{"id":"782b45af-460e-4dcc-bf0a-3c8a7fc3de11","arxiv_id":"2509.20730","paper_version":3,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":9,"one_line_summary":"Squeezing codes are new stochastic automata that stabilize memory without symmetry, with fluctuation-stabilized order and critical exponents (z≈1.38–1.93) below the detailed-balance bound.","lead":"This paper introduces new cellular automata, called squeezing codes, that store a bit of information by squeezing minority blobs until they vanish, even under noise. These models are counterintuitive: they get less stable in higher dimensions, and their phase transitions may belong to new, genuinely non-equilibrium universality classes.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Dynamic exponent z for F/M rests on a single-L fit outside the scaling window; non-equilibrium universality claim unproven.","rationale":"The paper has two central contributions: a rigorous proof of synchronous robustness (Theorem 1) via Toom's theorem, which appears sound assuming the damage-set bounds in App. A, and numerical evidence for asynchronous memory phases with exponentially diverging t_mem. Neither of these is threatened by my concern. The most novel and advertised claim is that the zero-bias transitions realize new, intrinsically non-equilibrium universality classes, supported primarily by z=1.44 (F) and z=1.38 (M), below the detailed-balance bound z≥2 (Tab. II and Sec. VI C). This claim is load-bearing because the abstract highlights it and it is the main reason the paper is interesting beyond the memory-phase result. The measurement of z uses Eq. (55) and a single L=300 quench; the authors themselves state that for M the run time exceeds L^z for the quoted z, and Fig. 14 shows a slow onset. Since θ=β/(νz), a finite-time effective θ differing from the asymptotic value translates directly into an incorrect z. A full finite-size scaling collapse at larger L would distinguish a genuine z<2 from a crossover artifact. Thus I agree with the reader's weakest_assumption and recommend keeping the CONDITIONAL verdict.","tokens_in":41321,"tokens_out":8749,"duration_ms":65337,"concrete_test":"Perform critical quenches for M (and F) at L=512 and 1024, with t_max≥5L^{z_guess} for each L, and test the scaling collapse ⟨m(t,L)⟩=L^{-β/ν}f(t/L^z) with z as a free parameter, using β/ν fixed from the static collapse (e.g., 0.23 for M). If the best global collapse requires z≥2, the claimed non-equilibrium universality class is a finite-size artifact; if z≈1.38 collapses all sizes, the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The headline claim of intrinsically non-equilibrium universality classes (z=1.44 for F and z=1.38 for M, Tab. II) rests on the dynamic exponent z extracted in Sec. VI C from the scaling form ⟨m(t)⟩∼t^{-θ}, θ=β/(νz) (Eq. 55), valid only for 1≪t≲L^z. For M, the authors run to t_max=50L=1.5×10^4 at L=300, but with the quoted z=1.38, L^z≈2.6×10^3, so the fitted window extends well beyond the scaling regime. The slow onset of the power law (Fig. 14) means the fitted θ is an effective, not asymptotic, exponent; a downward drift of θ at larger times/sizes would yield a larger z, potentially ≥2. Since the z<2 values are the only evidence that F and M are 'fundamentally non-equilibrium' (abstract), this is the load-bearing point. No error bars or L-dependence checks for θ are given, so one cannot rule out corrections to scaling.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper introduces a family of probabilistic cellular automata ('squeezing codes') in d≥2 that stabilize a bit without conventional symmetry or equilibrium mechanisms. The authors define R, F, M, T via alternating ∧/∨ updates, prove synchronous robustness (Theorem 1) using Toom's monotone-eroder theorem, and present Monte Carlo evidence for asynchronous memory phases for R, F, M. They then develop Doi–Peliti/cluster mean-field equations predicting fluctuation-stabilized order: R2 has pc∼1/d^2, and R3 orders only in d=2. At the zero-bias transition, Binder and magnetization collapses give ν, β near but not equal to 2d model-A values, and relaxation quenches yield z≈2.10, 1.93, 1.44, 1.38 for R2, R3, F, M, with z<2 for F and M claimed as intrinsically non-equilibrium universality classes. The paper also discusses coarsening, flocking stripes, and synchronicity-protected memories.","tokens_in":41639,"tokens_out":6684,"duration_ms":50346,"significance":"If established, the paper's results would be important: they provide a new class of simple robust memories with a rigorous synchronous threshold, a concrete mechanism of fluctuation-stabilized order (including the analytically derived CVM prediction pc=1/(1+4d^2) and R3's dc=2), and candidate non-equilibrium critical points. The paper is careful to distinguish rigorous results (synchronous robustness, unique absorbing states) from numerical evidence, and the CVM calculations are derived rather than fitted. However, the headline claim of new intrinsically non-equilibrium universality classes rests entirely on dynamic exponent estimates that are not yet supported by the data as presented.","major_comments":[{"comment":"The values z_F=1.44 and z_M=1.38 are the sole evidence for the abstract's claim of intrinsically non-equilibrium universality classes. They come from a single L=300 quench. For M, tmax=50L=1.5×10^4, while the scaling form (55), valid for 1≪t≲L^z, gives L^z≈2.6×10^3 at z=1.38; the fit thus extends far beyond the scaling window. Fig. 14 also shows the local exponent θ(t) still drifting on the plotted range. Without multiple system sizes, fits restricted to t≲cL^z, or a corrections-to-scaling analysis, the z<2 result—and hence the central claim—is not established.","section":"Sec. VI C, Eq. (55), Figs. 13–14, Table II"},{"comment":"Theorem 1 is a central rigorous claim, but the proof reduces to unproved inclusions for the damage set. The text states 'We claim (A4)' and calls the verification 'straightforward', and analogous inclusions for F, M, T are asserted without derivation. The alternative zero-set proof is only illustrated in Fig. 16. Please either prove these inclusions as lemmas or provide a complete zero-set argument; as written, the eroder property—and therefore the theorem—cannot be fully checked.","section":"App. A1, Eqs. (A4)–(A7)"},{"comment":"The static exponents ν, β are obtained from Binder and magnetization collapses with no quantitative quality metric or uncertainty (Figs. 10–12). Because z is derived from θ=β/(νz), errors in β/ν propagate directly into the dynamic claim. The M data with model-A exponents shown in Fig. 12 is suggestive, but a formal comparison (e.g., collapse residuals, χ² values) and error bars are needed to support a distinct universality class.","section":"Sec. VI B, Table II"}],"minor_comments":[{"comment":"Typographical errors: 'Dol-Peliti' should be 'Doi-Peliti' (App. B); 'testible' should be 'testable' (App. D); 'loose' should be 'lose' (Sec. V).","section":"Throughout"},{"comment":"The sentence 'The proof is a consequence of a general result of Toom [6] about asynchronous eroders' should presumably read 'monotone eroders' or 'synchronous eroders'.","section":"Sec. III A"},{"comment":"For M, the text states that θ remains close to 0.17 up to t=15,000, but only t≤3000 is shown. A longer time panel would help the reader assess the claimed plateau.","section":"Fig. 14 caption/text"},{"comment":"The notation f^2_− (superscript position vs. subscript sign) is easy to confuse with a power; a parenthetical definition or a different symbol would improve readability.","section":"Eq. (47)"}],"recommendation":"major_revision","confidential_remarks":"To the editor: The paper is creative and likely important, but the abstract's strongest claim (z<2, new non-equilibrium universality classes) is currently backed by a single-system-size fit that extends outside the scaling window for M. The requested additions—multi-L scaling, error bars, fits within the asymptotic window—are standard and should be feasible. The rigorous synchronous-result part is likely solid after filling in the proof details. I would not recommend rejection; the manuscript needs a revision that brings the numerical claims in line with the evidence."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Worth your time: this is the cleanest new family of robust cellular-automaton memories since Toom's rule, and the synchronous robustness proof is real. The z<2 universality claim, on the other hand, is not yet backed by the data—for M the authors admit the fit window sits outside the scaling regime they quote.\n\nWhat's new and good. The R, F, M, T rules are a new construction: they erode minority domains by squeezing rather than by Toom-type majority voting, and the symmetry intertwining spin flip with spatial rotation is a genuinely different mechanism. Theorem 1 (synchronous robustness via Toom's monotone eroder theorem) is the paper's backbone and it holds up, modulo damage-set bounds in App. A1 that are stated rather than fully derived. The cluster mean-field analysis is also a real contribution: it produces parameter-free predictions (pc ~ 1/d^2, critical dimension dc = 2 for R3) that the numerics confirm, and it makes the counterintuitive \"fluctuation-stabilized order\" concrete—mean-field alone gives disorder, and short-range correlations restore order. That is worth reading even if you never touch the exponents.\n\nSoft spots, in proportion. The headline claim of intrinsically non-equilibrium universality classes rests entirely on the fitted dynamic exponents for F (z≈1.44) and M (z≈1.38). For M, the paper's own scaling form (55) requires 1≪t≲L^z; at L=300 with z=1.38, L^z≈2600, yet the fit runs out to t_max=50L=15000. That means the fitted θ is an effective exponent, and there is no L-dependence check and no error bars to rule out corrections to scaling. A downward drift of θ at larger sizes would push z back toward or above 2, which kills the \"fundamentally non-equilibrium\" claim. The authors are honest that the onset is slow, but the conclusion is still stated too strongly. Separately, they cite the z≥2 bound as applying to \"any local Markovian dynamics obeying detailed balance\"; the reference [38] is about critical frustration-free systems. That overstates a result used as a conceptual contrast.\n\nBottom line. The memory-phase result (synchronous proven, asynchronous numerically solid) is likely correct and is a real step in the taxonomy of robust memories. The universality-class claim is a promising but unproven extra. This deserves a serious referee: send it out, and focus the referee on Sec. VI C and the z≥2 citation. If I worked on non-equilibrium CAs or stable phases, I'd cite the central construction.","headline":"Squeezing codes are a genuinely new family of robust CA memories with a solid synchronous proof, but the z<2 critical exponent claim is not yet backed by the numerics.","tokens_in":42138,"tokens_out":3162,"would_cite":true,"duration_ms":62442,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Squeezing codes show that local stochastic dynamics can store a bit of information without any symmetry to protect it, with order that is stabilized by fluctuations rather than destroyed by them.","keywords":["squeezing codes","robust memories","cellular automata","fluctuation-stabilized order","non-equilibrium criticality","dynamic exponent","Toom's rule","synchronicity"],"falsifier":"Run M dynamics at the estimated critical noise on larger systems (L ≥ 1000) for times t ≫ L^z and compute the time-dependent exponent θ(t) of Eq. (56). If θ(t) keeps drifting so that z = β/(νθ) rises to ≥ 2, the z < 2 claim is refuted; alternatively, measure the critical magnetization autocorrelation time and test whether it grows as L^z with z ≥ 2.","tokens_in":41197,"feed_emoji":"🧲","tokens_out":6887,"duration_ms":63480,"temperature":0.7,"pith_summary":"This paper introduces squeezing codes, a family of locally-interacting stochastic dynamics that reliably store one bit of information against arbitrarily biased noise, despite lacking the conventional symmetries usually thought necessary for robust memory. Errors are corrected by 'squeezing': one spin species spreads along one axis while the other contracts along the perpendicular axis, so any flipped region is ballistically eroded. The authors prove robustness rigorously for synchronous updates and present numerical evidence for asynchronous ones, showing that the ordering is fluctuation-stabilized—naive mean-field predicts only disorder, while including short-range correlations restores the ordered phase, with one rule ordering only in two dimensions. At the zero-bias transition out of the memory phase, the measured dynamic exponents z for the F and M rules fall below the z ≥ 2 bound that detailed balance imposes, indicating intrinsically non-equilibrium critical points.","feed_headline":"Squeezing codes store a bit with no symmetry to protect it","feed_subtitle":"Local dynamics squeeze away errors; order is fluctuation-stabilized and critical points have z < 2.","key_machinery":"The carrying mechanism is the alternating squeezing update (e.g. R: even steps AND with vertical neighbors, odd steps OR with horizontal neighbors), which squeezes minority domains into thin strips that vanish. The dynamics are symmetric under XC, a spin flip combined with a spatial rotation (R), reflection (F, M), or identity (T); this symmetry plus monotonicity lets Toom's monotone-eroder theorem prove robustness under synchronous updates. Fluctuation-stabilization is captured via the cluster variational method (CVM), which keeps short-range correlation fields and yields a phase diagram matching numerics, including the d=2-only order of R3.","core_discovery":"Central claim: robust memories can be built by 'squeezing'—alternating AND/OR updates along orthogonal axes ballistically erode minority domains, with spin-flip-plus-rotation/reflection symmetry replacing ordinary Z2. Synchronously, R, F, M, T are proven robust via Toom's monotone-eroder theorem; asynchronously, R, F, M retain memory numerically. Order is fluctuation-stabilized: mean-field shows disorder, but cluster-variation with short-range correlations restores it; R3 orders only in d=2. At the zero-bias transition, dynamics are not model-A: z ≈ 1.93 (R3), 1.44 (F), 1.38 (M) vs z_A=2.167, with F and M below the rigorous z≥2 bound for detailed balance—intrinsically non-equilibrium critica","pith_inferences":["The non-reciprocal advection terms in the Langevin equations (e.g., γ∂x m + λm∂y m for M) are the likely relevant perturbations driving z below 2; one could try to construct a renormalization group about the model-A fixed point where these operators change the dynamic exponent.","The same XC-symmetry-breaking between spin and space appears in active matter; fluctuation-stabilized ordering may be a general route to noise-robust order in nonreciprocal systems.","The synchronicity transition at αc ≈ 0.385 resembles percolation; measuring its critical exponents (e.g. β, ν) would test whether it falls in the percolation universality class or defines a different one.","The static vs dynamic dichotomy (similar ν,β but very different z for R2 vs R3) suggests the static critical behavior may be shared while dynamics split; verifying conformal invariance at the critical point would sharpen this."],"forward_implications":["Robust memory does not require any symmetry: the squeezing mechanism alone can protect one bit against arbitrarily biased noise.","Higher dimension hurts rather than helps: R3 orders only in d=2, and the critical noise for R2 scales as pc = 1/(1+4d^2) in the cluster analysis.","The zero-bias transitions of F and M have z < 2, so no local equilibrium/detailed-balance model can describe them; these are genuinely non-equilibrium dynamic universality classes.","Some rules are synchronicity-protected: T (and squeezing codes in odd d) lose their memory when updates are asynchronous, with a sharp transition at αc ≈ 0.385.","Quenches into the memory phase of F and M can get stuck in long-lived propagating bands ('flocks'), while R always relaxes to a logical state."],"fun_headline_variants":["Squeezing codes: robust memory with no symmetry","Order from fluctuations: memory without symmetry","Squeezing dynamics store bits, no symmetry needed","No symmetry? No problem for squeezing memory","Fluctuation-stabilized order breaks symmetry rules"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"For F and M the claim of intrinsically non-equilibrium critical points rests on fitted dynamic exponents z ≈ 1.44 and 1.38; the paper notes the scaling regime for M is not fully reached at L = 300, so if the true z is ≥ 2 that claim fails.","fun_headline_variants_meta":{"raw":{"variants":["Squeezing codes: robust memory with no symmetry","Order from fluctuations: memory without symmetry","Squeezing dynamics store bits, no symmetry needed","No symmetry? No problem for squeezing memory","Fluctuation-stabilized order breaks symmetry rules"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000176,"raw_usage":{"total_tokens":1073,"prompt_tokens":635,"completion_tokens":438,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":379,"completion_tokens_details":{"reasoning_tokens":368}},"tokens_in":379,"tokens_out":438,"duration_ms":4683,"temperature":1.0,"reasoning_tokens":368,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T15:13:47.705863+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run M dynamics at the estimated critical noise on larger systems (L ≥ 1000) for times t ≫ L^z and compute the time-dependent exponent θ(t) of Eq. (56). If θ(t) keeps drifting so that z = β/(νθ) rises to ≥ 2, the z < 2 claim is refuted; alternatively, measure the critical magnetization autocorrelation time and test whether it grows as L^z with z ≥ 2.","supporting_citations":[],"review_version":1}