{"id":"6d2b3fbc-c28d-4f9f-9c89-215511b1de30","arxiv_id":"2607.07411","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":7,"one_line_summary":"Across four noisy particle systems, entropy production rate peaks precisely at the hyperuniformity threshold, and a path-integral derivation shows this follows from the non-invertibility of the diffusion tensor at the center-of-mass-conserving noise limit.","lead":"This paper shows that when noisy particle systems become hyperuniform—suppressing large-scale density fluctuations like a crystal while staying disordered like a gas—they simultaneously reach maximum entropy production, meaning they are as thermodynamically irreversible as possible. A smart generalist might read it because it connects structure (what you see) to energetics (what you pay) in self-assembling materials, with implications for designing better photonic or biomimic","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"The numerical EPR peak relies on a 4-state Markov coarse-graining whose sensitivity to state-space resolution is untested; state definitions based on relative motion may capture more irreversibility at cs=-1 where pairwise dynamics are most structured.","rationale":"The reader identified the Pt issue (state-dependent power term potentially counteracting the prefactor ψ) as the weakest assumption. This is a valid concern about the analytical theory, but it is not the most load-bearing: the numerical EPR provides independent evidence for the peak, so even if the analytical argument has this gap, the central claim can still hold. The more load-bearing concern is whether the numerical evidence itself is robust to the choice of coarse-graining. The 4-state Markov approach is a specific methodological choice, and the state definitions (relative direction of motion for pairs, speed/persistence for individual particles) could systematically capture more irreversibility at the hyperuniform point where dynamics are most structured. The consistency across 4 systems is reassuring but does not rule out a shared bias from the same coarse-graining methodology applied to all systems. The analytical theory cannot serve as an independent check because it operates on a different quantity (global EPR with uncontrolled DT) and its connection to the numerical EPR is only qualitative. That said, the concern is about untested robustness rather than an identified error — the 4-state EPR is a standard technique, the monotonic increase is consistent across systems, and the D non-invertibility mechanism is mathematically sound. The CONDITIONAL verdict with MODERATE confidence is appropriate; I would not change it because the concern does not identify a specific failure but rather an untested sensitivity that, if it landed, would weaken but not necessarily overturn the claim.","tokens_in":59356,"tokens_out":8001,"duration_ms":514373,"concrete_test":"Recompute the numerical EPR for RO at cs ∈ {-1, -0.75, -0.5, -0.25, 0} using (i) an 8-state Markov process (splitting each of the 4 original states by a secondary criterion such as kick magnitude or local coordination number) and (ii) a continuous-variable EPR estimator based on forward/backward path probability ratios using k-nearest-neighbor density estimation. If the normalized EPR σ̃(cs) = σ(cs)/σ(0) at any cs value changes by more than 20% relative to the 4-state result, or if the monotonicity in cs is lost, the peak's robustness is in question.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim rests primarily on numerical EPR computed from a 4-state Markov process (Methods, Eq. 13). For spatial-noise systems, the four states are defined by the relative direction of motion within overlapping particle pairs (Fig. 2b). As cs→-1, pairwise kicks become perfectly anti-correlated (equal magnitude, opposite direction), making the relative motion within pairs highly structured. This means the coarse-graining is most informative precisely at the hyperuniform point, potentially inflating the measured EPR there relative to cs=0 where pairwise motion is less structured. The EPR of a coarse-grained process is a lower bound on the true EPR, but the tightness of this bound depends on the coarse-graining choice, and there is no reason it should be uniform across cs. The paper does not test whether the peak survives with a different number of states or different state definitions. The analytical theory (Eq. 3-6) provides a mechanism (D non-invertibility at cs=-1), but it operates on a 'global' EPR with an uncontrolled regularization parameter DT, qualitatively distinct from the 'local' 4-state numerical EPR. The connection between the two is not established quantitatively, so the analytical mechanism cannot serve as an independent check on the numerical coarse-graining.","agreement_with_reader":"partial"},"referee_report":{"model":"glm-5.2","summary":"This paper investigates the connection between hyperuniformity and time irreversibility across several non-equilibrium particle systems (RO, BRO, SGD with spatial noise correlations, and passive particles in an active bath with temporal noise correlations). The authors compute entropy production rates (EPR) numerically via a 4-state Markov coarse-graining and find that the EPR peaks precisely at the hyperuniform point (c_s = -1 for spatial noise, c_t = -1/2 for temporal noise) in all systems studied. They also develop a path-integral (MSRJD/Onsager-Machlup) derivation showing that the EPR diverges at these points because the diffusion tensor D becomes non-invertible when the noise conserves the center of mass. The SI contains a detailed and extensive derivation.","tokens_in":59529,"tokens_out":2109,"duration_ms":62186,"significance":"The paper addresses a genuinely novel question—the thermodynamic cost of hyperuniform self-assembly—and provides a clear, falsifiable claim: maximal irreversibility coincides with hyperuniformity. The path-integral derivation is thorough and follows standard methods, with explicit results for both spatial and temporal noise classes. The study of multiple microscopically distinct systems (RO, BRO, SGD, active bath) lends weight to the universality claim. The SI is extensive, including explicit inversions of the diffusion tensor and the noise kernel, TRS checks, and a generalized chain rule.","major_comments":[{"comment":"The central numerical claim rests on the 4-state Markov coarse-graining (Methods, Eq. 13; Figs. 2b-c, 3b). The EPR of a coarse-grained process is a lower bound on the true EPR, but the tightness of this bound depends on the coarse-graining choice and need not be uniform across the noise correlation parameter c. At c_s = -1, pairwise kicks become perfectly anti-correlated, making the relative-motion states (on which the coarse-graining is based) maximally structured; this could inflate the measured EPR at the hyperuniform point relative to c_s = 0. The paper does not test whether the EPR peak survives with a different number of states or alternative state definitions. This is load-bearing for the central claim and should be addressed, e.g., by repeating the analysis with 2-state, 8-state, or continuously-resolved coarse-grainings and showing that the peak location is robust.","section":null},{"comment":"The analytical EPR (Eq. 3-6) captures only the prefactor ψ(c, D_T/D_A), which is state-independent, while the full EPR also includes the state-dependent power term P_t[X_t] (Eq. 3). The authors justify focusing on ψ alone (Theory section), but if P_t has a non-trivial dependence on c that counteracts ψ, the full analytical EPR might not peak at the hyperuniform point. While the numerical EPR does show the peak independently, the claim that the analytical theory 'explains' the observations (Discussion) is not fully substantiated without at least an estimate or bound on the c-dependence of P_t.","section":null},{"comment":"The connection between the analytical (global) EPR and the numerical (local) EPR is acknowledged as qualitative (Theory section: 'the analytical EPR does not carry the same status as the numerical EPR measurements'). However, the analytical theory introduces an uncontrolled regularization parameter D_T (the effective thermal bath), and the EPR divergence is recovered only in the limit D_T → 0. The paper does not specify what value of D_T/D_A is relevant to the simulations, nor whether the analytical ψ(c, D_T/D_A) at that value would still peak at c = -1. A quantitative comparison—even at the level of showing that ψ(c_s, D_T/D_A) is monotonic in c_s for a physically motivated range of D_T/D_A—would strengthen the link between theory and numerics.","section":null}],"minor_comments":[{"comment":"Several typos: 'defnied' (p. 3, BRO section), 'heed must be taken' (SI Sec. I.A.1, should be 'heed must be paid' or 'care must be taken'), 'Boltzmnann' (SI Eq. S202 context).","section":null},{"comment":"In the Methods, the state definitions for the 4-state Markov process are described qualitatively (e.g., 'relative direction of motion within each particle pair') but the precise binning thresholds (e.g., what angle separates the states) are not specified. This hinders reproducibility of the numerical EPR.","section":null},{"comment":"Figures 2g-i and 3d show normalized EPR σ̃ = σ(c)/σ(c=0), but the absolute values of σ are not reported. It would be useful to know the magnitude of the EPR to assess physical relevance.","section":null},{"comment":"The packing fraction ϕ = 8.0 is used for all systems. It would be helpful to briefly justify this choice and note whether the EPR peak at the hyperuniform point persists at other packing fractions.","section":null},{"comment":"Reference [17] is listed as an arXiv preprint (arxiv:2601.23098). If published by the time of revision, the reference should be updated.","section":null}],"recommendation":"major_revision","confidential_remarks":"The paper presents an interesting and novel connection between hyperuniformity and irreversibility. The main concern is whether the numerical EPR peak is an artifact of the coarse-graining choice, which is testable but currently unaddressed. The analytical theory is sound in its derivation but only partially connects to the numerics. I believe the core claim is likely correct but needs the robustness check on the coarse-graining to be convincing. The authors' own prior work [16, 17] establishing the hyperuniformity conditions is independently grounded and does not raise circularity concerns."},"author_rebuttal":{"model":"glm-5.2","summary":"We thank the referee for a careful and constructive report. The referee raises three major comments concerning (1) the robustness of the EPR peak to the choice of coarse-graining, (2) the potential c-dependence of the state-dependent power term P_t in the analytical EPR, and (3) the role of the regularization parameter D_T and the quantitative link between analytical and numerical results. We address each point below. In brief: we will add a robustness check of the EPR peak against alternative coarse-grainings (2-state, 8-state, and continuously-resolved), we will provide an estimate/bound on the c-dependence of P_t, and we will add a quantitative comparison between the analytical prefactor psi and the numerical EPR at a physically motivated value of D_T/D_A. We agree these revisions strengthen the paper and will incorporate them.","responses":[{"response":"The referee is correct that the tightness of the coarse-grained EPR lower bound can depend on the coarse-graining choice, and that this is load-bearing for our central claim. We will address this by repeating the EPR measurement with alternative coarse-grainings: (i) a 2-state Markov process, (ii) an 8-state process, and (iii) a continuously-resolved estimate (e.g., via a k-nearest-neighbor or binning-based estimator of the time-reversal asymmetry). We will show the results for at least one representative system from each class (e.g., RO for spatial noise, active bath for temporal noise). If the peak location is robust across these choices, as we expect based on the fact that the underlying microscopic dynamics become genuinely irreversible at the hyperuniform point (as shown analytically), this will substantially strengthen the claim. We note that the analytical result provides an independent argument: the divergence of the EPR at c = -1 (spatial) or c = -1/2 (temporal) arises from the non-invertibility of the diffusion tensor D, which is a property of the microscopic dynamics and not of any coarse-graining. Nevertheless, we agree that a direct numerical demonstration of robustness is essential and will include it.","revision_made":"yes","referee_comment":"The central numerical claim rests on the 4-state Markov coarse-graining... The paper does not test whether the EPR peak survives with a different number of states or alternative state definitions... should be addressed, e.g., by repeating the analysis with 2-state, 8-state, or continuously-resolved coarse-grainings and showing that the peak location is robust."},{"response":"This is a fair point. Our justification for focusing on psi alone is that psi is state-independent and would survive any averaging over trajectories, while P_t is intractable in general many-body dynamics. However, the referee is correct that if P_t has a non-trivial c-dependence that counteracts psi, the full analytical EPR might not peak at the hyperuniform point, weakening the claim that the theory 'explains' the numerics. We will address this in two ways. First, we will provide a physical argument that P_t does not counteract the peak: P_t = -X_t dot nabla V(X_t) is the instantaneous dissipated power, and in the systems we study, the steady-state energy (and hence the typical force magnitudes) does not vary dramatically with c, so P_t is not expected to have a sharp c-dependence that could cancel the divergence of psi. Second, we will provide a quantitative estimate or bound on the c-dependence of P_t in at least one tractable limit (e.g., the 1d RO-like example used in Eq. 5, where the system is analytically tractable). This will allow us to show that the full analytical EPR, including P_t, still peaks at the hyperuniform point. We will revise the Discussion to state these arguments explicitly and to qualify the claim appropriately.","revision_made":"partial","referee_comment":"The analytical EPR (Eq. 3-6) captures only the prefactor psi(c, D_T/D_A), which is state-independent, while the full EPR also includes the state-dependent power term P_t[X_t]... if P_t has a non-trivial dependence on c that counteracts psi, the full analytical EPR might not peak at the hyperuniform point... the claim that the analytical theory 'explains' the observations (Discussion) is not fully substantiated without at least an estimate or bound on the c-dependence of P_t."},{"response":"We agree that the link between the analytical theory and the numerics would be strengthened by a quantitative comparison at a physically motivated value of D_T/D_A. In the simulations, there is no explicit thermal bath, so D_T is formally zero; however, the effective regularization arises from the discreteness of the dynamics and finite-size effects, which play a role analogous to a small D_T. We will make this connection explicit in the revised manuscript. Furthermore, we will show that psi(c_s, D_T/D_A) is monotonically increasing as c_s -> -1 for any fixed D_T/D_A > 0, and that the peak location remains at c_s = -1 (or c_t = -1/2) for all finite D_T/D_A, with the peak height diverging only as D_T/D_A -> 0. This means that for any physically motivated range of D_T/D_A, the analytical prefactor peaks at the hyperuniform point, consistent with the numerics. We will add a figure or table showing psi(c, D_T/D_A) for several values of D_T/D_A to make this quantitative. We will also clarify in the text that D_T is a regularization parameter whose role is to model the effect of unobserved degrees of freedom (following the Zwanzig-Mori formalism, as stated in the manuscript), and that the qualitative conclusion (peak at the hyperuniform point) is robust to its value.","revision_made":"yes","referee_comment":"The analytical theory introduces an uncontrolled regularization parameter D_T... the paper does not specify what value of D_T/D_A is relevant to the simulations, nor whether the analytical psi(c_s, D_T/D_A) at that value would still peak at c = -1. A quantitative comparison... would strengthen the link between theory and numerics."}],"tokens_in":59334,"tokens_out":1326,"duration_ms":140577,"standing_objections":[]},"desk_editor":{"model":"glm-5.2","letter":"The main thing to know: this paper establishes a new and physically meaningful connection between hyperuniformity and maximal entropy production. The core observation — that EPR peaks at the hyperuniform point across four microscopically distinct systems — is clean, numerically consistent across all of them, and supported by an independent analytical mechanism. That mechanism (center-of-mass conservation → diffusion tensor non-invertibility → EPR divergence) is a real insight, not a restatement of known results. The path-integral derivation in the SI is detailed and follows standard MSRJD/Onsager-Machlup methods carefully, including a thorough treatment of multiplicative noise discretization conventions and a proof that no generalized time-reversal symmetry exists for c ≠ 0. The TRS analysis in SI Sec. V is the kind of careful work that earns trust in the rest of the derivation. The hyperuniformity conditions themselves come from the authors' own prior work, but those are independently grounded results published in Nature Communications with detailed derivations, so there's no circularity issue here. The numerical EPR is computed via a 4-state Markov coarse-graining, which is a standard and legitimate approach. The stress-test concern about coarse-graining resolution potentially inflating EPR at cs = -1 is worth raising but I don't think it undermines the result. The state definitions are based on relative direction of motion within pairs, and while pairwise motion is indeed most structured at cs = -1, the same monotonic trend appears in the temporal-noise systems where the coarse-graining is defined differently (speed and velocity angle for individual particles). Having two independent coarse-graining schemes both peak at the hyperuniform point is reassuring. That said, the concern is not fully discharged — a robustness check with different state resolutions would strengthen the paper. The more substantive gap is the one the reader flags: the analytical EPR captures only the prefactor ψ, not the full state-dependent EPR, and the connection between the divergent analytical result (regularized by the uncontrolled parameter D0) and the finite numerical EPR is qualitative, not quantitative. The authors are transparent about this — they explicitly note the analytical EPR is a global quantity while the numerical one is local. But a reader is left wanting at least a scaling argument connecting the two. This is the main soft spot, and it's real but not fatal. The numerical evidence stands on its own; the analytical theory provides a mechanism even if the quantitative bridge is missing. Who benefits: researchers in soft matter, active matter, and nonequilibrium thermodynamics who care about the cost of self-assembly. The conceptual framing — that structural order and thermodynamic cost are directly linked — is genuinely useful and will prompt follow-up work. This deserves a serious referee. The referee should push hardest on the analytical-numerical connection and on coarse-graining robustness, but the central claim is well-supported enough to warrant full review.","headline":"Genuine conceptual link between hyperuniformity and irreversibility, with a real but bridgeable gap between analytical and numerical EPR","tokens_in":60127,"tokens_out":685,"would_cite":true,"duration_ms":56494,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"glm-5.2","headline":"Hyperuniform states are maximally irreversible","keywords":[],"falsifier":"If one could compute or numerically estimate the full state-dependent EPR (including the P_t term) and show that it does not peak at the hyperuniform point, or if one found a non-equilibrium system where hyperuniformity emerges without maximal EPR, the central claim would be falsified.","tokens_in":59429,"feed_emoji":"🔥","tokens_out":1048,"duration_ms":95172,"temperature":0.7,"pith_summary":"The paper claims that across several non-equilibrium particle systems with tunable noise correlations, the entropy production rate (EPR) — a measure of time irreversibility and energetic cost — reaches its maximum precisely when the system becomes hyperuniform. Hyperuniformity is the suppression of large-scale density fluctuations, and it can be induced by tuning noise correlations to specific limits: anti-correlated inter-particle kicks (c_s = -1) or maximally anti-correlated temporal noise (c_t = -1/2). The authors verify this numerically across four systems (random organization, biased random organization, stochastic gradient descent, and passive particles in an active bath) using a four-state Markov coarse-graining of the dynamics. They then construct a path-integral derivation of the EPR from the microscopic dynamics, showing that the EPR depends on a prefactor involving the inverse of a diffusion tensor D. At the hyperuniform point, D becomes non-invertible (its row and column sums vanish), causing the EPR to diverge in the absence of thermal regularization. The mechanism is that hyperuniformity requires the noise to conserve the center of mass, which constrains the noise structure and makes it maximally structured — and structured noise is energetically expensive to sustain.","feed_headline":"Hyperuniformity peaks thermodynamic cost","feed_subtitle":"Across four non-equilibrium systems, entropy production is maximal exactly when density fluctuations are most suppressed — and a path-integr","key_machinery":"The diffusion tensor D, which couples the system's velocity to the noise structure. Its invertibility controls the EPR prefactor ψ. At the hyperuniform point (c_s = -1 or c_t = -1/2), D becomes doubly stochastic with vanishing row/column sums, making it non-invertible and causing the EPR to diverge.","core_discovery":"The paper identifies a universal link between emergent long-range spatial order (hyperuniformity) and maximal thermodynamic irreversibility. The central mechanism is that the same noise correlation structure that produces hyperuniformity — center-of-mass conservation in spatial noise, or caged diffusion in temporal noise — simultaneously makes the diffusion tensor D non-invertible, which in turn makes the entropy production rate diverge. The EPR increases monotonically with the crossover length scale of hyperuniform order and peaks at the hyperuniform limit across all systems studied.","pith_inferences":["If the EPR diverges at the hyperuniform point in the absence of thermal noise, then any physical realization of hyperuniformity must involve some regularization mechanism (thermal bath, finite system size, or other noise sources) that caps the EPR — the divergence is a singular limit rather than a directly observable infinity.","The monotonic relationship between crossover length and EPR suggests a scaling law: the energetic cost of hyperuniform order may scale with the volume over which density fluctuations are suppressed, which could be tested by measuring EPR as a function of system size.","If the state-dependent power term P_t in the full EPR formula has non-trivial dependence on the noise correlation c, it could either amplify or partially offset the divergence of the prefactor ψ — the numerical simulations suggest it does not offset it, but a direct analytical bound on P_t near the hyperuniform point would strengthen the claim."],"forward_implications":["Hyperuniform self-assembly in non-equilibrium settings carries a thermodynamic cost that grows with the length scale of the hyperuniform order, implying that large-scale hyperuniform materials require continuous energy input.","Biological systems that spontaneously exhibit hyperuniformity (e.g., cell tissues, leaf vein networks) may be operating near maximal irreversibility, constraining their energetic budgets.","Designing synthetic hyperuniform materials requires engineering noise correlation structures that are thermodynamically expensive, which sets practical limits on self-assembly strategies.","The link between conservation laws (center-of-mass conservation) and irreversibility may extend to other non-equilibrium ordered states beyond hyperuniformity."],"fun_headline_variants":["Hyperuniform states maximize entropy production","Hyperuniformity and peak irreversibility coincide","Maximal thermodynamic cost tags hyperuniform assembly","Entropy production diverges at hyperuniform limit","Hyperuniform order tracks peak time irreversibility"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The analytical theory focuses on a state-independent prefactor ψ of the EPR, which is tractable, while the full EPR also contains a state-dependent power term P_t that is intractable in many-body dynamics. The authors argue that ψ alone captures the peak because it survives trajectory averaging, but if P_t has a non-trivial dependence on the noise correlation that counteracts ψ, the full analytical EPR might not peak exactly at the hyperuniform point.","fun_headline_variants_meta":{"raw":{"variants":["Hyperuniform states maximize entropy production","Hyperuniformity and peak irreversibility coincide","Maximal thermodynamic cost tags hyperuniform assembly","Entropy production diverges at hyperuniform limit","Hyperuniform order tracks peak time irreversibility"]},"model":"glm-5.2","effort":"low","cost_usd":0.0,"raw_usage":{"total_tokens":567,"prompt_tokens":501,"completion_tokens":66,"prompt_tokens_details":null},"tokens_in":501,"tokens_out":66,"duration_ms":15134,"temperature":1.0,"reasoning_tokens":null,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-09T11:30:47.212336+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"If one could compute or numerically estimate the full state-dependent EPR (including the P_t term) and show that it does not peak at the hyperuniform point, or if one found a non-equilibrium system where hyperuniformity emerges without maximal EPR, the central claim would be falsified.","supporting_citations":[],"review_version":1}