{ "schemaVersion": "1.6", "slug": "waiting-time-entropy-bound-counterexamples", "title": "Counterexamples to a conjectured waiting-time bound on entropy production in periodically driven Markov networks", "shortTitle": "Waiting times can fake dissipation", "url": "https://evidencepress.org/releases/waiting-time-entropy-bound-counterexamples/", "oneLine": "A forward-only waiting-time statistic conjectured to be a lower bound on the entropy production of periodically driven systems can report dissipation where there is none, and can overstate it by more than 31 per cent (certified) in the very model class where the conjecture was tested.", "abstract": "Entropy production measures how far a small system, such as a molecular motor, a single molecule pulled in an experiment or a colloid in a driven optical trap, is from equilibrium. When only some transitions are visible, it is estimated from the visible jumps and the waiting times between them. For periodically driven Markov networks, Maier, Degünther, van der Meer and Seifert (Journal of Statistical Physics 191, 104, 2024) conjectured that a statistic computed from the forward experiment alone, with the phase of the drive averaged out, never exceeds the true entropy production. This candidate shows that the conjecture is false. In a three-state network with one observed edge, two hidden barriers modulated out of phase keep the system in equilibrium at every instant, so the entropy production is exactly zero, while the statistic is strictly positive (analytic proof; about 1e-5 per unit time). In the class with fixed kinetic prefactors, where the conjecture was tested numerically, small-amplitude driving provably obeys the bound, but slow large-amplitude driving violates it: interval arithmetic certifies a ratio of at least 1.3179 for a switching protocol, smooth counterexamples exist, and smooth protocols reach about 2 numerically. The statistic is nonnegative and, at equilibrium, measures how distinguishable the protocol is from its time reverse. For a fixed network and fastest rate its excess grows at most like the logarithm of one over the slowest rate, and that growth is attained. Estimators that use the reversed protocol remain valid. AI-assisted, unrefereed candidate; no external peer review, independent reproduction or formal verification is claimed.", "datePublished": "2026-10-21", "dateModified": "2026-10-21", "version": "1.0.0-candidate", "doi": "10.5281/zenodo.23541298", "doiUrl": "https://doi.org/10.5281/zenodo.23541298", "conceptDoi": null, "pdfUrl": "https://github.com/ipitchford/waiting-time-entropy-bound-counterexamples/releases/download/v1.0.0-candidate/waiting-time-entropy-bound-counterexamples-1.0.0-candidate.pdf", "altPdfUrl": null, "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/23541298", "repoUrl": "https://github.com/ipitchford/waiting-time-entropy-bound-counterexamples", "releaseUrl": "https://github.com/ipitchford/waiting-time-entropy-bound-counterexamples/releases/tag/v1.0.0-candidate", "markdownUrl": "https://evidencepress.org/releases/waiting-time-entropy-bound-counterexamples/index.md", "bibtexUrl": "https://evidencepress.org/releases/waiting-time-entropy-bound-counterexamples/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/waiting-time-entropy-bound-counterexamples.mp3", "imageUrl": "https://evidencepress.org/assets/og/waiting-time-entropy-bound-counterexamples.png?v=982c69f61d", "coverArtUrl": "https://evidencepress.org/assets/art/waiting-time-entropy-bound-counterexamples.svg?v=ba72ae67ef", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/waiting-time-entropy-bound-counterexamples.mp3", "name": "Plain-English audio briefing", "description": "Communication aid; not additional research evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/waiting-time-entropy-bound-counterexamples.txt" } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Unrefereed AI-assisted mathematical candidate. Every theorem has a written proof; one violation in the fixed-prefactor class is certified by interval arithmetic; numerical and exploratory results are labelled. Internal AI-assisted and adversarial review only. External specialist review, independent reproduction and formal verification have not been established." }, "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/23541298", "note": "The versioned research package and manuscript are publicly archived on GitHub and Zenodo. Provider asset parity was checked against frozen local files; availability does not establish correctness or peer review." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "verify_all.py regenerates every saved JSON output byte for byte. A separate clean replay of the full suite reproduced all outputs. The frozen archive's code and results are byte-identical to that replay copy except five files changed only in comments, docstrings and one output key; the affected fast checks and all exact and certified checks were rerun on the frozen archive before release." }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed", "note": "No unaffiliated rerun has been established." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "partial", "note": "Key numbers were cross-checked by two numerical methods (Magnus stepping and DOP853) and exact rational recurrences inside the producer workflow; this is implementation diversity, not external reproduction." }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed", "note": "No proof-assistant formalisation. The switching-protocol certificate uses Arb ball arithmetic, which is rigorous numerics, not formal verification." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "AI-assisted internal and adversarial review is not external 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 external editorial peer review." }, { "dimension": "dataEnvironmentReproducibility", "label": "Data and environment reproducibility", "question": "Are data and computational environment pinned well enough to rebuild?", "state": "partial", "note": "Checks need Python 3.13 with NumPy, SciPy, Matplotlib and python-flint; pinned versions are recorded but not locked in a container. The full suite takes about 15–25 minutes." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "AI-assisted mathematical research", "AI-assisted internal review, verification re-review and adversarial review (including OpenAI gpt-5.6-sol)", "Claude Code: publication packaging and release" ], "humanRole": "Publication authority only; no human research contribution inferred.", "reviewStatus": "Internal AI-assisted and adversarial review, not external peer review." }, "attributionBoundary": null, "problem": null, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "canonical-theorem", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "Prepared from the approved revised manuscript and its exact check outputs. Versioned GitHub and Zenodo research archives are checked against the frozen release assets on 21 October 2026.", "grandfatheredAtSchemaVersion": null, "keywords": [ "stochastic thermodynamics", "entropy production", "thermodynamic inference", "waiting-time distributions", "periodic driving", "Markov jump processes", "counterexample", "molecular motors" ], "keyResults": [ "Zero-dissipation counterexample (Theorem 3.1, proof with exact sign certificate): in a three-state network observed only on the edge 1–2, hidden barriers 1 + ε cos ωt and 1 + ε sin ωt keep every rate equal to its reverse, so σ = 0, but the forward-only phase-averaged statistic S of Maier et al. is strictly positive (S ≈ 1.13×10⁻⁵ per unit time at ε = 1/2, ω = 1). The upper inequality of their conjecture fails.", "Fixed-prefactor class, where the conjecture was tested: small-amplitude driving obeys S < σ (Theorem 4.1, proof); a 128-piece switching protocol with A = 4, ω = 1/20 has S ≥ 0.0036867 > σ = 0.0027973, a ratio of at least 1.3179 (Theorem 4.2, certified by interval arithmetic); smooth counterexamples exist (Corollary 4.3); smooth circular protocols reach S/σ ≈ 2.03 (numerical).", "What S measures (proofs): S = D(Q‖Q^R) − D(r‖r̄) ≥ 0; at equilibrium the reversed-protocol estimator is zero and S = D(Q‖Q̃) measures how distinguishable the protocol is from its time reverse. Time-symmetric driving gives S = 0 at equilibrium (proof); with a cycle affinity a numerical witness has S ≈ 1.10 σ.", "Size of the error (proofs): S ≤ K[(m+1) log(k_max/k_min) + log m! + m d] + R_max for every network, so at fixed graph and fastest rate the excess grows at most like log(1/k_min); a pulse family with σ = 0 attains that growth. No logarithmic claim is made as k_max grows.", "Safe alternatives: the reversed-protocol estimator of Maier et al. and its coarse-grainings, their σ_pk estimator, and the corrected inequality of Harunari, Fiore and Barato remain valid." ], "reviews": [], "evidencePackage": "Written proofs of every theorem; an exact rational Taylor and sign certificate for the zero-dissipation counterexample; an Arb ball-arithmetic certificate (160 bits) for the fixed-prefactor switching protocol; exact symbolic coefficients for the single-knob and zero-area results; labelled deterministic numerics with convergence evidence and an exploratory finite-sample simulation; every saved output regenerated byte for byte by verify_all.py.", "openProblems": [], "mathObjects": [], "relatedWorks": [ { "citation": "Maier, Degünther, van der Meer and Seifert (2024), Inferring kinetics and entropy production from observable transitions in partially accessible, periodically driven Markov networks. Journal of Statistical Physics 191, 104 (arXiv:2406.12657).", "url": "https://doi.org/10.1007/s10955-024-03315-7", "doi": "10.1007/s10955-024-03315-7" }, { "citation": "Harunari, Fiore and Barato (2024), Inference of entropy production for periodically driven systems. Physical Review E 110, 064126.", "url": "https://doi.org/10.1103/PhysRevE.110.064126", "doi": "10.1103/PhysRevE.110.064126" }, { "citation": "van der Meer, Ertel and Seifert (2022), Thermodynamic inference in partially accessible Markov networks: a unifying perspective from transition-based waiting time distributions. Physical Review X 12, 031025.", "url": "https://doi.org/10.1103/PhysRevX.12.031025", "doi": "10.1103/PhysRevX.12.031025" }, { "citation": "Harunari, Dutta, Polettini and Roldán (2022), What to learn from a few visible transitions' statistics? Physical Review X 12, 041026.", "url": "https://doi.org/10.1103/PhysRevX.12.041026", "doi": "10.1103/PhysRevX.12.041026" }, { "citation": "Skinner and Dunkel (2021), Estimating entropy production from waiting time distributions. Physical Review Letters 127, 198101.", "url": "https://doi.org/10.1103/PhysRevLett.127.198101", "doi": "10.1103/PhysRevLett.127.198101" }, { "citation": "Martínez, Bisker, Horowitz and Parrondo (2019), Inferring broken detailed balance in the absence of observable currents. Nature Communications 10, 3542.", "url": "https://doi.org/10.1038/s41467-019-11051-w", "doi": "10.1038/s41467-019-11051-w" }, { "citation": "Rahav, Horowitz and Jarzynski (2008), Directed flow in nonadiabatic stochastic pumps. Physical Review Letters 101, 140602.", "url": "https://doi.org/10.1103/PhysRevLett.101.140602", "doi": "10.1103/PhysRevLett.101.140602" }, { "citation": "Gu (2026), Counterexamples to the conjectured ordering between the waiting-time bound and the thermodynamic uncertainty bound on entropy production. arXiv:2601.04039.", "url": "https://arxiv.org/abs/2601.04039", "doi": "10.48550/arXiv.2601.04039" } ], "claimAssurance": null, "operatingModel": { "version": "1.0", "workId": "ep-work:waiting-time-entropy-bound-counterexamples", "attemptIds": [ "ep-attempt:waiting-time-entropy-bound-counterexamples-publication" ], "aims": [ "science" ], "artifactRoles": [ "research-output", "communication" ], "lineageId": null, "accelerationPrimitives": [ "counterexample-proxy-first", "adversarial-controls", "assurance-vector", "agent-readable-research-object" ], "decisionObject": { "type": "other", "description": "A refutation of the upper inequality in the Maier et al. (2024) conjecture for the forward-only phase-averaged waiting-time statistic, with a certified violation in the fixed-prefactor class, a logarithmic error bound and a map of which estimators remain safe.", "scope": "Finite continuous-time Markov jump processes with periodic rates and a symmetric set of observed transitions, as in Maier et al.; mathematical results about model networks, not an experimental claim." }, "bottleneckTargeted": [ "assurance", "publication" ], "semanticBridge": { "state": "explicit", "description": "The statistic and the conjecture are reproduced from arXiv:2406.12657v1 Eqs. (1), (3), (5), (13)–(16) and App. A.2, with the journal numbering (Eqs. (17)–(18)) recorded in the package's source manifest; the fixed-prefactor class is the one used in the source's numerical tests.", "remainingRisks": [ "The journal version of record was compared through its equation numbering and source manifest; the arXiv v1 text is the quoted source.", "The pulse-family constant and most magnitudes are numerical, not certified.", "The bounded literature check is not exhaustive priority certification." ] }, "humanJudgmentGates": [ "Publication is authorised; scientific endorsement is not inferred.", "External specialist review and independent reproduction remain open." ], "parentLinks": [], "assuranceTarget": { "dimensions": [ "independentRerun", "specialistReview", "formalVerification", "priorityAssessment" ], "nextAction": "Have a specialist in stochastic thermodynamics check the reduction of the statistic to the repeat densities F and G (Section 3) and the fixed-prefactor switching certificate (Theorem 4.2).", "claimCeiling": "The upper inequality of the Maier et al. conjecture is false, including in the fixed-prefactor class (one certified witness); stated proofs and bounds hold for the model class defined in the paper. 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