{"id":"f3ab7f55-1666-4090-ae7f-563f95a896dd","arxiv_id":"2507.03485","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"For the first time, a one-dimensional cellular automaton is proven ergodic under both very low and very high uniform noise while being non-ergodic at an intermediate noise rate.","lead":"This paper builds a one-dimensional cellular automaton whose behavior under random noise changes twice: with very little noise it forgets its initial state, at an intermediate noise level it remembers it, and with lots of noise it forgets again. It is the first example of a probabilistic cellular automaton with two phase transitions for ergodicity, a question raised by Gács's famous non-ergodicity counterexample.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The synchronized-zone barrier in the proof of Proposition 5.8 is asserted rather than proved: the no-error conditions defining G_t_i do not rule out deterministic arrows entering the zone, so the claim that 'no arrows on the F-layer cross this region' is not established and the MAC drift does not…","rationale":"The central theorem has two independent legs. High-noise ergodicity is standard. Non-ergodicity at ε2 depends on the fidelity of Theorem 2.1 to Gács's published theorem; this is under-specified but probably reparable by citing a numbered theorem. The low-noise leg is the genuinely new part: it must show that synchronized zones block information for all sufficiently small ε. That conclusion is obtained only through Proposition 5.8 and Proposition 5.11, which translate the drift of the Markov additive chain into collision of dependence borders. Both propositions rely on the informal claim that a synchronized zone is an impenetrable wall. I checked the local definitions: G_t_i in Section 5.2.3 and F_t_i(l) in Section 5.9.1 are events about noise symbols E being absent in prescribed spacetime regions; they are silent about arrows already present to the left of the zone. The F-layer update rule gives precedence to a left-arrow input, so an arrow moves into the zone deterministically, without any E ≠ ⊥. The proof does not show why such an arrow cannot punch a moving hole through every 0-projection wall. Inequalities (5.4) and (5.8) control only the coarse MAC, not this potential leak. This is exactly the reader's weakest_assumption, and I agree with it. I recommend keeping the CONDITIONAL verdict: the concern is concrete and central, but it is checkable, and the construction may survive a repaired barrier lemma. There is no formalization or reproducible code to de-risk the asymptotic computations, which further supports conditional acceptance rather than full acceptance.","tokens_in":26135,"tokens_out":17369,"duration_ms":229492,"concrete_test":"Test the barrier assertion directly for small parameters, e.g. k = 5 and a = 4. Iterate the deterministic F-layer rule on a finite interval starting from a 2k-cell block with equal clock values and a single arrow ↗_s immediately to its left, with no noise. Mark every spacetime cell whose F-output is 0 or an arrow with timer 0 mod k. Then search, by dynamic programming on the spacetime grid, for a causal path of speed at most 1 from a cell left of the block to a cell right of the block that avoids all marked cells. If such a path exists, the claim that no arrow crosses the synchronized zone is false and the barrier used in Proposition 5.8 is not established. Repeat the same search for the right-border configuration of Section 5.9. If no such path exists, the test supports turning the informal barrier paragraph into a standalone lemma.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is the success case of Proposition 5.8 (Section 5.3, after Eq. (5.3)) and its mirror image for the right border in Section 5.9.2. The proof asserts that a synchronized F-layer zone of width greater than 2k is an impenetrable barrier because 'no arrows on the F-layer cross this region,' so every G-layer path from left to right must hit a simultaneously projected 0_G. But the event G_t_i defined in Section 5.2.3 only requires that the noise variables E are equal to ⊥ in the green cells and the yellow triangle; it says nothing about other arrows already present to the left of the zone or created before time t_n. The F-layer rule is deterministic and gives priority to a left arrow: if z_{i-1} = ↗_s, then cell i becomes ↗_{s+1} regardless of its current value. Thus an arrow located to the left of a synchronized block moves into the block without triggering any checked E ≠ ⊥. Such an arrow is a moving defect: on the next synchronized 0-projection, the cell occupied by the arrow need not project 0_G, so the vertical wall has a hole. Since Lemma 5.5 permits information to travel at speed 1, a causal path could in principle ride this moving hole. Proposition 5.8 is the only bridge from the Markov additive chain drift computed in inequality (5.4) to l_t → +∞ and hence to p_t(T_ϵ) → 1. If this barrier lemma fails, the low-noise ergodicity leg of Theorem 3.1 does not follow. The same gap appears in the right-border treatment around Proposition 5.11. The paper contains no standalone proof of the barrier, and no formalization or code that could certify it.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs a one-dimensional probabilistic cellular automaton T_ε by coupling Gács's non-ergodic CA G with an auxiliary 'clock' layer F of counters modulo k and right-moving arrow particles. The main result (Theorem 3.1) asserts that there are parameters k,a and rates 0<ε1<ε2<ε3<1 such that T_ε is uniformly ergodic for ε∈(0,ε1), non-ergodic at ε=ε2, and uniformly ergodic for ε>ε3. The non-ergodicity regime is obtained by showing that at ε2 the errors injected into the G-layer by the F-layer plus noise satisfy the conditional probability condition of Gács's theorem (Theorem 2.1). The low-noise ergodicity regime is approached via coupling-from-the-past and dependence cones; the boundaries of the dependence cone are controlled by Markov additive chains whose positive (resp. negative) drift is computed in Sections 5.4–5.10. The high-noise regime follows from a known percolation argument. The conclusion would be the first example of two phase transitions for ergodicity in a one-dimensional PCA.","tokens_in":1719,"tokens_out":1711,"duration_ms":259562,"significance":"If the proof is completed, this is a substantial result: it answers a natural open question, shows that the ergodicity region in noise parameter can be disconnected, and provides a promising template for constructing multi-phase-transition PCAs by coupling a reliable automaton with a controlling layer. The paper also has clear strengths: precise definitions, a coherent overall architecture, and explicit asymptotic computations of the Markov additive chain drifts. The use of Gács's automaton as a black box is methodologically appealing. However, the significance is conditional on closing the gap concerning the synchronized-zone barrier described below.","major_comments":[{"comment":"The proof of Proposition 5.8 asserts that a synchronized F-layer zone of width greater than 2k is an impenetrable barrier because 'no arrows on the F-layer cross this region.' This is not a consequence of the events G_i^t, B_i^t, O_i^t defined in Section 5.2.3: those events are formulated entirely in terms of the noise variables E_i^t being ⊥ in the green cells and the yellow triangle, and they do not constrain the F-layer configuration inherited from time -t. The F-rule gives priority to a left arrow: if z_{i-1}=↗_s then cell i becomes ↗_{s+1} regardless of its previous value. Hence an arrow initially to the left of the zone (or created by noise outside the checked region) can move into the zone without triggering any E≠⊥. When the synchronized cells next project 0_G, the cell occupied by this arrow need not project 0_G, so the wall has a moving hole and a radius-1 dependence path could cross. Since Proposition 5.8 is the only bridge from the Markov additive chain drift (inequality (5.4)) to l_t→+∞, the low-noise ergodicity leg of Theorem 3.1 does not follow. The same gap affects the right-border version (Section 5.9.2, inequality (5.8)). The authors should either prove the barrier property from the F dynamics for all configurations compatible with the noise, or enlarge the event G_i^t to exclude crossing arrows and recompute α_ε, β_ε, and the drift inequalities.","section":"Section 5.3 (Proposition 5.8) and Section 5.9.2 (Proposition 5.11)"},{"comment":"Theorem 2.1 is stated as 'a direct consequence' of the main result of [4], but no theorem or lemma number from [4] is given, and the condition 'for all finite S and all events H in the past, P(error in each cell of S | H) ≤ ε_c^{|S|}' is a strong conditional-uniform error condition. Because Section 4.4 concludes non-ergodicity of T_{ε2} from this theorem, the authors need to provide a precise reference (theorem number) and a derivation, or prove the statement. As it stands, a reader cannot verify that Gács's theorem applies to the error process produced by the F-layer plus uniform noise.","section":"Section 2.2 (Theorem 2.1)"},{"comment":"The sentence 'if T_{ε2} is non-ergodic on this layer, then it is non-ergodic globally' is asserted without proof. The proof that follows bounds the probability of errors in the G-coordinate, not the full-symbol errors of T_{ε2}; since Theorem 2.1 is stated for Gács's CA on alphabet G, the logical step from the G-coordinate bound to non-ergodicity of the joint system T_{ε2} on A=G×F needs to be supplied. One possible route is to argue that a G-coordinate error implies a full-symbol error and to apply a version of Gács's theorem to the joint system, but this is not written out. Please clarify this implication.","section":"Section 4, opening paragraph"}],"minor_comments":[{"comment":"The sentence 'In all known examples of CA robust to noise, there is critical value ϵc such that the perturbed cellular automaton is ergodic for any ϵ < ϵc' appears to contradict the surrounding discussion; it should presumably read 'non-ergodic for any ϵ < ϵc'.","section":"Section 1"},{"comment":"In the definitions of G_i^t and B_i^t, the condition 'n-j ≥ t-i' in the yellow triangle is not translation-invariant and is likely a typo; since these events should be invariant under spacetime translations, please replace it with the intended condition (for example n ≥ j).","section":"Section 5.2.3"},{"comment":"The sentence 'We claim that the proof of Proposition 5.8 still stands for the next proposition' should be replaced by an actual proof, especially given the issue raised in the first major comment.","section":"Section 5.9.2"},{"comment":"The title and abstract contain a typo: 'PER TURBED' should be 'PERTURBED'.","section":"Title and Abstract"},{"comment":"The text contains the typo 'ergordic' where 'ergodic' is intended.","section":"Section 2.2"}],"recommendation":"major_revision","confidential_remarks":"The main issue is the unproved barrier property in Section 5; this is a serious gap in the low-noise argument and cannot be fixed by wording alone. The non-ergodicity leg also needs a precise mapping to Gács's theorem. I believe the result may be true and the overall strategy promising, so I recommend major revision rather than rejection. The authors should also consider providing a more formal treatment of the barrier property, possibly with a standalone lemma and proof."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is the first construction of a positive-rates 1D PCA whose ergodicity region in noise space is disconnected, and the mechanism is genuinely new. The non-ergodicity leg at ε2 looks sound. The low-noise ergodicity leg, however, has a load-bearing gap around Proposition 5.8 that the stress-test note identifies correctly.\n\nWhat is new: the F-layer with mod-k clocks and finite-lifetime arrows, noise-created synchronized blocks that destroy memory at low noise, and the use of Gács as a black box to get non-ergodicity at an intermediate rate. The theorem is stated cleanly. Section 4's five-case decomposition is a serious piece of work; the bounds on S1–S5 are exactly the kind of computation that can be checked and they appear coherent. Section 5's dependence-cone/MAC machinery is ambitious and mostly well explained. The conclusions are honest about not knowing the full phase diagram.\n\nThe soft spot is not minor. In the success case of Prop 5.8, the proof asserts that no arrows cross the synchronized zone, hence the zone is an impenetrable 0_G wall. But the events G_t_i and O_t_i only require that the noise variables E are ⊥ in the green cells and yellow triangle. They do not rule out an old deterministic arrow sitting to the left of the zone, created by noise long before the checked window. The F rule gives priority to a left arrow: z_{i-1}=↗_s forces z_i to become ↗_{s+1} regardless of current value. So such an arrow marches into the synchronized block, and on the next synchronized 0-projection the cell it occupies need not put out 0_G. The wall has a moving hole. Lemma 5.5 lets information travel at speed 1, so a causal path can ride the hole, and the entire bridge from the MAC drift in (5.4) to l_t→+∞ collapses. The same gap appears in the right-border treatment around Prop 5.11.\n\nI don't think this is a throwaway detail; it is the main missing lemma of the low-noise proof. It might be repairable by enlarging the checked region so that the success event also excludes arrows entering from the left, or by redefining the chain to jump far enough to clear old arrows. But as written the low-noise leg is incomplete. Everything else is in much better shape. The abstract also overstates point (2): the proof gives one ε2, not an interval.\n\nWho this is for: specialists in probabilistic CA and reliable computation. It deserves a serious referee — desk rejection would be wrong — but the referee should be sent in with the assignment of pressure-testing the barrier lemma. If that lemma gets written down and proved, this becomes a cornerstone result; until then, treat the low-noise ergodicity claim as unproven.","headline":"First two-phase-transition PCA construction with a genuinely new mechanism, but the low-noise ergodicity proof has a load-bearing gap around the synchronized-zone barrier in Proposition 5.8.","tokens_in":27085,"tokens_out":5993,"would_cite":false,"duration_ms":73297,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37B15","60K35"],"pacs":[],"model":"deepseek-v4-flash","headline":"A one-dimensional cellular automaton whose noise-perturbed version is ergodic, then not ergodic, then ergodic again as the noise rate rises.","keywords":["probabilistic cellular automaton","ergodicity","phase transition","positive rates conjecture","noise robustness","Markov additive chain","dependence cone"],"falsifier":"Run the perturbation for the two chosen parameter regimes on a large finite ring: in the low-noise regime, initialize a lone synchronized block of length $2k+1$ and inject a single nonzero $G$-signal on one side immediately after a $0$-projection; if the signal appears on the other side before the next projection, the wall property is false. For the intermediate regime, check numerically whether the conditional error probability for a finite set $S$ at $\\epsilon_2$ exceeds $\\epsilon_c^{|S|}$ as Theorem 2.1 requires; a single counterexample to that bound would break the non-ergodicity leg.","tokens_in":25834,"feed_emoji":"🎲","tokens_out":11022,"duration_ms":113052,"temperature":0.7,"pith_summary":"This paper constructs a one-dimensional cellular automaton $T$ whose uniform-noise perturbation $T_\\epsilon$ has two phase transitions for ergodicity: for very small noise rates it is ergodic, at an intermediate rate it is not, and for rates close to one it is ergodic again. This is the first example in which the set of noise rates leading to non-ergodicity is not an initial interval, and it refutes the intuition that low noise is always the regime where information survives. The construction shows that the positive-rates conjecture is false in a stronger sense: non-ergodicity can appear in a middle window while both very low and very high noise are mixing. A sympathetic reader should care because the result changes what is known about the possible phase structure of one-dimensional probabilistic cellular automata.","feed_headline":"Ergodicity turns on, off, on as noise rises","feed_subtitle":"A single one-dimensional rule shows the ergodic-noise region need not be an interval.","key_machinery":"The workhorse is a two-layer update rule $T(x)_i=(G(y_{i-1},y_i,y_{i+1}),F(z)_i)$ with the caveat that when the $F$-layer produces $0$ or an arrow of type $0\\bmod k$, the $G$-layer is replaced by the fixed symbol $0_G$. The $F$-layer itself is a clock: values increment mod $k$ each step, and an arrow particle $\\nearrow_s$ moves right at speed 1 and resets the cells it passes to synchronized states, living at most $ak$ steps. Synchronized blocks longer than $2k$ cells reset all their $G$-layers to $0_G$ simultaneously every $k$ steps, so no radius-1 information can cross them; the paper treats this as an impenetrable wall. To turn this into a proof of ergodicity, the left and right borders of the dependence cone are bounded by Markov additive chains $(L_n,J_n)$ and $(R_n,K_n)$; the positive drift of $L_n$ and negative drift of $R_n$ follow from barrier-overcoming probabilities $\\beta_\\epsilon$ and success probabilities $\\alpha_\\epsilon$, with the barrier height $H=\\ln(a)/s(\\epsilon)$ chosen to make successive checks independent. Non-ergodicity at $\\epsilon_2$ uses Theorem 2.1, a percolation-style condition on the error field: if every finite set $S$ has error probability at most $\\epsilon_c^{|S|}$ conditional on the past, the perturbed system is non-ergodic.","core_discovery":"The central claim is Theorem 3.1: there exist integers $k,a$ and rates $0<\\epsilon_1<\\epsilon_2<\\epsilon_3<1$ such that the uniform $\\epsilon$-perturbation of a specific radius-1 cellular automaton $T$ is uniformly ergodic for all $0<\\epsilon<\\epsilon_1$, is not ergodic at $\\epsilon_2$, and is uniformly ergodic for all $\\epsilon>\\epsilon_3$. The automaton is a product of two layers: an $F$-layer clock that increments modulo $k$, with right-moving arrow particles that synchronize the clocks they pass over, and a $G$-layer that is forced to a fixed symbol whenever the $F$-layer outputs $0$, so that synchronized blocks of length larger than $2k$ act as walls that stop information flow. At an intermediate rate $\\epsilon_2 \\sim 1/\\ln\\ln\\ln k$, the clock layer decorrelates and the forced errors are sparse enough for the $G$-layer's built-in error correction (from the classical positive-rates counterexample, used as a black box) to preserve distant information forever, making the process non-ergodic. In the low-noise regime, arrows produced by rare errors live long and synchronize large zones, which force the dependence cone of each cell to shrink to a point almost surely; in the high-noise regime, the standard percolation argument for any perturbed cellular automaton gives ergodicity.","pith_inferences":["A direct numerical test: for the $k,a$ of the construction, the survival time of arrows in the $F$-layer should grow as $\\epsilon$ shrinks, and the crossing of the dependence-cone borders should occur at times consistent with the drift inequalities; measuring these in finite simulations would test the mechanism without waiting for a full proof.","If the wall property is the weakest point, then a small modification of the $F$-layer (for example, allowing occasional errors in a synchronized block) might destroy low-noise ergodicity, which would imply the phenomenon is sensitive to the exact synchrony rather than a general principle.","The open question posed by the authors — which subsets of $[0,1]$ can be realized as the ergodicity set of a perturbed CA — is sharpened by this example; iterating the clock-layer idea might produce three or more phase transitions, though the paper notes the construction is hard to adapt because the base automaton's invariant measures are not understood."],"forward_implications":["If Theorem 3.1 is correct, the set of noise rates for which a positive-rate one-dimensional PCA is ergodic can be a non-monotone subset of $[0,1]$, not just an interval above a critical value.","The low-noise regime is not automatically the regime where information is best preserved: here it is the regime where synchronized clock blocks scramble all transmitted information and force ergodicity.","Because every perturbed cellular automaton is ergodic at noise rates close to 1, the new content is that non-ergodicity can occupy a middle window strictly between two ergodic regimes."],"supporting_citations":[{"why":"Supplies the non-ergodic one-dimensional automaton and the percolation-style error condition (Theorem 2.1) used as a black box for the intermediate-noise regime.","marker":"[4]"},{"why":"Provides the coupling-from-the-past ergodicity criterion (Proposition 5.2) and the high-noise ergodicity result (Remark 3.2) used for the $\\epsilon>\\epsilon_3$ leg and for the finite-window ergodicity argument.","marker":"[10]"},{"why":"Provides the Markov additive chain law of large numbers that turns the drift inequalities (5.4) and (5.8) into the almost-sure divergence of the dependence-cone borders.","marker":"[1]"}],"fun_headline_variants":["Ergodicity on-off-on as noise increases","Double ergodicity phase transition in one automaton","Noise toggles ergodicity twice","Ergodic, then not, then ergodic again"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The low-noise proof assumes that a contiguous block of more than $2k$ cells whose clock layer is synchronized is an impenetrable wall that no radius-1 information can cross, and this wall property is argued in a paragraph rather than isolated as a formal lemma.","fun_headline_variants_meta":{"raw":{"variants":["Ergodicity on-off-on as noise increases","Double ergodicity phase transition in one automaton","Noise toggles ergodicity twice","Ergodic, then not, then ergodic again"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000652,"raw_usage":{"total_tokens":3015,"prompt_tokens":999,"completion_tokens":2016,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":615,"completion_tokens_details":{"reasoning_tokens":1966}},"tokens_in":615,"tokens_out":2016,"duration_ms":19584,"temperature":1.0,"reasoning_tokens":1966,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T20:09:37.505916+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the perturbation for the two chosen parameter regimes on a large finite ring: in the low-noise regime, initialize a lone synchronized block of length $2k+1$ and inject a single nonzero $G$-signal on one side immediately after a $0$-projection; if the signal appears on the other side before the next projection, the wall property is false. For the intermediate regime, check numerically whether the conditional error probability for a finite set $S$ at $\\epsilon_2$ exceeds $\\epsilon_c^{|S|}$ as Theorem 2.1 requires; a single counterexample to that bound would break the non-ergodicity leg.","supporting_citations":[{"cited_title":"Reliable Cellular Automata with Self-Organization","cited_arxiv_id":null,"evidence_quote":"Supplies the non-ergodic one-dimensional automaton and the percolation-style error condition (Theorem 2.1) used as a black box for the intermediate-noise regime."}],"review_version":1}