{"id":"68e74204-5d72-47ad-8b7c-7c6d572dc504","arxiv_id":"2412.02675","paper_version":3,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":3.0,"correctness_risk":"low","formal_verification":"none","parameter_count":2,"one_line_summary":"A perspective arguing that time-irreversibility and entropy production inferred from coarse-grained observations depend critically on the type of coarse graining, with milestoning able to create false irreversibility signals.","lead":"This perspective explains why detecting the arrow of time from coarse-grained measurements is subtle: lumping or milestoning observations can either hide or fake signs of irreversibility. It argues that the assumptions behind how we reduce data decide whether inferred entropy production is physically meaningful.","discovery_kind":"review","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 'bulletproof' saturated-order estimator in Eq. (22) is undermined by the Outlook: rigorous results are claimed for n-th order semi-Markov processes in §III.B.1, but the Outlook states they were never proved for n≥2.","rationale":"The central mathematical claim—that milestoning and time-reversal do not commute, so naive time reversal can yield ΔS_inf(q)>0 while the microscopic dynamics obeys detailed balance—is supported by peer-reviewed counterexamples (Refs. 62, 64, 67, 119) and by the log-sum inequality argument for lumped dynamics that makes the milestoned case special. We do not dispute this. The reader's verdict of CONDITIONAL is appropriate. Our stress-test identified a different, internal concern not emphasized by the reader: the paper's constructive remedy, the saturated-order estimator, is claimed to be reliable for n-th order semi-Markov processes, but the Outlook explicitly states that rigorous results for n≥2 are missing. This is not merely a nuance; it is a contradiction between §III.B.1 and the Outlook. If the rigorous result only holds for renewal processes, then for typical milestoned trajectories with higher-order memory the estimator may not be guaranteed to be a lower bound or to saturate, weakening the paper's practical advice. The concrete test—examining the cited preprint and running a numerical check on an n=2 equilibrium milestoned process—can settle which statement is correct. In the meantime, the central warning about false positives stands, but the paper's own internal inconsistency supports (and possibly strengthens) the CONDITIONAL verdict rather than changing it.","tokens_in":21958,"tokens_out":9567,"duration_ms":97152,"concrete_test":"Read arXiv:2410.11819 (Ref. 113) and determine whether its proofs establish saturation and lower-bound properties for all n or only for n=1 (renewal processes). If only n=1 is rigorous, simulate a simple n=2 semi-Markov process generated by milestoning a reversible overdamped diffusion (e.g., a three-well potential with asymmetric milestone placements), estimate ˙S_est^2 from Eq. (22) from long trajectories, and check whether it is zero within statistical error. A positive value for an equilibrium process would falsify the 'bulletproof' claim; a zero value would indicate the missing proof is a technical gap rather than a numerical failure.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section III.B.1 asserts that for an n-th order semi-Markov process, the estimator in Eq. (22) satisfies ˙S_est^{n+k}=˙S_est^n and ˙S_est^n ≤ ˙S_tot, citing arXiv:2410.11819. The Outlook then states: 'rigorous results are only available for renewal dynamics. The results for the thermodynamic entropy production rate for general n ≥ 2 semi-Markov processes was never proved rigorously.' This is a direct internal contradiction. Because milestoned trajectories of overdamped diffusions are generally not renewal processes—waiting times are history-dependent—the paper's central constructive advice to use saturated-order estimators as the 'presumably only bulletproof' method is not supported for n≥2 by the paper's own account. The existence counterexamples for false-positive entropy production remain valid, but the recommended remedy is less secure than claimed. The paper does not reconcile the two statements or explain which one is intended to be authoritative.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript is a perspective on how coarse-graining affects the inference of time-reversal symmetry and entropy production from observed trajectories. It contrasts lumping (deterministic many-to-one maps) with milestoning (state changes triggered by crossing hypersurfaces), arguing that only milestoning can produce Markovian coarse-grained kinetics, yet milestoning breaks the commutativity between time reversal and coarse-graining. The paper shows, citing published work, that a milestoned equilibrium system can exhibit positive informatic entropy production under the naive backward reading of the trajectory, and that lower bounds and antisymmetric-observable tests valid for lumped observables can fail for milestoned ones. It proposes saturated-order estimators (Eq. 22) as the only generally reliable inference route and singles out unproved assumptions in the literature.","tokens_in":22177,"tokens_out":6838,"duration_ms":71563,"significance":"If the central assertions are correct, the paper delivers an important cautionary message for experimental inference of dissipation: apparent time-reversal asymmetry in coarse-grained signals is not proof of microscopic broken detailed balance, and the type of coarse-graining is decisive. The mathematical framework for overdamped Markovian dynamics, the log-sum inequality bound, and the equivalence of informatic and thermodynamic entropy production are standard and correctly presented. The milestoning counterexamples are anchored in the authors' prior peer-reviewed work, which is a legitimate basis for a perspective. The main weakness is that the practical recommendation built around Eq. (22) is contradicted by the paper's own Outlook; as written, that advisory claim is not internally consistent. The perspective nature does not excuse unresolved tension in the central constructive message, but the issue is localized and fixable.","major_comments":[{"comment":"There is a direct internal contradiction about the status of the saturated-order estimator. Section III.B.1 states that if (qτ) is an n-th order semi-Markov process then Ṡ_est^{n+k} = Ṡ_est^n for all k>0 and Ṡ_est^n ≤ Ṡ_tot, citing Ref. [113]. The Outlook then states that 'rigorous results are only available for renewal dynamics' and that the thermodynamic entropy production rate 'for general n ≥ 2 semi-Markov processes was never proved rigorously.' These statements cannot both be true. Because §III.B.2 instructs readers to use saturated-order estimators in Eq. (22) as 'the presumably only (bulletproof) thing to do in general,' this inconsistency is load-bearing: the recommended remedy is not supported by the manuscript's own account for n≥2. Please specify which statement is authoritative, and if the theorem remains unproved, weaken the 'bulletproof' wording and clearly mark the n≥2 property as conjectural.","section":"III.B.1, Eq. (22), and IV (Outlook)"},{"comment":"The claim that experimental imperfections such as finite resolution, detector blind spots, signal intensity modulation, and thresholding 'behave effectively as milestoning' is asserted without a formal or even worked-out justification. This assertion carries the practical relevance of the no-go message, since the milestoning functional is defined as a state change triggered by crossing a specified hypersurface, which is not the same as deleting or blurring parts of the trajectory. Please either prove or illustrate this equivalence with concrete examples, or state it explicitly as a conjecture and discuss the differences between milestoning and missing-data/blurring coarse-graining.","section":"III.B.2"}],"minor_comments":[{"comment":"The sentence containing 'implies implies' should be corrected to a single 'implies'.","section":"III.B.2"},{"comment":"The notation \\hatγ_k^{(f)} is introduced informally; define it explicitly as the final state of the subsequence before using it in the log ratio.","section":"III.B.1, Eq. (22)"},{"comment":"The trajectory panels are labeled (a)-(h), but the text refers to '(a, e)', '(b-d)', and '(f-h)'; please make the panel references consistent and add exact landmarks for the lumping boundary and milestone positions.","section":"Fig. 2 and caption"},{"comment":"Several of the works on which the central claims rest are arXiv preprints (notably Refs. [113] and, in part, [119]); provide published versions or explicit status notes where available, since the manuscript attributes a theorem to Ref. [113] that the Outlook later disavows.","section":"References"},{"comment":"The appendix could benefit from a one-sentence summary of the Wong-Zakai discretization point for readers unfamiliar with Refs. [130-132], to make the uniqueness argument more self-contained.","section":"Appendix A"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is largely a review of the authors' own prior work, with the central counterexamples cited rather than re-derived; this is acceptable for a perspective. The refereeing should, however, verify that the theorem attributed to Ref. [113] in Section III.B.1 matches the actual proven content of that preprint, because the Outlook appears to disavow it. The paper is suitable for the journal after the contradiction is resolved and the 'bulletproof' wording is adjusted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"I've read the perspective carefully. The central warning is correct: milestoning and time reversal do not commute, so naive time-reversal of milestoned trajectories can produce positive informatic entropy production even when the microscopic dynamics is equilibrated. That claim is well supported by the authors' earlier work (Hartich & Godec PRX 2021, PRR 2023; Blom et al PNAS 2024) and by the explicit counterexamples they cite. The paper also does a good service by contrasting lumping and milestoning with a simple double-well simulation; the point becomes tangible. The appendix on the Onsager–Machlup action is a nice clarification of a recurring misconception and is mathematically sound.\n\nThe problem is an internal contradiction that the paper never reconciles. Section III.B.1 states that for an n-th order semi-Markov process the estimator in Eq. (22) satisfies Ṡ_est^{n+k} = Ṡ_est^n and Ṡ_est^n ≤ Ṡ_tot, citing arXiv:2410.11819. The Outlook then says, explicitly, that rigorous results are only available for renewal dynamics and that the results for general n ≥ 2 were never proved rigorously. Both cannot be right. Since milestoned dynamics are generally not renewal processes, the paper's recommendation that saturated-order estimators are the 'presumably only bulletproof' method is unsupported for exactly the cases the perspective cares about. The false-positive counterexamples stand, but the remedy is less secure than the main text claims.\n\nTwo smaller quibbles. First, 'all aforementioned inference strategies are prone to fail for milestoned processes' is stronger than the evidence: the cited cases show existence of counterexamples, not failure for every milestoned process. Second, the assertion that real experimental signals are effectively milestoned is plausible but not demonstrated; it is a general concern, not a proven fact. These are minor qualifications.\n\nThe mathematics is standard and the central argument holds up. The internal contradiction is a referee-level issue, not fatal. I'd bring this to a reading group to discuss the limits of entropy-production inference from coarse-grained data, and I'd send it to peer review with a request that the authors reconcile the two statements about semi-Markov estimators or soften the recommendation. I would not cite it for the saturated-order estimator results—I'd cite the underlying papers—but as a perspective on why coarse graining matters, it earns its place.","headline":"A clear synthesis of why milestoning breaks time-reversal inference, but the internal contradiction about saturated-order estimators weakens its main remedy.","tokens_in":22689,"tokens_out":5612,"would_cite":true,"duration_ms":52328,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["82C31","60J60","60K15"],"pacs":["05.40.-a","05.70.Ln"],"model":"deepseek-v4-flash","headline":"Milestoning and time reversal do not commute, so coarse-grained observations can show apparent entropy production even when the microscopic dynamics obeys detailed balance.","keywords":["time-reversal symmetry","detailed balance","entropy production","coarse graining","milestoning","kinetic hysteresis","semi-Markov processes","stochastic thermodynamics"],"falsifier":"Simulate an overdamped diffusion with detailed balance in a double-well potential, for example $\\varphi(x)=x^4-3x^2+x/2$ as in the paper's Fig. 2, record milestoned trajectories with milestones at $x=\\pm 0.6$, and evaluate the waiting-time-based entropy estimator that includes the directional waiting-time log ratios $\\psi_{k|j}(t)$ versus $\\psi_{j|i}(t)$: if it returns a positive rate while the microscopic path measure is known to be time-reversal symmetric, and the rate vanishes when waiting-time contributions are discarded as in Eq. (22), the non-commutation claim is confirmed. On experimental data, threshold a detector signal from an equilibrium system at two different thresholds and look for a waiting-time asymmetry that appears at one threshold and disappears at the other.","tokens_in":21744,"feed_emoji":"🔁","tokens_out":8680,"duration_ms":83738,"temperature":0.7,"pith_summary":"This perspective argues that the time-reversal asymmetry seen in a coarsely observed trajectory is not always a reliable sign of microscopic dissipation, and that the distinction between lumping (merging many microscopic states into one observed state) and milestoning (recording only crossings of localized milestone surfaces) decides which inference methods are valid. For lumped observables, the informatic entropy production of the projection never exceeds that of the full dynamics, so a nonzero time-reversal asymmetry in lumped data does imply a nonequilibrium microscopic dynamics. For milestoned observables, however, milestoning and time reversal do not commute: the backward read of a milestoned trajectory is not the milestoning of the backward microscopic trajectory. The paper states that there exist examples with $\\Delta S_{\\mathrm{inf}}(q)>0$ while $\\Delta S_{\\mathrm{inf}}(x)=\\Delta S_{\\mathrm{tot}}(x)=0$, so apparent entropy production can be a pure artifact of the reduction; only estimators that intentionally discard waiting-time information are presented as generally safe.","feed_headline":"Milestoning and time reversal do not commute","feed_subtitle":"For milestone-based coarse graining, an equilibrium system can appear to produce entropy; only some estimators are safe.","key_machinery":"The load-bearing objects are the informatic entropy production $\\Delta S_{\\mathrm{inf}}$, defined as the Kullback-Leibler divergence between forward and time-reversed path measures; the overdamped Markovian diffusion model of Eq. (2); and the milestoning functional $F$, which maps a continuous trajectory to the sequence of milestones it crosses. For fully observed overdamped Markov dynamics, the informatic and thermodynamic entropy productions coincide, $\\Delta S_{\\mathrm{tot}}=\\Delta S_{\\mathrm{inf}}$, and for lumped observables the log-sum inequality gives $\\Delta S_{\\mathrm{inf}}(q)\\le \\Delta S_{\\mathrm{inf}}(x)$, so lumped time-reversal asymmetry is a genuine nonequilibrium signal. The argument turns on the failure of this logic under milestoning: the path weight of a milestoned process is not a marginal of the microscopic path weight, because the backwards-read milestoned trajectory differs from the milestoning of the backwards microscopic trajectory. The paper identifies the saturated-order estimator of Eq. (22), evaluated on state sequences with waiting-time information intentionally disregarded, as the only generally safe way to infer a lower bound on microscopic dissipation from such data.","core_discovery":"The paper's central claim is that milestoning and time reversal do not commute. For a milestoned observable $q_\\tau = F[(x_\\tau)_{0\\le \\tau\\le t}]$, one can have $(q_{t-\\tau})_{0\\le \\tau\\le t} \\neq F[(x_{t-\\tau})_{0\\le \\tau\\le t}]$, even when the underlying microscopic process is an overdamped diffusion obeying detailed balance. Consequently, applying the naive time reversal (reading the recorded trajectory backwards) to milestoned trajectories can produce a positive informatic entropy production rate even though the microscopic dynamics produces no thermodynamic entropy at all, i.e. $\\Delta S_{\\mathrm{inf}}(x)=\\Delta S_{\\mathrm{tot}}(x)=0$. This effect, called kinetic hysteresis, appears already for first-order semi-Markov processes arising as milestonings of thermodynamically consistent overdamped diffusions and extends to milestonings of Markov jump dynamics and of non-Markovian lumped dynamics. The paper concludes that time-antisymmetric correlation functions, thermodynamic uncertainty relations, and speed limits, which are valid for lumped observables, are not generally valid for milestoned observables, and that the safe route is a saturated-order semi-Markov estimator that deliberately ignores waiting times.","pith_inferences":["Implicit in the paper: real-world detectors that threshold or clip signals are a form of milestoning, so a nonzero waiting-time asymmetry in a single-molecule trace is not by itself evidence of dissipation; varying the detection threshold on the same data and watching the inferred entropy production shift would test this.","The non-commutation result suggests that memory kernels extracted from projected equilibrium trajectories cannot be classified as even or odd under time reversal without additional assumptions, which may help reconcile conflicting reports about non-Markovian entropy production.","A concrete extension the paper leaves open is an analogue of the saturated-order estimator for underdamped microscopic dynamics, where velocities have definite odd time-reversal parity; such an estimator could show whether the false-signal effect weakens or disappears."],"forward_implications":["Any inference method that reads a milestoned trajectory backwards and interprets the resulting asymmetry as dissipation can report $\\Delta S_{\\mathrm{inf}}(q)>0$ for a system at equilibrium; the false signal is caused by the coarse graining, not by the physics.","For lumped observables the chain $\\Delta S_{\\mathrm{inf}}(q)\\le \\Delta S_{\\mathrm{inf}}(x)=\\Delta S_{\\mathrm{tot}}(x)$ remains valid, so detecting nonequilibrium from time-antisymmetric observables of lumped data is on safe ground.","Estimators of entropy production from milestoned data must either use additional structural knowledge, such as Markovian transition states or transition-path times, or use saturated-order estimators that discard waiting-time contributions.","Experiments with finite resolution, thresholding, or detector blind spots effectively milestone every trajectory, so claims of broken detailed balance must justify the assumed coarse-graining procedure before they can be interpreted thermodynamically."],"supporting_citations":[{"why":"Defines the informatic entropy production as a Kullback-Leibler divergence and frames when it coincides with thermodynamic entropy production.","marker":"[37]"},{"why":"Supplies the thermodynamic entropy production for overdamped diffusions and Markov jump processes, including local detailed balance and the total entropy balance.","marker":"[38]"},{"why":"Introduces kinetic hysteresis and shows that milestoning an overdamped diffusion can yield positive entropy production estimates under naive time reversal despite detailed balance.","marker":"[62]"},{"why":"Extends the failure of local detailed balance under coarse graining to milestoning of Markov jump dynamics.","marker":"[64]"},{"why":"Provides milestoning-based estimators of dissipation at coarse resolution and gives examples with positive entropy production in equilibrium-like settings.","marker":"[67]"},{"why":"Derives the saturated-order semi-Markov estimators in Eq. (22) and shows how inconsistent time reversal produces artefactual scaling of inferred dissipation.","marker":"[113]"},{"why":"Gives explicit counterexamples where naive time reversal of milestoned trajectories produces erroneous entropy production estimates.","marker":"[119]"},{"why":"Establishes the log-sum inequality used to show that lumping cannot increase informatic entropy production, providing the contrast with milestoning.","marker":"[114]"}],"fun_headline_variants":["Milestoning invents entropy in equilibrium","Time reversal doesn't commute with milestoning","Coarse graining can fake time's arrow","Milestoning fakes broken detailed balance"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The counterexamples and the claimed safety of the saturated-order estimator assume the microscopic dynamics is an overdamped Markovian diffusion with additive noise, symmetric positive-definite diffusion proportional to temperature, and local detailed balance, as stated in Section II and used throughout Section III; if the real system is underdamped, has hidden slow degrees of freedom, or has additional sources of dissipation not expressible through the drift in Eq. (2), the specific conclusions may not carry over.","fun_headline_variants_meta":{"raw":{"variants":["Milestoning invents entropy in equilibrium","Time reversal doesn't commute with milestoning","Coarse graining can fake time's arrow","Milestoning fakes broken detailed balance"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000405,"raw_usage":{"total_tokens":2132,"prompt_tokens":994,"completion_tokens":1138,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":610,"completion_tokens_details":{"reasoning_tokens":1081}},"tokens_in":610,"tokens_out":1138,"duration_ms":10115,"temperature":1.0,"reasoning_tokens":1081,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T23:10:26.648148+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate an overdamped diffusion with detailed balance in a double-well potential, for example $\\varphi(x)=x^4-3x^2+x/2$ as in the paper's Fig. 2, record milestoned trajectories with milestones at $x=\\pm 0.6$, and evaluate the waiting-time-based entropy estimator that includes the directional waiting-time log ratios $\\psi_{k|j}(t)$ versus $\\psi_{j|i}(t)$: if it returns a positive rate while the microscopic path measure is known to be time-reversal symmetric, and the rate vanishes when waiting-time contributions are discarded as in Eq. (22), the non-commutation claim is confirmed. On experimental data, threshold a detector signal from an equilibrium system at two different thresholds and look for a waiting-time asymmetry that appears at one threshold and disappears at the other.","supporting_citations":[{"cited_title":"Nossal \\ and\\ author R","cited_arxiv_id":null,"evidence_quote":"Extends the failure of local detailed balance under coarse graining to milestoning of Markov jump dynamics."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Derives the saturated-order semi-Markov estimators in Eq. (22) and shows how inconsistent time reversal produces artefactual scaling of inferred dissipation."},{"cited_title":"Fodor , author C","cited_arxiv_id":null,"evidence_quote":"Establishes the log-sum inequality used to show that lumping cannot increase informatic entropy production, providing the contrast with milestoning."}],"review_version":1}