Pith. sign in

REVIEW 4 major objections 5 minor 12 references

Sparse ternary interventions steer binary triangular-lattice automata into dense, long-lived states and reset—rather than abolish—their eight-step binary crisis clock, placing the first major collapse at exactly 64 steps after the last inte

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-02 09:00 UTC pith:MXZDECNV

load-bearing objection Solid enumeration behind the density-gain claim, but the τ=64 crisis-clock reset is a likely selection artifact and should be treated as a hypothesis, not a result. the 4 major comments →

arxiv 2607.02207 v2 pith:MXZDECNV submitted 2026-07-02 nlin.CG nlin.CD

The Binary Crisis Clock: Controlled by Sparse Ternary Interventions

Ma{\l}gorzata Nowak-K\c{e}pczyk This is my paper
classification nlin.CG nlin.CD MSC 37B1568Q80
keywords modular discrete Laplacianbinary crisis clockternary interventionstriangular latticedensity collapseseed-return dynamicstrajectory selectionadditive cellular automata
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Modular Laplacian automata on a triangular lattice, running with a purely binary rule, repeatedly expand and then collapse: every eight steps the configuration decomposes into translated copies of its initial seed. The paper shows this 'binary crisis clock' is common to all tested masks and seeds, with the mask fixing the large-scale envelope and the seed–mask symmetry overlap fixing the internal organization. It then shows that replacing just two or three early updates with ternary (mod 3) steps—after which the rule reverts to binary—selects dramatically denser trajectories, lifting mean density from about 0.068 to about 0.47 over the 33–80 window. The ternary phase does not eliminate the clock: its phase is reset, and all 100 highest-ranked trajectories exhibit a detected crisis at τ=64, measured from the final intervention, with 95 showing a major crisis there that is the first major post-intervention collapse. The picture that emerges is a division of labor: geometry fixes the family of admissible morphologies, while sparse, precisely timed interventions choose a favorable trajectory within that family.

Core claim

On its own terms, the paper establishes a separation between spatial and developmental control in modular Laplacian evolution. For purely binary dynamics, every tested seed–mask combination on the triangular lattice displays the same temporal skeleton: growth phases end in density collapses at t=8k, where the configuration returns to translated copies of the initial seed; the neighborhood mask determines the geometric envelope, and the common spatial symmetry subgroup of mask and seed determines which symmetries persist. Replacing a small number of early binary updates with ternary updates acts as a developmental intervention: two or three well-placed interventions suffice to redirect the su

What carries the argument

The central object is the 'binary crisis clock' — the regular sequence of density collapses and seed-return events at t=8k (k=1,2,3,...) that the paper documents as a common mechanism across masks and seeds on the triangular lattice. The argument is carried by the modular Laplacian update rule (the next state is the integer sum of neighbour-minus-centre differences, reduced modulo 2 or 3), by the exhaustive/uniform search over intervention schedules ranked with the score J=ρ−0.5σρ, and by the crisis-detection algorithm (a local-minimum test with window w=4 and normalized score above κ=1.0). The clock gives the temporal reference against which 'reset' and 'first major collapse at τ=64' are de

Load-bearing premise

The load-bearing premise is that the eight-step binary crisis clock is a genuine, universal property of triangular-lattice modular Laplacian dynamics, not an artifact of the five tested masks, the few seeds, the finite simulation window, or the hand-set crisis detector; the paper explicitly states (Open Questions) that no complete analytical explanation of the rhythm is yet available.

What would settle it

Run the binary rule on a triangular lattice with a mask and seed not among the five masks and seeds tested here (for example a random asymmetric mask with a non-symmetric finite seed) for at least 160 iterations and inspect whether density minima appear at every t=8k; one tested combination lacking a collapse at any t=8k would refute the claimed universality. Similarly, re-running the paper's post-intervention crisis analysis with a wider detector window (w=6) or a lower threshold (κ=0.5) should leave the τ=64 concentration and the 8k-rhythm peaks essentially unchanged if the clock and its res

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If the eight-step crisis clock is universal on triangular lattices, any future study of these automata must treat t=8k as preferred times, extending the seed-return behaviour previously reported on square lattices.
  • The near-plateau of attainable density (≈0.47–0.49 for three or more interventions) and the enlarged high-density basin imply that a minimal recipe of two or three well-placed mod-3 steps suffices for trajectory shaping, and extra interventions mainly buy robustness.
  • Because the first major collapse after shaping occurs at exactly 64 binary steps, post-intervention dynamics become schedulable: the system stays dense and relatively crisis-free during the first 64 binary iterations.
  • The decoupling of mask-determined morphology and intervention-selected fate suggests that, in such additive automata, spatial and temporal control are separable design dimensions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A natural extension is to test whether the 'reset' delay is always a power-of-two multiple of 8 (here 8×8): if so, one could predict the next major collapse after any successful reset from the intervention's algebraic phase, without rerunning the automaton.
  • The crisis detector uses fixed parameters (w=4, κ=1.0, depth thresholds 0.05 and 0.40) with no robustness scan, so the exact counts (95/100, median depth 0.095) are parameter-dependent; the qualitative claim that the clock survives and is phase-reset appears more stable than those numbers.
  • The density plateau near 0.47–0.49 with tiny fluctuations may represent an upper bound for any sparse ternary shaping on the full hexagonal mask; intervening after t=32 or using n3>>10 could reveal whether this is an absolute ceiling for the mask.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper studies modular discrete-Laplacian cellular automata on triangular lattices. It first reports that, in all tested seed–mask combinations, purely binary dynamics exhibit a universal 'binary crisis clock': density collapses and seed-return events at t=8k, k=1,2,3,.... It then investigates sparse ternary interventions (replacing a few binary updates with modulus-3 updates during t≤32) as developmental switches that select denser, more stable binary trajectories. For n3=1,2,3 the intervention schedules are exhaustively enumerated (32, 496, 4960); for n3=4,...,10, 5000 unique schedules are sampled. The highest mean density over [33,80] rises from 0.068 (binary) to about 0.47 with two to three interventions and plateaus near 0.49. The final experiment analyzes the 100 highest-ranked and 100 random-control trajectories extended to t=160, reporting that the binary crisis rhythm persists relative to the final ternary intervention, with a prominent first major collapse at τ=64 and shallower intermediate crises than controls.

Significance. If the central claims hold, the paper offers a striking example of sparse, time-localized control selecting among long-lived trajectories in a purely deterministic additive cellular automaton, and it generalizes earlier square-lattice observations to triangular lattices. The density-gain claim is supported by complete enumeration for n3=1,2,3 and by internally consistent summary tables; the paper is also explicit that n3≥4 results are sampled optima and that no analytical proof is given for the 8k rhythm. These are real strengths. However, the headline 'crisis-clock reset' claim is measured on the same trajectories used for optimization and with hand-set detector parameters, making it vulnerable to selection artifacts; the paper's own Appendix A concedes the absence of post-selection significance tests. The risk is that the reset/attenuation conclusion may be a tautology of the ranking criterion rather than an independent dynamical property.

major comments (4)
  1. [§3.3.1 and §2.3] The central claim that the first major post-intervention collapse occurs at τ=64 is confounded by the selection procedure. The 100 highest-ranked trajectories are chosen by J=ρ33:80 − 0.5σ_ρ,33:80 over the evaluation window [33,80]. Since the final ternary intervention occurs at tl3 ≤ 32, a collapse at τ=64 translates to an absolute time tl3+64 between 87 and 96, i.e., outside the evaluation interval. High rank by construction requires high mean density and low fluctuation over [33,80], which excludes a major collapse before t=80. The 'first major post-intervention collapse' is therefore expected to occur after the evaluation window, and the concentration at τ=64 is measured on the very trajectories that J selected. To establish a genuine clock reset, the authors should compare against controls matched on final intervention time and on comparable density/fluctuation, or should test the τ
  2. [§2.2 and Table 3] The crisis detector parameters (w=4, κ=1.0, ε=10^-9, D_min=0.05, D_major=0.40) are fixed by hand and no robustness scan is reported. The qualitative conclusions—median depth 0.095 vs. 0.161 vs. 0.965, the count of detected crises, and especially the universality of the τ=64 event—may depend on these choices. The random-control group is matched only on n3, not on final intervention time or post-intervention density, so the comparison between highest-ranked and control groups is further confounded. The authors should vary the detector parameters and either match controls more tightly or control for density and tl3 in the comparison.
  3. [§3.1 and §4] The existence of a universal 'binary crisis clock' with period 8k is a load-bearing premise for the reset interpretation, but it is supported only by the finite set of masks and seeds tested in Experiment 1, and the paper explicitly concedes that no analytical explanation exists (Open Questions). The claim that the clock persists after intervention and is merely 'reset' requires the pre-intervention clock to be robustly established, not just asserted from a handful of simulations. The authors should either provide an algebraic argument for the 8k period (or at least for the tested masks) or clearly restrict the universality claim to the tested parameter range, acknowledging that the τ=64 result is conditional on that empirical premise.
  4. [Appendix A and §3.2.4] The paper's conclusion that 'timing is critical' rests largely on descriptive enrichment of preferred intervention motifs (e.g., (8,9), (16,17), (24,25) in Table A.4) with no post-selection significance tests, as the appendix itself states. Without correction for the multiple comparisons inherent in exhaustive enumeration and in selecting the top-50 schedules per n3, the apparent preference for 8k/8k+1 windows could be a chance fluctuation. The authors should report empirical p-values or permutation-based significance for these motif enrichments, or temper the claim accordingly.
minor comments (5)
  1. [Abstract and §3.3.1] The abstract states 'All 100 highest-ranked trajectories exhibited a detected crisis at τ=64' but does not specify how the 100 trajectories are composed across n3 strata. §2.4 says each n3 stratum contains 'the same number of trajectories as the highest-ranked group,' which is circular; please clarify the selection rule (e.g., top 10 per n3 for n3=1,...,10).
  2. [§2.2] The definition of σloc(t) uses ρ(t−w),...,ρ(t+w), but boundary handling is not specified for t<w or t+w beyond the simulation horizon. This is relevant for the crisis score near the start and end of a trajectory.
  3. [Figure 1/2] Figures 1 and 2 are referenced but not shown in the text; include the seed and mask diagrams or a link to supplementary material so the reader can follow the tests.
  4. [Table 3] The binary reference row is a single trajectory; reporting medians and percentiles for a one-point sample is misleading. Present the binary reference as a deterministic baseline with no distributional statistics, or include multiple binary seeds to form a reference distribution.
  5. [§4 / Open Questions] The paper is honest about the missing analytical explanation for τ=64, which is good, but the phrase 'resets the phase' in the conclusions is stronger than the evidence warrants given the points in the major comments. Consider softening the causal language throughout.

Circularity Check

1 steps flagged

Partial circularity: the crisis-attenuation claim is partly built into the trajectory-ranking score; the density-gain and timing results are independent.

specific steps
  1. fitted input called prediction [§2.3 (ranking score J) and §3.3.2 / Table 3 (crisis-depth comparison)]
    "Candidate schedules were evaluated over the post-developmental interval Ieval ={33,...,80}. For each trajectory, we calculated the mean density ρ33:80, its standard deviation σρ,33:80, and the ranking score J= ρ33:80−0.50σ ρ,33:80. ... The median relative crisis depth was0.095in the highest-ranked trajectories, compared with0.161in the n3-matched random controls and0.965in the binary reference."

    The highest-ranked group is defined by maximizing J, which explicitly penalizes density fluctuations over t=33..80. Crisis depth D is a local-density-drop measure in the same observation interval, so selecting for low σ automatically selects for shallower crises there. The reported attenuation ('median depth 0.095 vs 0.161') is therefore partly a restatement of the selection score, not an independent dynamical effect of ternary shaping. The random controls are matched only on n3, not on mean density or final intervention time, so the comparison cannot separate the selection constraint from a genuine mechanism. The paper's own Appendix A concedes that no post-selection significance tests are reported.

full rationale

The paper's main density-gain claim is a direct enumeration result: all schedules with n3=1,2,3 were exhaustively enumerated, so the finding that two or three well-placed interventions reach most of the attainable density is not circular. The 'timing is critical' claim is also descriptive of the ranked search landscape, not a prediction derived from the same optimization in a forced way. The binary crisis clock itself is asserted from simulations without an analytical proof, and the paper explicitly says no complete explanation exists; that is a weakness in evidence, but not circularity. The clearest circular element is the crisis-attenuation comparison: trajectories are selected to maximize ρ - 0.5σ over 33..80, and then the same selected group is said to have shallower crises than controls. Since D is measured over the same post-intervention window (including 33..80), low-fluctuation selection directly induces lower crisis depth; the control group is not matched on density or final intervention time, so the attenuation is at least partly constructed by the selection score. The τ=64 reset is not simply forced by the evaluation window: the window ends at t=80 and final intervention times vary (23-32), so a constant relative delay of 64 corresponds to absolute times 87-96 and is an independent, though unexplained, observation. Overall, one central claim (crisis attenuation) is partially circular by construction, while the density and timing claims retain independent content.

Axiom & Free-Parameter Ledger

4 free parameters · 4 axioms · 1 invented entities

The central claims rest on four hand-set numerical choices (crisis detector thresholds, ranking weight, intervention window, evaluation window) and on four background assumptions, the most load-bearing being the universality of the 8k seed-return rhythm, which the paper concedes it cannot explain. The only invented conceptual entity is the 'binary crisis clock,' which is a descriptive framing rather than a new physical object.

free parameters (4)
  • Crisis detector thresholds (w=4, κ=1.0, ε=1e-9, D_min=0.05, D_major=0.40) = w=4, κ=1.0, ε=1e-9, D_min=0.05, D_major=0.40
    Hand-set in §2.2. Every crisis claim in the paper—the 8k rhythm, the τ=64 first major collapse, and the attenuation statistics in §3.3—is defined through this detector. No robustness scan over detector parameters is reported.
  • Ranking weight 0.50 in J = ρ − 0.5σ_ρ = 0.50
    Chosen by hand in §2.3 to combine mean density and fluctuation. The 'highest-ranked' trajectory sets on which all §3.3 conclusions rest are defined by this exact weight; a different weight could select different trajectories and change the crisis statistics.
  • Developmental intervention interval T ⊆ {1,...,32} = t ≤ 32
    Interventions are restricted to the first 32 steps (§2.3). The reported density gains and the τ=64 reset time are conditional on this window; no sensitivity to the window boundary is analyzed.
  • Evaluation window Ieval = {33,...,80} and extended horizon t=160 = 33..80; 160
    The headline density numbers (0.068 → 0.47) are averages over steps 33–80 only; crisis analysis extends to t=160. Both bounds are arbitrary choices, and no horizon-sensitivity analysis is given.
axioms (4)
  • domain assumption The 8k binary seed-return rhythm (t=8k, k=1,2,3,...) is a genuine universal mechanism for triangular-lattice modular Laplacian dynamics
    Asserted from finitely many simulations in §3.1 ('The corrected simulations revealed a common binary mechanism across all tested seed–mask combinations'). The paper itself states no analytical proof exists (§4 Open Questions: 'a complete analytical explanation of the eight-step binary crisis rhythm on triangular lattices is still missing'). All Experiment 3 'clock reset' claims depend on this.
  • domain assumption The five tested masks (hex6, quadra, penta, tristar, tristar_rev) and seeds are representative of triangular-lattice behavior
    The claimed separation—'geometry determines the family of admissible morphologies, interventions select trajectories'—is induced from these five masks and a few seeds (§3.1). No broader sampling of masks is provided.
  • standard math Mask symmetry commutativity L_M(gu) = g L_M(u) for g ∈ G_M
    Stated in §3.1.1; standard equivariance of the discrete Laplacian under lattice symmetries, used to explain which seed–mask symmetries are preserved during evolution.
  • domain assumption Density normalized by the reachable set |R_t| is a valid measure of dynamical growth
    §2.1.1 normalizes active cells by R_t, the set reachable in ≤ t steps (with |R_t| = 1 + 3t(t+1) for hex6 and the one-point seed). Because |R_t| grows quadratically, density can fall even when the active set merely grows sub-quadratically; 'crisis depth' therefore conflates geometric dilution with actual collapse. This normalization shapes all reported crisis statistics.
invented entities (1)
  • Binary crisis clock (and binary crisis rhythm) no independent evidence
    purpose: Organizing framework: a phase structure that determines the timing of recurrent density collapses at multiples of 8 steps and that can be 'reset' by ternary interventions
    The paper treats the 8k rhythm as a mechanism ('the binary crisis clock is reset', Abstract; §3.3). It is an empirical regularity in the tested simulations, not an independently falsifiable handle: no out-of-sample prediction (e.g., a new mask must exhibit returns at 8k) is made and then tested, and the 'reset' claim is measured on the same trajectories the optimizer selected.

pith-pipeline@v1.3.0-alltime-deepseek · 10213 in / 19175 out tokens · 189908 ms · 2026-08-02T09:00:25.412132+00:00 · methodology

0 comments
read the original abstract

We investigate modular Laplacian automata on triangular lattices with evolution governed by binary and ternary moduli. Extending previous studies on square lattices, we examine how lattice geometry influences long-term growth, density, fragmentation, and the emergence of self-similar structures. We further investigate whether sparse ternary interventions can stabilize predominantly binary dynamics. The experiments reveal that mask geometry is the primary determinant of large-scale morphology. Full hexagonal masks generate recurrent density crises and fragmentation, whereas triangular masks support persistent growth and reveal a threshold phenomenon governed by growth-capable nuclei. Although seed symmetry influences transient behaviour, the asymptotic morphology is inherited mainly from the mask. To control binary fragmentation, we investigate sparse developmental ternary perturbations in which a small number of carefully timed occurrences of modulus 3 are inserted into an otherwise binary sequence. A Monte Carlo optimization demonstrates that as few as three interventions are sufficient to redirect the subsequent binary evolution toward substantially denser carpet-like configurations. The effectiveness of this strategy depends primarily on the timing of the interventions rather than on their number. Analysis of the post-intervention dynamics shows that ternary shaping does not replace binary evolution. Instead, it produces denser self-similar structures, substantially reduces crisis depth, and resets the phase of the binary crisis clock. The results suggest that geometry determines the family of admissible morphologies, whereas sparse developmental perturbations select favourable long-term trajectories within that family.

Figures

Figures reproduced from arXiv: 2607.02207 by Ma{\l}gorzata Nowak-K\c{e}pczyk.

Figure 1
Figure 1. Figure 1: Examples of seed configurations. 1.1.1. Automaton Evolution Let G = (V, E) denote the underlying lattice graph„ where V is the set of lattice points and E is determined by the chosen neighborhood structure. For p ∈ V , we denote by Ne(p) = {g ∈ V : (p, g) ∈ E} the set of neighboring vertices of p. Consider a sequence S = (k1, k2, k3, . . .), ki ∈ {2, 3}. The discrete graph Laplacian applied to νi is define… view at source ↗
Figure 2
Figure 2. Figure 2: Examples of neighborhoods. 1.1.2. Modular Arithmetic and Iterative Extensions The sequence S generalizes purely binary dynamics (S = (2, 2, 2, . . .)) to arbitrary mix￾tures of binary and ternary updates. Constant sequences correspond to purely binary or purely ternary evolution, whereas mixed sequences describe controlled alternations of the two moduli1 . 1.2. Quantitative measures To compare different la… view at source ↗
Figure 3
Figure 3. Figure 3: Influence of seed orientation on binary evolution for the asymmetric [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Influence of intervention timing on binary evolution generated by the one-point seed and the full [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 4
Figure 4. Figure 4: Influence of intervention timing relative to the binary seed-return cycle for the one-point seed and [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Experiment showing the effect of a single ternary insertion on the self-similar motif family [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 5
Figure 5. Figure 5: Redirection of a recurrent binary motif hierarchy by a single ternary intervention for the [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Best, worst, and random four-intervention schedules. The pale curves show 30 random controls. [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 6
Figure 6. Figure 6: Highest-ranked density outcomes as a function of the number of ternary interventions. The left [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Mean density and density fluctuation measured in the post-intervention window ( [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 7
Figure 7. Figure 7: Density trajectories for selected four-intervention schedules. Thick solid curves show the four [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The distributions reveal a rapid transition from binary collapse to a persistent carpet-like regime. [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 8
Figure 8. Figure 8: Mean post-intervention density versus density fluctuation for [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Mean density versus density fluctuation for 5000 Monte Carlo schedules with four ternary inter [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The highest-ranked trajectories exhibit pronounced concentrations near multiples [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 10
Figure 10. Figure 10: Distribution of crisis times measured relative to the last ternary insertion, [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
Figure 9
Figure 9. Figure 9: Survival distributions of mean post-intervention density for increasing numbers of ternary inter [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Distribution of post-intervention crisis times measured relative to the final ternary intervention, [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗

Review history (2 revisions) →

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

12 extracted references · 6 canonical work pages

  1. [1]

    Cornelius, William L

    Sean P. Cornelius, William L. Kath, and Adilson E. Motter. Realistic control of network dynamics.Nature Communications, 4:1942,

    1942

  2. [1952]

    George M

    doi: 10.1098/rstb.1952.0012. George M. Whitesides and Bartosz Grzybowski. Self-assembly at all scales.Science, 295 (5564):2418–2421,

    arXiv 1952

  3. [1984]

    Małgorzata Nowak-Kępczyk

    doi: 10.1007/ BF01223745. Małgorzata Nowak-Kępczyk. Frobenius revivals in laplacian cellular automata: Chaos, repli- cation, and reversible encoding.arXiv preprint, 2025a. URLhttps://arxiv.org/abs/ 2511.17389. Małgorzata Nowak-Kępczyk. Stabilization and regaining periodicity in modular laplacian dynamics.arXiv preprint, 2025b. URLhttps://arxiv.org/abs/250...

    arXiv

  4. [1985]

    Mihai Prunescu

    doi: 10.1007/BF01010423. Mihai Prunescu. Self-similar carpets over finite fields.European Journal of Combinatorics, 30(4):866–878,

    work page doi:10.1007/bf01010423

  5. [1992]

    doi: 10.1016/0022-0000(92)90007-6. Alan M. Turing. The chemical basis of morphogenesis.Philosophical Transactions of the Royal Society of London. Series B, 237(641):37–72,

    work page doi:10.1016/0022-0000(92)90007-6

  6. [2002]

    doi: 10.1126/science.1070821. 21

    doi:10.1126/science.1070821

  7. [2009]

    Osamu Suzuki, Małgorzata Nowak-Kępczyk, and Michał Ławrynowicz

    doi: 10.1016/j.ejc.2008.08.002. Osamu Suzuki, Małgorzata Nowak-Kępczyk, and Michał Ławrynowicz. Binary and ternary structures of the evolutions in the universe (2×3×2×···-world) v. non-commutative galois theory of evolution. In(Hyper)Complex Seminar 2021 in Memoriam of Prof. Julian Ławrynowicz, chapter II. Bull. Soc. Sci. Lettres Łódź,

    work page doi:10.1016/j.ejc.2008.08.002 2008

  8. [2010]

    Christiane Hadlich, Klaus Gürlebeck, and Osamu Suzuki

    doi: 10.46298/dmtcs.2759. Christiane Hadlich, Klaus Gürlebeck, and Osamu Suzuki. Fixed point theorem, peri- odicity theorem and binomial and trinomial sequences for iteration dynamical systems of discrete laplacians on the plane lattice.Research Institute for Mathematical Sci- ences Kôkyûroku, 1742:68–81,

    work page doi:10.46298/dmtcs.2759

  9. [2013]

    Johannes Gütschow, Vincent Nesme, and Reinhard F

    doi: 10.1038/ncomms2939. Johannes Gütschow, Vincent Nesme, and Reinhard F. Werner. The fractal structure of cel- lular automata on abelian groups.Discrete Mathematics & Theoretical Computer Science, DMTCS Proceedings vol. AL, Automata 2010:5,

    doi:10.1038/ncomms2939 2010

  10. [2014]

    Olivier Martin, Andrew M

    doi: 10.14492/hokmj/1416837570. Olivier Martin, Andrew M. Odlyzko, and Stephen Wolfram. Algebraic properties of cellular automata.Communications in Mathematical Physics, 93(2):219–258,

    arXiv

  11. [2018]

    Norman H

    doi: 10.1142/S0219887818501876. Norman H. Packard and Stephen Wolfram. Two-dimensional cellular automata.Journal of Statistical Physics, 38:901–946,

    work page doi:10.1142/s0219887818501876

  12. [2026]

    20 Małgorzata Nowak-Kępczyk, Julian Ławrynowicz, Osamu Suzuki, and Andrzej Niemczynow- icz

    doi: 10.1142/S0218348X26500714. 20 Małgorzata Nowak-Kępczyk, Julian Ławrynowicz, Osamu Suzuki, and Andrzej Niemczynow- icz. Fractals and chaos related to ising–onsager lattices. ternary approach versus binary approach.International Journal of Geometric Methods in Modern Physics, 15(11),

    work page doi:10.1142/s0218348x26500714