{"id":"0fbcebc8-ae99-406a-8365-2da7985c04cc","arxiv_id":"2505.22679","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A probabilistic cellular automaton with overlapping-tile templates and noise injection can evolve stable, non-touching loop patterns on 2D grids under fixed boundaries.","lead":"The authors built a set of local rules that make a cellular automaton reliably form stable loop-shaped patterns (closed paths that never touch) on a 2D grid. The work shows how complex global shapes can emerge from simple local matching plus injected randomness, relevant for pattern formation in natural and engineered systems.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"As-printed formal definition of exception condition E in §3.5.1 contradicts §4.3; reported loop patterns may be produced by a different rule than the one specified.","rationale":"The paper presents a detailed constructive approach and numerous illustrative simulations; I read it as an experimental/existence claim, not a theorem. The reader's identified weakness (no convergence proof, per-size rule selection) is real but acknowledged by the authors in §5.2. A more direct threat is the inconsistency in the rule definition itself. The exception condition E controls when uncovered zero-cells are stable; the formal formula in §3.5.1 and the prose in §4.3 specify different logics. Since the entire paper is about one CA rule, the rule must be unambiguous. This is testable by implementing both variants, so it does not warrant rejection; it supports the existing CONDITIONAL verdict. Credit is due for the explicit pseudocode, stated probabilities, and extensive pattern catalog, and I do not see evidence of intentional obfuscation. The concern is about precision and reproducibility.","tokens_in":17385,"tokens_out":7098,"duration_ms":74489,"concrete_test":"Re-implement Algorithm 1 with the exact Section 3.5.1 rule, running both variants of E: (i) E=(E2∧E3)∨Decision(πE) as printed, and (ii) E=(E2∨E3)∧Decision(πE), or E=E2∨E3 with πE=0.5, as described in §4.3. For each variant, run 300 trials from random initial states for n=5,6,7 under Rule2 and for n=7,11,32 under Rule4, using the stated probabilities. Compare the fraction of trials reaching stable loop patterns, the all-zero absorbing state, and large uncovered-zero states, and compare the final patterns with Figs. 8, 10, 11, and 12. If variant (i) fails while variant (ii) reproduces the figures, the printed formal rule does not support the central claim and a corrigendum is required.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3.5.1 (Noise Injection) specifies the exception that suppresses noise for uncovered zero-cells as E=(E2∧E3)∨Decision(πE), where E_i says a cell at distance i has h1=3. As written, E is true with probability πE for every uncovered zero-cell, regardless of distance, and otherwise requires both distance-2 and distance-3 path neighbors. The explanatory text and Section 4.3 say Rule2 activates 'the condition E=E2∨E3' with πE=0.5 and that uncovered zero-cells are stabilized only at distance 2 or 3 from a path cell. These are different rules. An implementer following the formal definition would suppress noise injection in arbitrary uncovered areas, likely changing the set of absorbing configurations and possibly allowing large zero regions or the all-zero state to persist; an implementer following the prose would reproduce the reported patterns. Because the central claim is about the behavior of this specific CA rule, the rule as printed is underspecified at the exact mechanism that decides whether uncovered zero-cells are tolerated. The lack of a convergence proof is acknowledged by the authors, but this is a more basic correctness/reproducibility issue: the simulations in Section 4 support the corrected version, not necessarily the formula in Section 3.5.1.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper presents a probabilistic, asynchronously updated cellular automaton rule for evolving stable loop patterns on n x n grids with fixed zero boundaries. A loop is a closed path of one-cells whose path cells have exactly two one-neighbors and whose corners are enclosed by a zero hull; a loop pattern is a set of mutually non-touching loops. The rule derives 38 local templates from six overlapping tiles (four corner tiles and two line tiles), adjusts a cell to the center of a matching template, and injects noise when the path condition or other constraints fail. Five rule variants (Rule0 through Rule4) are defined by enabling different noise-injection conditions. The authors report simulations for field sizes n = 3 to 7, selected results for n = 11 and n = 32, and a short discussion of dependencies and open problems, explicitly leaving a convergence proof as future work.","tokens_in":17819,"tokens_out":8682,"duration_ms":88224,"significance":"If the rule is stated precisely and the simulations are reproducible, the paper is a useful constructive demonstration that local template matching plus probabilistic noise injection can assemble non-trivial global structures such as separated loops. The use of overlapping tiles with cover level 3 to encode a hull around a path is elegant, and the explicit enumeration of all small loop patterns for n = 5 is a valuable check. The paper does not provide a theorem or a proof of convergence, and its evidence is experimental; its contribution is a design scheme together with a systematically described set of observed behaviors, not a formal guarantee. The strengths are the clear tile/template construction, the uniform rule schema with five named variants, and the authors' honest listing of open issues.","major_comments":[{"comment":"The formal definition of the exception condition E in Section 3.5.1, namely E = (E2 ∧ E3) ∨ Decision(πE), contradicts the accompanying prose and the Rule2 description in Section 4.3, where the condition is stated as E = E2 ∨ E3 with πE = 0.5. Taken literally, the displayed formula makes Decision(πE) true with probability 0.5 for every uncovered zero-cell regardless of distance from a path cell, because the random decision is independent of E2 and E3. The condition Q0 = (h0 = 0) ∧ NOT(E) would then suppress noise in arbitrary uncovered regions, not only at distance 2 or 3 as the text claims. This changes the set of absorbing configurations and can change which patterns are stabilized. Since the central claim is about the behavior of this specific CA rule, the printed formula must agree exactly with the rule used in the simulations; otherwise the reported patterns in Section 4 are not reproducible from the formal specification.","section":"§3.5.1 and §4.3"},{"comment":"There is a second internal contradiction in the rule specification. Section 3.5.1 states that when several templates match, 'the first hitting template is used for adjustment,' while the Summary in Section 5.1 states that 'If there are several concurrent h0 and h1 hits, then the state adjustment is random.' These two tie-breaking rules are not equivalent, and under asynchronous updating the difference can affect both transient behavior and the resulting stable patterns. The authors should state a single unambiguous tie-breaking rule and confirm which one was implemented in the simulations.","section":"§3.5.1 and §5.1"},{"comment":"Rule4 is described as forbidding configurations with cover level v = 8, but its formal condition is Q1 = (h0 = 8) ∧ Decision(πQ1), where h0 is the number of zero-template hits rather than the cover level. The paper itself notes in Section 4.3 that hit values converge to cover levels only when the pattern stabilizes, so during transients h0 and v can differ. If the CA rule only reads template hits from the local neighborhood, it cannot literally test the cover level v = 8 at the selected cell. The authors should clarify whether v is part of the cell state available to the rule, or replace Q1 with the intended condition and report that the simulations used it.","section":"§3.5.1 and §3.5.2, Rule4"},{"comment":"The abstract's claim that the CA rule can 'securely evolve stable loop patterns' is stronger than the evidence presented. The rule variant is selected as the field size increases (the introduction to Section 4 says the rule complexity is increased from Rule0 to Rule4 as n grows), Rule0 has the all-zero configuration as a fixed point, Rule1 can leave toggling cells, and the statistics in Section 4.8 are based on 300 runs for n ≤ 15 and 100 runs for n = 16 to 20, with no error bars or confidence intervals. For n = 11 and n = 32 only selected patterns are shown, and no success rates are given. A more systematic per-variant, per-size success-rate table, or an explicit qualification that the claim covers only the simulated cases, is needed to support the central claim.","section":"§4, §4.8, and §5.2"}],"minor_comments":[{"comment":"The caption says 'A 2×3 square loop is build by overlapping tiles'; the loop shown is a 3×3 square, and 'build' is a typo for 'built'.","section":"§3.3, Fig. 3 caption"},{"comment":"The sentence about uncovered cells says they have a distance 'orthogonally or vertically' from a path cell; this should be 'orthogonally or diagonally.'","section":"§4.4"},{"comment":"The figures plot averages without error bars or per-point dispersions. Given that the number of runs is stated, adding standard errors would make the reported dependencies easier to interpret, especially since the text speculatively calls the growth exponential.","section":"§4.8, Figs. 15 and 16"},{"comment":"The symbol H is used in the bullet about cover-level threshold but is not formally defined in the displayed formula; please define H and state its value explicitly when Rule4 is discussed.","section":"§3.5.1, Q1"},{"comment":"The condition P2 that 'two consecutive (h0 = 0)-cells at distance 2 and 3' exist is described in words but not written in set notation; a short formal definition would remove ambiguity about which directions and which cells are included.","section":"§3.5.1, P2"}],"recommendation":"major_revision","confidential_remarks":"The contradictions in the formal rule definition are likely fixable, but they are not merely typographical for the purposes of this paper because Section 4's simulation evidence is the only support for the central claim. The authors should provide exact pseudocode or the simulation source, correct the E and Q1 conditions, and unify the tie-breaking statement. I would not reject the paper on these grounds, but as printed the rule is not fully reproducible. The paper fits the scope of a journal on cellular automata and the tile-overlap idea is worth publishing once the rule definition is made precise."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Rolf and Mariusz extend their template-matching probabilistic CA approach from earlier point, domino, and loop-on-torus work to loop patterns on fixed zero-boundary grids. The genuinely new pieces are the simpler corner/line tile set, five rule variants, the enumeration of 5x5 loop patterns (24 up to symmetry, 33 with shifts), and the field-size statistics on convergence time, uncovered cells, and loop density. The simulations are numerous and the tile design is coherent. That is real work, and I believe the experimental claims as far as they go.\n\nThe main soft spot is the one the stress-test note caught: §3.5.1 defines E=(E2∧E3)∨Decision(πE), while §4.3 and the rule descriptions consistently use E=E2∨E3, with E2/E3 individually stabilizing uncovered zero-cells at distance 2 or 3 from a path cell. This is not a minor typo. E is the exception that suppresses noise injection for uncovered zero-cells, so changing E changes the absorbing configurations. An implementer following the formal formula could suppress noise in arbitrary uncovered areas and see different fixed points; only following the prose reproduces the reported patterns. The paper needs an explicit correction and, ideally, pseudocode or shipped code for the exact rule used.\n\nThe other weaknesses are proportionate. Rule variants are chosen as the field size grows, probabilities are hand-set to 0.5, no error bars or raw data are provided, and no code is shipped. The authors themselves list convergence as an open issue, so the abstract's \"securely evolve\" is stronger than what is demonstrated. I would soften that to \"empirically converges for tested sizes.\" The related-work section leans on the authors' own series, but that is understandable because this is a direct extension, and the claimed difference from the earlier torus paper is stated clearly.\n\nWho gets value: readers interested in CA-based pattern generation, synthetic reticulated or crack-like structures, and unconventional-computing motifs will find a concrete baseline and a clear method. It is an engineering/experimental contribution, not a theorem paper. It deserves a serious referee, not a desk reject, but the review should ask for: (1) a corrected E definition, (2) code or detailed pseudocode plus run statistics, and (3) a carefully scoped statement of what is proven versus empirically observed. With those changes this becomes a solid niche contribution.","headline":"A useful empirical extension of the authors' template-plus-noise CA scheme to fixed-boundary loop patterns, but the printed rule has a formal inconsistency in the E condition that needs correction before the results are fully reproducible.","tokens_in":18199,"tokens_out":2995,"would_cite":false,"duration_ms":32794,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37B15","68Q80"],"pacs":[],"model":"deepseek-v4-flash","headline":"A probabilistic cellular-automaton rule built from six overlapping tiles and template-matching noise injection can evolve stable loop patterns on fixed-boundary grids up to 32×32.","keywords":["cellular automata","loop patterns","overlapping tiles","templates","noise injection","probabilistic cellular automaton","asynchronous updating","pattern formation"],"falsifier":"Run Rule4 on a grid of, say, n=200 from many random starting patterns and let it run for a large fixed number of updates; if any run ends with cells still flipping or with no loop at all, rather than a configuration in which every cell keeps its state, then the claim that the rule securely evolves stable loops is falsified for that size.","tokens_in":17181,"feed_emoji":"🔁","tokens_out":6623,"duration_ms":62073,"temperature":0.7,"pith_summary":"The paper claims that a local, probabilistic, asynchronously updated cellular-automaton rule can form stable loop patterns: closed paths of one-cells, each cell with exactly two orthogonal one-neighbors, wrapped in a hull of zero-cells, on an n×n grid with fixed zero borders. The rule tests every site against a set of 38 templates derived from six overlapping tiles (four corner shapes and two straight segments), and injects random noise wherever the local pattern is missing or violates the 'loop path condition.' Five rule variants, from 'path-only' to 'forbid 3×3 loops,' select which loop patterns are reachable; simulations show convergence to stable patterns for sizes from 3×3 to 32×32, with the all-zero fixed point avoided by the more complex variants. The interest is that global reticulated structures—loop networks of the kind seen in cracks, veins, and street patterns—can be produced by purely local rules without any global coordination.","feed_headline":"Overlapping tiles let a cellular automaton grow stable loops","feed_subtitle":"Template matching plus injected noise forms closed non-touching paths on grids up to 32×32.","key_machinery":"The load-bearing object is the tile set: four 3×3 corner tiles (A0–A3) and two line tiles (B0–B1), each with a central anchor pixel. From these, 38 templates are derived by shifting each tile so that each valid pixel becomes the template center. The CA rule runs three steps per cell—TEST (count matching templates), ADJUST (set the cell to the center value of the first hit), and NOISE INJECTION (flip the cell randomly under conditions such as h1≠3 for path cells or h0=0 for uncovered zeros). The key mechanism is that the required overlap level v=3 enforces the loop path condition: three tiles' one-pixels must overlap at each path site, so a line or corner is surrounded by zeroes and a closed path has a zero hull. The five rule variants are simply different subsets of noise-injection conditions selected from the general rule.","core_discovery":"The central claim is that one can solve the loop-pattern formation problem by combining overlapping tiles with a template-matching, noise-injection CA rule. The tiles are chosen so that maximal overlap (cover level v=3) realizes the loop path condition: every one-cell has two one-neighbors in NESW and every convex corner has an outer-diagonal zero, so loops are closed, non-touching, and separated by a hull. Hits h0 and h1 count zero-centered and one-centered template matches; if any hit exists the cell adjusts to the template's center, and if not (or if the path condition or other selected conditions fail) random noise is injected. The paper demonstrates by simulation that the general rule, specialized as Rule0 through Rule4, evolves stable loop patterns for field sizes 3×3 through 32×32, including a closed space-filling curve for 11×11, and it leaves as an explicit open question whether convergence to stable loops can be proven for all field sizes.","pith_inferences":["Editorial inference: the same template-match-plus-noise recipe should work for any locally checkable tiling constraint with a finite overlap set; the domino and sensor-coverage cases already appear in the paper, but the loop case suggests a general design rule: pick tiles whose maximal overlap encodes the constraint, then inject noise on every violation.","Editorial inference: if convergence is eventually proven, the rule becomes a distributed asynchronous algorithm for constructing loop and space-filling structures, which would connect to self-assembly and programmable matter; a testable extension is to run the same rule under synchronous updating to see whether the noise-injection conditions still converge or instead produce persistent oscillation","Editorial inference: the exceptional peaks at n=9 and n=16 in both convergence time and loop density hint that the tile geometry resonates with certain grid sizes; one could test this by varying the tile shapes and checking whether the peaks shift or disappear."],"forward_implications":["If the central claim is right, then any tested grid size admits a local rule that reaches a stable pattern containing at least one closed loop, while avoiding the all-zero fixed point once Rule1 or a stronger variant is used.","The five rule variants give graded control over the pattern class: Rule0 allows all loop patterns plus the all-zero fixed point; Rule1 eliminates uncovered zero cells but permits toggling; Rule2 stabilizes uncovered cells at distance 2 or 3; Rule3 excludes patterns with pairs of uncovered cells at distance 2 and 3; Rule4 forbids 3×3 loops by injecting noise at cover level v=8.","For n=11 with Rule4, closed space-filling curves appear among the evolved patterns, and the paper notes such curves can be constructed for fields of size (4k−1)×(4k−1).","The average number of micro time-steps per cell needed to reach a stable pattern grows superlinearly with the number of cells, and the paper suggests the growth is exponential, which sets a practical limit on large grid sizes.","When Rule4 is applied to a 32×32 grid, patterns with 2 to 11 loops are observed, and single-loop patterns are rare, indicating the rule's output distribution is biased toward multi-loop configurations."],"supporting_citations":[{"why":"Establishes the prior loop-pattern case on the torus and the tile-search tool, which the current tile set simplifies; the current paper extends it to fixed boundaries.","marker":"[2]"},{"why":"Supplies the probabilistic CA with noise injection used to form point patterns, the method the loop rule builds on.","marker":"[3]"},{"why":"Supplies the template-based probabilistic CA for placing maximal dominoes, including the exponential-time behavior cited for large fields.","marker":"[4]"},{"why":"Shows the same probabilistic CA approach solving sensor-network coverage, supporting the general pattern-formation framework.","marker":"[5]"},{"why":"Shows noise injection controlling the space between dominoes, the basis for the distance-2/3 uncovered-cell conditions.","marker":"[7]"},{"why":"Provides the theoretical treatment of overlapping one-dimensional tiles that motivates the overlap definition used here.","marker":"[10]"}],"fun_headline_variants":["Overlapping tiles help CA form stable, non-touching loops","Noise-assisted CA rule yields closed loop patterns up to 32x32","Loop patterns from cellular automata with overlapping tile rules","CA tiles plus noise injection: stable loop generation","Stable loop patterns in 2D CA via tile-match and noise"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The rule must settle into a stable loop pattern instead of collapsing to all zeros or toggling forever; this is observed in simulations for the tested sizes but not proved.","fun_headline_variants_meta":{"raw":{"variants":["Overlapping tiles help CA form stable, non-touching loops","Noise-assisted CA rule yields closed loop patterns up to 32x32","Loop patterns from cellular automata with overlapping tile rules","CA tiles plus noise injection: stable loop generation","Stable loop patterns in 2D CA via tile-match and noise"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000306,"raw_usage":{"total_tokens":1730,"prompt_tokens":897,"completion_tokens":833,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":513,"completion_tokens_details":{"reasoning_tokens":746}},"tokens_in":513,"tokens_out":833,"duration_ms":8260,"temperature":1.0,"reasoning_tokens":746,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T21:25:27.071775+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run Rule4 on a grid of, say, n=200 from many random starting patterns and let it run for a large fixed number of updates; if any run ends with cells still flipping or with no loop at all, rather than a configuration in which every cell keeps its state, then the claim that the rule securely evolves stable loops is falsified for that size.","supporting_citations":[{"cited_title":"Generating Loop Patterns with a Genetic Algorithm and a Probabilistic Cellular Automata Rule","cited_arxiv_id":null,"evidence_quote":"Establishes the prior loop-pattern case on the torus and the tile-search tool, which the current tile set simplifies; the current paper extends it to fixed boundaries."},{"cited_title":"A cellular automata rule placing a maximal number of dominoes in the square and diamond.J","cited_arxiv_id":null,"evidence_quote":"Supplies the template-based probabilistic CA for placing maximal dominoes, including the exponential-time behavior cited for large fields."},{"cited_title":"Cellular automata rules solving the wireless sensor network coverage problem.Nat","cited_arxiv_id":null,"evidence_quote":"Shows the same probabilistic CA approach solving sensor-network coverage, supporting the general pattern-formation framework."},{"cited_title":"Minimal Covering of the Space by Domino Tiles","cited_arxiv_id":null,"evidence_quote":"Shows noise injection controlling the space between dominoes, the basis for the distance-2/3 uncovered-cell conditions."},{"cited_title":"On languages of one-dimensional overlapping tiles","cited_arxiv_id":null,"evidence_quote":"Provides the theoretical treatment of overlapping one-dimensional tiles that motivates the overlap definition used here."}],"review_version":1}