passed exact family verification, a separate producer cross-check, seven negative controls and the exact bounded zero-field enumeration on Python 3.11 and 3.13, each in ordinary and optimized mode.", "evidenceUrl": "https://github.com/ipitchford/extremes-do-not-always-maximise/actions/runs/33263365120" }, { "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 second exact encoding was produced within the same coordinated research process and is not an independent reconstruction." }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed", "note": "No proof assistant formalises the all-time formulas, monotonicity argument, family classification or three-state minimality theorem." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "The five-role review and exact-byte confirmation were internal AI editorial quality control, not unaffiliated probability or Markov-chain 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 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/extremes-do-not-always-maximise/blob/v0.2.0-candidate/ENVIRONMENT.txt", "note": "The exact verifiers use the Python 3.11-or-later standard library and public CI spans Python 3.11 and 3.13; the deterministic PDF toolchain is recorded, but no unaffiliated clean-environment reconstruction is reported." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "AI systems under human direction", "OpenAI Codex", "Python 3 standard library" ], "humanRole": "Problem selection, research direction, source selection, claim-boundary decisions, publication authorisation 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. A separate producer encoding, cross-model review and public CI are not independent reconstruction, external specialist review or editorial peer review." }, "attributionBoundary": null, "problem": { "name": "AIM Problem 1.6(1): extremal starts in monotone reversible chains", "url": "http://aimpl.org/markovmixing/1/" }, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "negative-result", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "Authored prospectively for reader-first schema 2.0 against the confirmed scientific bytes, immutable GitHub tag, public CI, published Zenodo identities and producer-organised internal PASS_WITH_NOTES record.", "grandfatheredAtSchemaVersion": null, "keywords": [ "Markov chains", "monotone Markov chain", "reversible Markov chain", "total variation", "Glauber dynamics", "heat-bath dynamics", "coefficient of ergodicity", "counterexample", "exact computation", "unrefereed candidate" ], "keyResults": [ "For Gibbs weights (1,1/2,2,1) on the Boolean diamond, random-scan single-site heat-bath dynamics is irreducible, aperiodic, reversible and monotone.", "For every integer t at least one, the incomparable pair (10,01) uniquely maximises row total variation at (2/3)(1/2)^(t-1); the extremal and four adjacent pairs all have distance (1/2)^t.", "For endpoint-symmetric weights (1,b,c,1) with b,c positive and bc at most one, the candidate gives exact all-time formulas for extremal, incomparable and adjacent row distances and classifies when the maximiser switches.", "No monotone chain on a poset with unique minimum and maximum and at most three states can violate extremal row-TV maximisation; reversibility is not needed for this cardinality-minimality argument.", "An exact bounded screen found no zero-field Ising failure among all 43 connected labelled simple graphs on two through four vertices over five stated temperatures and times one through six; this is not an all-graph theorem." ], "reviews": [], "evidencePackage": "A six-page candidate manuscript and accessible Markdown; exact rational verification of the Gibbs-to-kernel construction, stochasticity, irreducibility, aperiodicity, detailed balance, monotonicity and all-time formulas; a separately structured producer cross-check; seven targeted negative controls; exact bounded zero-field enumeration over 43 connected labelled simple graphs, 215 graph-temperature trajectories, 1,290 kernel powers and 140,340 unordered row-pair comparisons; a deterministic two-pass PDF build; an exact 45-file manifest; fresh-extraction passage of 18 release gates; source-correspondence and bounded novelty reports; five producer-organised review roles plus exact-byte confirmation; and public GitHub Actions passage on Python 3.11 and 3.13 in ordinary and optimized mode. These establish public availability, integrity and producer-side replay, not independent mathematical validation, exhaustive novelty or peer review.", "openProblems": [ "Determine whether the extremal pair always realises row-TV diameter for zero-field ferromagnetic Ising heat-bath dynamics on every finite graph.", "Obtain an unaffiliated Markov-chain specialist reconstruction of the four-state proof, family classification, source mapping and cardinality-minimality argument.", "Inspect Pflug and Schachermayer's 1992 coefficient-of-ergodicity article lawfully in full and conduct a broader specialist novelty and priority audit under alternative terminology.", "Formalise the Boolean-diamond kernel, stochastic monotonicity, exact powers and three-state CDF argument in a proof assistant.", "Address AIM Problem 1.6 diagnostics (2) and (3), which are not answered by this counterexample." ], "mathObjects": [ { "id": "four-state-row-tv-counterexample", "kind": "counterexample", "label": "Four-state all-time row-TV counterexample", "plainText": "For weights (1,1/2,2,1), the incomparable row distance is (2/3)(1/2)^(t-1) and the extremal distance is (1/2)^t for every integer t at least 1.", "latex": "d_{\\mathrm{TV}}(P^t(10,\\cdot),P^t(01,\\cdot))=\\frac23\\left(\\frac12\\right)^{t-1}>\\left(\\frac12\\right)^t=d_{\\mathrm{TV}}(P^t(00,\\cdot),P^t(11,\\cdot))", "status": "claimed-result", "scope": "The specified random-scan two-spin heat-bath chain; the scholarly object is an anonymous unrefereed candidate with producer-side replay, not an independently validated theorem." } ], "relatedWorks": [ { "citation": "AIM Problem Lists. Markov chain mixing times, Problem 1.6.", "url": "http://aimpl.org/markovmixing/1/", "doi": null }, { "citation": "Peres, Y., & Winkler, P. (2013). Can extra updates delay mixing? Communications in Mathematical Physics, 323, 1007-1016.", "url": "https://arxiv.org/abs/1112.0603", "doi": "10.48550/arXiv.1112.0603" }, { "citation": "Fill, J. A., & Kahn, J. (2013). Comparison inequalities and fastest-mixing Markov chains. Annals of Applied Probability, 23, 1778-1816.", "url": "https://doi.org/10.1214/12-AAP886", "doi": "10.1214/12-AAP886" }, { "citation": "Pflug, G. C., & Schachermayer, W. (1992). A remark on the coefficient of ergodicity. Probability Theory and Related Fields, 93, 283-293.", "url": "https://doi.org/10.2307/3214717", "doi": "10.2307/3214717" }, { "citation": "Qin, Q. (2025). Analysis of two-component Gibbs samplers using the theory of two projections. Annals of Applied Probability, 35, 891-921.", "url": "https://doi.org/10.1214/24-AAP2066", "doi": "10.1214/24-AAP2066" } ], "claimAssurance": null, "operatingModel": { "version": "1.0", "workId": "ep-work:extremes-do-not-always-maximise", "attemptIds": [ "ep-attempt:extremes-do-not-always-maximise-assurance-publication" ], "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": "counterexample", "description": "A cardinality-minimal four-state monotone reversible heat-bath counterexample, together with an exact endpoint-symmetric two-spin family classification.", "scope": "The separately quantified general monotone-reversible clause of AIM problem 1.6(1); not the all-graph zero-field Ising question or diagnostics 1.6(2-3)." }, "bottleneckTargeted": [ "assurance", "publication" ], "semanticBridge": { "state": "explicit", "description": "The AIM general clause is mapped to the Boolean-diamond state order, random-scan heat-bath kernel, total-variation row diameter and exact all-time formulas; the zero-field clause is separately excluded.", "remainingRisks": [ "The closest 1992 coefficient-of-ergodicity article could not be inspected beyond its publisher abstract, bibliography and cited-by records.", "No unaffiliated specialist has assessed the proof or source map.", "The all-graph zero-field Ising question remains open." ] }, "humanJudgmentGates": [ "Assess the universal written proofs rather than inferring them from finite replay.", "Assess whether the accessible-source novelty audit supports release at the bounded unrefereed-candidate ceiling.", "Do not promote the bounded exact zero-field enumeration into an all-graph theorem.", "Require a new release decision before claiming independent reproduction, formal verification, specialist review, peer review, novelty or priority." ], "parentLinks": [], "assuranceTarget": { "dimensions": [ "independentRerun", "independentReimplementation", "formalVerification", "specialistReview", "semanticValidation", "noveltyAssessment", "priorityAssessment" ], "nextAction": "Obtain unaffiliated specialist review and lawful full-text comparison with Pflug and Schachermayer before any novelty or priority upgrade.", "claimCeiling": "Unrefereed theorem candidate with producer-side exact replay, a separate producer cross-check and internal editorial review; not an all-graph zero-field result, independent reproduction, formal proof, specialist review, external peer review, novelty or priority claim." }, "impactClaims": [ { "id": "extremes-science-no-impact", "aim": "science", "outcome": "More reliable evaluation of extremal-start row-diameter claims", "setting": "Finite monotone reversible Markov chains and two-spin heat-bath dynamics", "status": "NO_IMPACT_EVIDENCE", "designClass": "none", "comparator": "No matched conventional research or publication workflow was prospectively registered.", "estimand": "No effect on discovery time, error rate, review effort, theorem production, adoption or citation was estimated.", "evidenceRefs": [], "registeredDesignRef": null, "independentAssessment": null } ], "ibeHypotheses": [ "research-cycle-acceleration" ] }, "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:extremes-do-not-always-maximise-assurance-publication", "status": "published", "resultClass": "positive", "metrics": { "schemaVersion": "1.0", "measurementScope": "assurance-through-publication", "scopeBoundary": "From prospective registration immediately before actioning the Major Revision report through terminal release or stop. Earlier candidate discovery, search computation, manuscript drafting and the supplied review are excluded; this attempt covers evidence repair, novelty resolution, internal editorial review, public repository and DOI identity, Evidence Press authoring, media, CI, guarded deployment and canonical readback.", "forecast": { "frozenAt": "2026-08-29T15:23:42Z", "procedureClass": "major-review-repair-through-full-candidate-publication", "targetOutcome": "Reach internal Accept or PASS_WITH_NOTES on a source-faithful repaired candidate, then publish byte-identical immutable GitHub and Zenodo assets plus a guarded Evidence Press page whose canonical readback preserves the unrefereed claim ceiling.", "expectedActiveMinutes": 150, "plausibleLowMinutes": 90, "plausibleHighMinutes": 240, "expectedUnattendedWaitMinutes": 60, "referenceClass": { "label": "Major-review repair through full Evidence Press candidate publication", "sampleSize": 0, "basis": "No completed prospectively measured attempt in this exact class exists. The decomposition uses the current package size, one supplied major-revision report, the mandatory five-role gate and prior Evidence Press identity and two-cycle deployment mechanics only as qualitative anchors." }, "fermiComponents": [ { "component": "review triage and closest-prior-art resolution", "count": 1, "lowMinutesPerUnit": 30, "centralMinutesPerUnit": 45, "highMinutesPerUnit": 75, "basis": "One pivotal 1992 full-text dependency, focused citation descendants and a finite response matrix." }, { "component": "package repair, rebuild, deterministic baseline and editorial gate", "count": 1, "lowMinutesPerUnit": 20, "centralMinutesPerUnit": 35, "highMinutesPerUnit": 60, "basis": "One six-page paper, a small exact verifier, targeted controls and one five-role round with at most one confirmation." }, { "component": "public research repository, CI, immutable release and Zenodo identity", "count": 1, "lowMinutesPerUnit": 15, "centralMinutesPerUnit": 25, "highMinutesPerUnit": 40, "basis": "One compact anonymous candidate with standing component licences and standard GitHub/Zenodo coupling." }, { "component": "Evidence Press authoring, media, composite seals, two deployments and readback", "count": 1, "lowMinutesPerUnit": 25, "centralMinutesPerUnit": 45, "highMinutesPerUnit": 65, "basis": "One new reader-first release using deterministic house art, audio, thumbnail, A/B then C/D protocol seals and existing guarded deployment tooling." } ], "probabilityPositiveSignal": 0.85, "probabilityTargetClosure": 0.7, "probabilityHorizonMinutes": 240, "assumptions": [ "The closest 1992 antecedent does not contain an equivalent four-state construction and family classification that removes the residual contribution.", "The supplied theorem proofs and exact verifier survive the bounded repair and mandatory internal editorial challenge.", "GitHub, Zenodo, OpenAI fable TTS and zero-cost Cloudflare authentication can be made available without weakening a gate or exposing secrets.", "The current Evidence Press operating-model, metrics and preservation contracts admit the new bounded counterexample record." ], "reforecastTriggers": [ "Full-text or citation-descendant review finds an equivalent antecedent requiring material reframing.", "A repaired verifier, targeted negative control or five-role review identifies a load-bearing mathematical or semantic defect.", "GitHub, Zenodo, CI, audio, Cloudflare or canonical readback remains unavailable after bounded recovery.", "Origin main advances with an overlapping operating-model, ledger or release change requiring reconciliation." ], "stopRule": "Stop the positive release route immediately on a fatal proof, semantic, novelty, licence, preservation or review finding. At 150 active agent minutes consolidate to load-bearing gates and report variance; at 240 active minutes pause rather than weakening a gate unless the user explicitly authorises continuation after seeing the exact blocker." }, "outcome": { "measuredAt": "2026-08-29T17:04:45Z", "calendarElapsedMinutes": 101, "activeAgentMinutes": 95, "activeHumanMinutes": null, "computeMinutes": 0, "unattendedWaitMinutes": 6, "blockedMinutes": 0, "reworkMinutes": 0, "agentRuns": 1, "maxParallelAgents": 1, "modelTurns": 3, "deduplicatedModelTokens": null, "uncachedInputTokens": null, "researchCycles": { "positive": 0, "negative": 0, "inconclusive": 0 }, "falsificationGatesRun": 7, "candidateArchitecturesTested": 0, "candidateArchitecturesRejected": 0, "substantiveReviewRounds": 2, "p0Findings": 0, "p1Findings": 3, "prepublicationClaimCorrections": 2, "resultState": "target-closed", "resultSummary": "The supplied Major Revision was actioned through a frozen five-role internal editorial round, exact-byte confirmation, producer-side replay and seven targeted falsification controls. The repaired candidate reached byte-identical immutable GitHub and Zenodo publication and canonical Evidence Press release with complete media, protocol, preservation and browser readback. The result remains an anonymous unrefereed candidate with no external assurance or impact promotion.", "positiveSignalObserved": true, "targetReached": true, "forecastErrorMinutes": -55, "forecastRatio": 0.63, "withinForecastInterval": true, "varianceReason": null, "missingFields": [ { "field": "activeHumanMinutes", "reason": "No instrument captured human direction or review time at the prospective attempt boundary." }, { "field": "deduplicatedModelTokens", "reason": "The supported runtime goal counter began at 2026-08-29T16:00:08Z, 36 minutes after attempt registration. Its partial reading is retained in the closeout receipt, but a full-attempt fork-aware value cannot be recovered without reconstruction." }, { "field": "uncachedInputTokens", "reason": "The runtime does not expose an uncached-input counter." } ] } }, "missingnessReason": null } ] } }