{ "schemaVersion": "1.6", "slug": "thermal-cumulant-sign-counterexamples", "title": "Thermal heat-current cumulant ratios can be arbitrarily negative: three-state counterexamples to two sign conjectures", "shortTitle": "Negative heat cumulants without quantum effects", "url": "https://evidencepress.org/releases/thermal-cumulant-sign-counterexamples/", "oneLine": "Two conjectured sign rules for heat-current fluctuations in classical thermal networks are false: three states with a rarely visited trap make the third and fourth cumulant ratios arbitrarily negative at fixed thermodynamic data, while variance-based bounds remain intact.", "abstract": "Ptaszyński (Physical Review E 106, 024119, 2022) conjectured, from random-network calculations, that the heat-current cumulant ratios C3/C1 and C4/C2 are nonnegative in classical Markov systems driven only by temperature differences, so that a negative value could certify non-thermal or non-classical driving. This candidate gives exact three-state counterexamples that obey bathwise local detailed balance with two baths. After a common rescaling of all rates, which leaves both ratios unchanged, a fully connected family keeps the energies, temperatures, stationary probabilities, mean heat current and entropy production fixed while both ratios tend to minus infinity like ε^-2. In the standard state-jump model with one counted bath, two states can never violate either sign. The mechanism is classical slow switching through a weakly coupled trap state, and the divergence tracks the variance: the variance-normalised ratios tend to -9/4 and -3/4 in that family, and the thermodynamic uncertainty relation is untouched. For positive mean and variance a kinetic lower bound in terms of the spectral separation of the stationary eigenvalue captures the ε^-2 rate. With six states even the normalised ratios are unbounded below. Violations need rate separations of order 10^2, which is why random sampling rarely finds them. AI-assisted, unrefereed candidate; no external peer review, independent reproduction or formal verification is claimed.", "datePublished": "2026-10-27", "dateModified": "2026-10-27", "version": "1.0.0-candidate", "doi": "10.5281/zenodo.23689958", "doiUrl": "https://doi.org/10.5281/zenodo.23689958", "conceptDoi": "10.5281/zenodo.23689957", "pdfUrl": "https://github.com/ipitchford/thermal-cumulant-sign-counterexamples/releases/download/v1.0.0-candidate/thermal-cumulant-sign-counterexamples-1.0.0-candidate.pdf", "altPdfUrl": null, "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/23689958", "repoUrl": "https://github.com/ipitchford/thermal-cumulant-sign-counterexamples", "releaseUrl": "https://github.com/ipitchford/thermal-cumulant-sign-counterexamples/releases/tag/v1.0.0-candidate", "markdownUrl": "https://evidencepress.org/releases/thermal-cumulant-sign-counterexamples/index.md", "bibtexUrl": "https://evidencepress.org/releases/thermal-cumulant-sign-counterexamples/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/thermal-cumulant-sign-counterexamples.mp3", "imageUrl": "https://evidencepress.org/assets/og/thermal-cumulant-sign-counterexamples.png?v=279e4b887a", "coverArtUrl": "https://evidencepress.org/assets/art/thermal-cumulant-sign-counterexamples.svg?v=917bf0d9a1", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/thermal-cumulant-sign-counterexamples.mp3", "name": "Plain-English audio briefing", "description": "Communication aid; not additional research evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/thermal-cumulant-sign-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 result has a written proof with exact computer checks; the random census is numerical evidence only. Internal AI-assisted review, verification re-review, a targeted proof audit and an outside-model correctness audit, whose findings were fixed by the editor without a further review round. 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/23689958", "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": "All six scripts were rerun in README order from a clean copy containing only the code and the uv lock file; every certificate, the figure data and the random census reproduced the shipped files byte for byte, and eleven semantic negative controls rejected corrupted rates, laws and coefficients. GitHub CI repeats the exact checks on Linux." }, { "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": "Three separately written exact verifiers and an internal proof audit with its own full-counting-statistics code recomputed the key coefficients; this is producer-side implementation diversity, not unaffiliated reimplementation." }, { "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 results are proved in prose and checked in exact arithmetic." }, { "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": "The exact certificates need only the Python standard library; the SymPy, figure and census scripts use versions pinned in uv.lock, not a locked container. The census takes about ten minutes." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "AI-assisted mathematical research", "AI-assisted internal review, re-review, proof audit and outside-model correctness audit", "Claude Code: final editor fixes, 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 final manuscript (version 2.0 with final editor fixes, 27 October 2026, 13 pages), its exact certificates and the frozen release archive. The versioned GitHub and Zenodo research archives were checked against the frozen release assets on 27 October 2026.", "grandfatheredAtSchemaVersion": null, "keywords": [ "stochastic thermodynamics", "heat current", "full counting statistics", "cumulants", "skewness and kurtosis", "Markov jump processes", "local detailed balance", "counterexample", "telegraph noise" ], "keyResults": [ "Counterexample in the conjectured class (Section 3, exact certificate): a three-state, two-bath network obeying local detailed balance has C3/C1 ≈ −6415 and C4/C2 ≈ −54732 at ε = 10⁻³; K < 0 for ε < 0.0667 and S < 0 for ε < 0.0069.", "Weak-trap mechanism (Theorem 4.3, proof): a trap reached through a weak edge gives C3/C1 and C4/C2 ∝ −ε⁻² exactly when (3 − √5)/2 < U/V < 1; every thermal two-state conductor with a trap at the energy of its lower state, joined by an unobserved-bath edge with equal in and out prefactors, lands in this region (Corollary 4.4).", "Fixed thermodynamic data (Theorem 5.1, proof and exact certificate): with distinct energies (0, 2, 1), all twelve bath-resolved rates positive, stationary law (40, 5, 18)/63, C1 = 40/63 and fixed entropy production, S = −16/(21609 ε²) + O(ε⁻¹) and K = −16/(64827 ε²) + O(ε⁻¹). No lower bound depending only on these data exists.", "What survives (Section 6, proofs): the divergence tracks the variance; the normalised ratios C3C1/C2² and C4C1²/C2³ tend to −9/4 and −3/4 in that family, and in the trap family the normalised kurtosis limit is at least −15/4 (attained). The thermodynamic uncertainty relation holds throughout. With six states both normalised ratios are unbounded below.", "Kinetic bound (Theorem 6.3, proof): |Cn| ≤ n!(g/2)(E_max/ℓ)ⁿ in terms of the spectral separation g; if C1 > 0 (resp. C2 > 0) this bounds S (resp. K) below with the correct ε⁻² rate, so very negative ratios require slow kinetics.", "Minimality (Proposition 7.1, proof): in the standard state-jump model with one counted bath, two states give C2 > 0, C4 > 0 and C3/C1 ≥ E²/4." ], "reviews": [], "evidencePackage": "Written proofs of every theorem and proposition; exact rational certificates (characteristic-polynomial series, rational Poisson recursion and an independent scaled-determinant audit, standard library only) for the three-state examples and the two-state identities; SymPy checks of the trap-family coefficients, normalised limits, six-state construction and kinetic bound; a seeded random census labelled numerical evidence only; eleven semantic negative controls; every saved output regenerated byte for byte from a clean copy.", "openProblems": [], "mathObjects": [], "relatedWorks": [ { "citation": "Ptaszyński (2022), Bounds on skewness and kurtosis of steady-state currents. Physical Review E 106, 024119. The source of the two sign conjectures (its equations (19) and (20)).", "url": "https://doi.org/10.1103/PhysRevE.106.024119", "doi": "10.1103/PhysRevE.106.024119" }, { "citation": "Barato and Seifert (2015), Thermodynamic Uncertainty Relation for Biomolecular Processes. Physical Review Letters 114, 158101. The variance-based bound, which these examples obey.", "url": "https://doi.org/10.1103/PhysRevLett.114.158101", "doi": "10.1103/PhysRevLett.114.158101" }, { "citation": "Gingrich, Horowitz, Perunov and England (2016), Dissipation Bounds All Steady-State Current Fluctuations. Physical Review Letters 116, 120601.", "url": "https://doi.org/10.1103/PhysRevLett.116.120601", "doi": "10.1103/PhysRevLett.116.120601" }, { "citation": "Barato and Seifert (2015), Skewness and Kurtosis in Statistical Kinetics. Physical Review Letters 115, 188103.", "url": "https://doi.org/10.1103/physrevlett.115.188103", "doi": "10.1103/physrevlett.115.188103" }, { "citation": "Wampler and Barato (2022), Skewness and kurtosis in stochastic thermodynamics. Journal of Physics A: Mathematical and Theoretical 55, 014002.", "url": "https://doi.org/10.1088/1751-8121/ac3b0c", "doi": "10.1088/1751-8121/ac3b0c" }, { "citation": "Salazar (2022), Thermodynamic skewness relation from detailed fluctuation theorem. Physical Review E 106, L042101.", "url": "https://doi.org/10.1103/PhysRevE.106.L042101", "doi": "10.1103/PhysRevE.106.L042101" }, { "citation": "Jordan and Sukhorukov (2004), Transport statistics of bistable systems. Physical Review Letters 93, 260604. The classical telegraph (switching) statistics behind the mechanism.", "url": "https://doi.org/10.1103/PhysRevLett.93.260604", "doi": "10.1103/PhysRevLett.93.260604" }, { "citation": "Bagrets and Nazarov (2003), Full counting statistics of charge transfer in Coulomb blockade systems. Physical Review B 67, 085316.", "url": "https://doi.org/10.1103/PhysRevB.67.085316", "doi": "10.1103/PhysRevB.67.085316" }, { "citation": "Flindt, Fricke, Hohls, Novotný, Netočný, Brandes and Haug (2009), Universal oscillations in counting statistics. Proceedings of the National Academy of Sciences 106, 10116–10119.", "url": "https://doi.org/10.1073/pnas.0901002106", "doi": "10.1073/pnas.0901002106" }, { "citation": "Di Terlizzi and Baiesi (2019), Kinetic uncertainty relation. Journal of Physics A: Mathematical and Theoretical 52, 02LT03.", "url": "https://doi.org/10.1088/1751-8121/aaee34", "doi": "10.1088/1751-8121/aaee34" }, { "citation": "Saryal, Friedman, Segal and Agarwalla (2019), Thermodynamic uncertainty relation in thermal transport. Physical Review E 100, 042101.", "url": "https://doi.org/10.1103/PhysRevE.100.042101", "doi": "10.1103/PhysRevE.100.042101" }, { "citation": "Ptaszyński and Esposito (2025), Critical heat current fluctuations in Curie-Weiss model in and out of equilibrium. Physical Review E 111, 034125.", "url": "https://doi.org/10.1103/PhysRevE.111.034125", "doi": "10.1103/PhysRevE.111.034125" }, { "citation": "Bruderer, Contreras-Pulido, Thaller, Sironi, Obreschkow and Plenio (2014), Inverse counting statistics for stochastic and open quantum systems: the characteristic polynomial approach. New Journal of Physics 16, 033030.", "url": "https://doi.org/10.1088/1367-2630/16/3/033030", "doi": "10.1088/1367-2630/16/3/033030" }, { "citation": "Anonymous (2026), Counterexamples to a conjectured waiting-time bound on entropy production in periodically driven Markov networks. Evidence Press release. A companion counterexample to a conjectured thermodynamic-inference bound for Markov networks.", "url": "https://evidencepress.org/releases/waiting-time-entropy-bound-counterexamples/", "doi": null } ], "claimAssurance": null, "operatingModel": { "version": "1.0", "workId": "ep-work:thermal-cumulant-sign-counterexamples", "attemptIds": [ "ep-attempt:thermal-cumulant-sign-counterexamples-publication" ], "aims": [ "science" ], "artifactRoles": [ "research-output", "communication" ], "lineageId": null, "accelerationPrimitives": [ "counterexample-proxy-first", "adversarial-controls", "assurance-vector", "agent-readable-research-object" ], "decisionObject": { "type": "counterexample", "description": "Exact three-state, two-bath thermal networks with local detailed balance refute Ptaszyński's (2022) conjectures C3/C1 ≥ 0 and C4/C2 ≥ 0 for heat currents; a full-support family drives both ratios to −∞ at fixed energies, temperatures, stationary law, mean heat current and entropy production.", "scope": "Finite classical continuous-time Markov jump processes with bathwise local detailed balance and heat counted into one bath by state-energy jumps, as in Ptaszyński (2022), Section III. Long-time cumulants only; the source's other inequalities and quantum models are not addressed." }, "bottleneckTargeted": [ "assurance", "publication" ], "semanticBridge": { "state": "explicit", "description": "Table 1 of the paper matches each hypothesis of the source (its equations (65)–(67) and Section III summary) to where it holds; the source's counting field equals the paper's tilting, so its S_h and K_h coincide with S and K.", "remainingRisks": [ "The source does not name its sampling distribution; the census replication samples uniformly.", "The source's other inequalities are untested.", "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": [ "specialistReview", "independentRerun", "formalVerification", "priorityAssessment" ], "nextAction": "Have a specialist in full counting statistics check the root selection in Theorem 5.1 and the conditional kinetic bound of Theorem 6.3.", "claimCeiling": "Proved counterexamples to both sign conjectures in the stated class, with fixed-data divergence, normalised limits, a conditional kinetic bound and two-state minimality. 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