Pith. sign in

REVIEW 3 major objections 5 minor 26 references

Predictability can be dynamically constructed in deterministic systems

T0 review · 3 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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

Pith's one-line read In a fully deterministic cellular automaton, the final pattern is fixed from the start yet unpredictable from it, because the topological structures that make the outcome legible are constructed only as the dynamics unfold.

desk verdict Solid computational study with a genuine topological mechanism; the late-time predictability evidence is real but partly retrospective, so the 'forward prediction' claim needs softening. read the letter →

arxiv 2604.01088 v2 pith:2JIM5ZRT submitted 2026-04-01 physics.bio-ph cond-mat.stat-mechnlin.PS

Lars Koopmans , Elinor M. Kay , Hyun Youk This is my paper · ORCID
classification physics.bio-phcond-mat.stat-mechnlin.PS
keywords deterministicpredictabilitycellularautomataself-organizationtopologicaldefectsvorticesnon-contractibleloopsmachinelearningphasefield
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that in deterministic, non-chaotic many-body systems, the future can be fixed yet practically unpredictable, because the structures that make the future legible—topological defects such as vortices and loops—do not exist at the start and are built during the dynamics. Using a generalized cellular automaton of secreting and sensing cells, the authors show that six families of machine-learning models, trained on up to 850,000 runs, cannot tell from the initial 14x14 configuration whether a run will end static or dynamic. By recoding cell states as a discrete phase field, they uncover charged vortices connected by strings that can wrap the torus as non-contractible loops; the loss of these loops announces vortex annihilation and hence static or rectilinear-wave outcomes, while spiral waves form abruptly with no advance signature. The load-bearing conclusion is that predictability is dynamically constructed and arrives late and asymmetrically, closing differently for annihilation-bound versus spiral-bound trajectories. If right, this reframes self-organization as the physical construction of predictive information, not the unfolding of a pre-encoded outcome.

What carries the argument

The key machinery is the recoding of the four cell states as phase vectors (0, π/2, π, 3π/2) on a toroidal lattice, turning the dynamics into a discrete phase field. This reveals three classes of vortices—+1, −1, and 0—whose total topological charge is conserved even though no update rule states it. Each vortex pair is threaded by same-state strings; a string that winds around the torus is a non-contractible loop (NCL). NCL strings carry the predictive signal: a sliding-window decline in their count announces imminent annihilation (static or rectilinear fate), while a near-one count accompanies spiral formation without warning. A supporting tool is an effective Brownian-particle model of ran

What would settle it

Retrain a graph neural network or other permutation-equivariant classifier on the same 850,000 training examples, preserving the frozen test set; if balanced accuracy on static-versus-dynamic classification exceeds chance using only the initial 14x14 configuration, the key claim—that predictive structure is not present in any accessible representation at t=0—is falsified.

Watch

Extended reading notes

Core claim

Central claim: a deterministic system's macroscopic fate can be fixed by the initial state and still be unreadable from it, because the predictive degrees of freedom are collective topological structures that do not exist at t=0. Recoding the four cell states as phase vectors reveals charged vortices joined by strings of like-state cells, some of which form non-contractible loops around the toroidal lattice. The dynamics of these loops carry the result: as the last vortex pair approaches annihilation the loops vanish, making static and rectilinear fates readable in forward time; spiral-wave fates keep their loops until the wave forms abruptly. Since six machine-learning models and a mutual-i

Load-bearing premise

The paper's claim that no predictive structure exists in the initial configuration rests on the assumption that the six tested model classes and the per-cell mutual information cover all practically accessible ways of reading the lattice; the authors' own Supplementary Note 2.8 concedes that permutation-invariant and graph architectures were not tested, so if any such representation-aware learner extracted above-chance signal from the initial state, the strong 'absent at t=0'

Editorial extensions

If this is right

  • Determinism alone does not guarantee practical predictability: a complete set of update rules can coexist with the absence of any tractable way to read the outcome from the current state.
  • The negative machine-learning result is not a data or capacity failure inside the tested classes: more data, deeper models, and spatial inductive biases all saturate at chance, and feature-wise mutual information collapses to the shuffled baseline.
  • Vortex annihilation, and therefore static and rectilinear-wave outcomes, is predictable in forward time from the progressive loss of non-contractible loop strings; spiral-wave formation is not foreshadowed and remains heterogeneous down to the final timestep.
  • Globally distinct histories collapse to a few hundred exact terminal profiles in the final ~20 timesteps, a 97.4% reduction in trajectory diversity, showing the late-time dynamics are far more constrained than the full evolution.
  • The same minimal ingredients—cyclic internal states, finite-range interactions, and topological constraints—should make late-constructed predictability a general class property across excitable media, coupled oscillators, and lattice models with discrete rotational symmetry.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The authors' caveat implies a testable falsifier: if a permutation-equivariant or graph neural network trained on the same data beat chance from the initial configuration, the conclusion would weaken to 'predictive structure exists but is hard to find' rather than 'predictive structure is absent at t=0.'
  • The Brownian-vortex-gas analogy suggests annihilation times in larger lattices should follow first-passage statistics of diffusing annihilating particles; pattern-forming biological tissues could be searched for analogous late-time topological signatures as early-warning signals.
  • If the mechanism generalizes, the common modeling assumption that macroscopic outcome is encoded in initial conditions may need to be replaced by a distinction between formal determination and physically accessible determination, with the latter depending on when topological structures coalesce.
  • For machine learning on physical data, the paper implies that static snapshot classification can be fundamentally limited even with perfect labels; adding temporal or topological features may help, but the usefulness of any added feature will itself be time-dependent.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies a deterministic generalized cellular automaton of secrete-and-sense cells on a triangular lattice, whose trajectories end in one of three macroscopic fates (static configurations, rectilinear waves, or spiral waves). The authors report three main findings: (i) a battery of machine-learning classifiers, trained on up to 850,000 runs, cannot predict the binary static-versus-dynamic fate from the initial 14×14 configuration above chance; (ii) recoding the four cell states as a discrete phase field reveals topological objects—charged vortices, strings, and non-contractible loops (NCLs)—whose late-time dynamics differ among the three fate classes; and (iii) trajectory-similarity analysis of the total vortex-core-size time series shows that distinct histories collapse onto a small number of terminal profiles only in the final ~20 timesteps. The paper concludes that predictability is not a property of the initial condition but is dynamically constructed along the trajectory through self-organizing topological modes.

Significance. If the central claim were established with a consistent predictive protocol, the paper would make a significant conceptual contribution: a deterministic, non-chaotic many-body system in which practical predictability has a genuine temporal onset, carried by emergent topological degrees of freedom. The strengths are substantial. The machine-learning analysis is exemplary in design: a frozen test set of 100,000 runs, learning curves spanning three orders of magnitude in training data, threshold calibration on validation data, six model classes including spatially structured CNNs, and a model-agnostic mutual-information control. The late-time analyses are also internally consistent and replicated across 19–20 independent trajectory sets. The topological mechanism—pairwise vortex annihilation enforced by toroidal charge neutrality, with a proof in Supplementary Note 5—is a concrete, falsifiable structural finding. However, the paper's strongest interpretive claim is currently ahead of the evidence, because the late-time 'predictability' is demonstrated retrospectively, not with a forward prediction protocol, and the t=0 negative result carries an explicit caveat in Supplementary Note 2

major comments (3)
  1. [Results, 'Non-contractible loops predict final pattern types' (Fig. 4e-f)] The statement that 'an observer, without knowing how many timesteps remain, can use the forward-time dynamics of NCL strings to predict' (text near Fig. 4f) is not demonstrated. Fig. 4e aligns all runs by the final event (last vortex pair perishes or spiral forms), so the observer in this plot knows when each run ends. Fig. 4f is a colored scatter in (mean NCL count, slope) with no classifier, no held-out runs, and no reported balanced accuracy or ROC-AUC as a function of time-to-event. The opacity encoding (1–30 vs 31–60 timesteps before the final event) again uses future information. To support the forward-time claim, the authors should train a temporal classifier on features available at time t (e.g., sliding windows ending at t, without any final-event alignment) and evaluate its out-of-sample accuracy on held-out runs as a function of remaining time. This is a load-bearing gap becau
  2. [Results, 'Fate-dependent microstructures emerge only in the final timesteps' (Fig. 5b-e)] The trajectory-similarity analysis is also retrospective. Every backward-time window [1, τ] is anchored at the final configuration, so the analysis requires knowing when the run ends. The collapse to 23 connected components by τ=20 and to 977 exact terminal profiles in the pooled set is a descriptive statement about trajectory geometry, not a demonstration that a predictor operating online—without knowledge of the terminal time—can classify fate. In addition, no quantitative cluster-purity or classification metric is reported for the components in Fig. 5c; the statement that clusters are 'segregated by final pattern type' is based on visual inspection. A forward metric (e.g., nearest-neighbor classification using only data up to time t, or a time-indexed balanced accuracy) is needed to establish that predictability is dynamically constructed rather than merely retrospectively described.
  3. [Supplementary Note 2.8; Discussion] The t=0 half of the central claim is explicitly provisional. Supplementary Note 2.8 concedes that permutation-invariant architectures and graph neural networks were not tested and that 'we cannot exclude the possibility that alternative representations or substantially different model classes could uncover predictive structure not detected here.' The Discussion nevertheless states that 'the degrees of freedom that carry predictive signatures are not present in any accessible representation initially (at t = 0).' This overstates the evidence: the ML and NMI results support the weaker claim that no predictive signal was found within the tested model classes, representations, and data scales. If any representation-aware learner extracts above-chance signal from the initial configuration, the conclusion degrades from 'predictive structure is absent at t=0' to 'predictive structure exists at
minor comments (5)
  1. [Throughout] There are repeated typographical artifacts, e.g., 'Y et' at the start of the Introduction and in a later paragraph. These should be corrected.
  2. [Fig. 4 caption] The definition of 'final event' in panel (e) is given in the caption but should also appear in the main text where the panel is first referenced. As written, the reader must infer that the alignment uses information not available to an online observer.
  3. [Fig. 5b caption] The caption states that the color bar at right indicates final pattern type, but the mapping from colors to the three fate classes is not given in the caption or figure. Please add a legend.
  4. [Methods, 'Unique-trajectory analysis'] The padding of shorter trajectories beyond their actual duration is described, but it would be helpful to state explicitly that all analyses are restricted to times within the actual duration for each trajectory, so padding cannot create spurious matches.
  5. [Discussion] The sentence 'predictability is constructed late and asymmetrically' is repeated almost verbatim in the following paragraph. Consider consolidating to avoid redundancy.

Circularity Check

2 steps flagged · score 3.0 of 10

Mild circularity: spiral fate and persistent vortex pairs are the same phase-field object; the Brownian termination-time 'prediction' uses CA-fitted parameters; the static/linear NCL results and ML t=0 negative retain independent content.

  1. self definitional [Results, 'Universal three-stage dynamics' (Fig. 2b); 'Non-contractible loops as hidden topological structures of vortex pairs' section]
    "We discovered that the final pattern was determined by whether any vortices survived: when all vortices vanished, either a static configuration or rectilinear waves always emerged... whereas whenever vortices persisted, spiral waves always and eventually emerged. ... Since spiral waves arise only when at least one vortex pair survives indefinitely--whereas rectilinear waves and static configurations require complete vortex annihilation--we wondered whether... NCL strings might foreshadow which runs ultimately form spiral waves."

    Within the phase-field recoding, a spiral wave is a rotating pattern whose core is a phase singularity (vortex). The fate label 'spiral' and the claimed predictive degree of freedom (a persistently surviving +1/-1 vortex pair) are therefore the same object under two descriptions. Saying vortex persistence 'determines' or 'foreshadows' spiral fate is a restatement of what the spiral class is, not an independent predictor uncovered in the dynamics. This partially undercuts the 'predictability is dynamically constructed' claim for the spiral class, though the static-vs-rectilinear distinction via NCL strings and the trajectory-collapse results are empirically independent and limit the damage.

  2. fitted input called prediction [Methods, 'Brownian particle model of vortex dynamics'; Fig. 3d caption]
    "We inferred sigma from the empirical distribution of vortex velocity increments... An annihilation threshold distance r_threshold was estimated from the distribution of pairwise vortex separations at the moment of annihilation in the cellular automaton. ... generated annihilation timescales of the same order of magnitude, with predicted and observed termination times strongly correlated across lattice sizes (Fig. 3d; Pearson rho = 0.95)."

    Both free parameters (diffusion coefficient sigma and annihilation radius r_threshold) are fitted directly to the cellular automaton's own vortex trajectories, and the 'predicted' termination times are then compared with those same automaton runs. The rho = 0.95 agreement is a consistency check of a coarse-grained model whose inputs were taken from the target data, not an independent first-principles prediction. The paper itself concedes it 'did not derive Brownian motion from first principles'; this step is peripheral to the central predictability claim.

full rationale

The paper's central claim -- that fate-defining degrees of freedom are absent at t=0 and dynamically constructed -- rests on three evidentiary pillars. (i) The ML negative result at t=0 is a genuine empirical finding, but it is qualified by the paper's own Supplementary Note 2.8 ('we cannot exclude the possibility that alternative representations or substantially different model classes could uncover predictive structure not detected here'), so the Discussion's phrase 'not present in any accessible representation initially (at t = 0)' overstates the tested scope; this is a completeness over-claim, not a circularity. (ii) The late-time NCL-string and trajectory-collapse analyses use causal sliding windows and unsupervised clustering, but the figures align trajectories backward from the known final event and color points by the future fate label; no out-of-sample forward-time classifier or balanced accuracy versus time-to-event is reported, so the asserted observer 'without knowing how many timesteps remain' protocol is retrospective rather than demonstrated. This is a methodological gap in the predictive protocol, not a reduction of a prediction to its inputs. (iii) The self-citation to prior work (ref 16, Dang et al. 2020) supplies the CA model, but the model is fully specified in Supplementary Note 1, so the citation is not load-bearing; the topological charge-neutrality proof is self-contained in Supplementary Note 5. The genuinely circular content is limited to the spiral-vortex coupling (a persistent vortex pair and a spiral wave are the same phase-field object, so 'vortex persistence predicts spiral fate' is partly definitional) and the minor Brownian-model 'prediction' whose parameters were fitted to the target data. Because the static-vs-rectilinear predictability via NCL strings, the trajectory collapse, and the t=0 ML failure have independent empirical content, the paper is not fundamentally circular; score 3 reflects these two identified reductions without claiming the whole derivation is forced.

Assumptions & free parameters 4 free parameters · 5 assumptions · 3 invented entities

The ledger tracks what the paper adds versus what it assumes. Free parameters: two Brownian-model coefficients (sigma, r_threshold) fitted from the same CA data, and two analysis choices (tau^1/2 graph threshold, W=100 window). Axioms: quasi-steady-state diffusion, the Z4 phase recoding on which all vortex and NCL entities rest, torus topology and discrete Stokes, the p~0.5/I~0 ensemble, and the completeness of the tested machine-learning probe. Invented entities: vortices, strings and NCLs, and effective Brownian particles - all descriptive constructs with internal validation only (exact charge conservation, cross-set reproducibility), no externally falsifiable handle.

free parameters (4)
  • Diffusion coefficient sigma of Brownian vortex model = inferred from CA vortex velocity increments (4-timestep differences)
    Supp Note 4.5; estimated from the same automaton data the model is compared against. The rho=0.95 termination-time agreement (Fig. 3d) is in-sample with respect to this parameter.
  • Annihilation threshold r_threshold = estimated from pairwise vortex separations at annihilation in CA runs
    Supp Note 4.5; second fitted parameter of the Brownian model; affects the staircase structure and termination times.
  • Trajectory-similarity threshold d_thr(tau) = tau^(1/2) = tau^(1/2)
    Methods, Graph-based clustering; hand-chosen (about one vortex-core cell RMS difference per timestep). Controls the fate-segregation and cluster-count claims.
  • NCL sliding-window length W = 100 timesteps
    Methods, NCL quantification; used for mean NCL-string count and slope estimates in Fig. 4e-f; Supp Fig. 22 shows robustness across endpoints but W itself is a choice.
assumptions (5)
  • domain assumption Quasi-steady-state approximation: diffusion relaxes much faster than cellular state updates
    Supp Note 1; underpins the discrete synchronous threshold update rule; a standard separation-of-timescales assumption.
  • domain assumption The four cell states are correctly recoded as phases on a Z4 cycle (0, pi/2, pi, 3pi/2)
    Main text Fig. 2a; supported by cell-level statistics (Supp Fig. 1: 90.69% unidirectional transitions), but all vortex and NCL claims live inside this recoding. If the recoding is not the right dual description, the topological entities are representation artifacts.
  • standard math Torus topology from periodic boundary conditions; discrete Stokes theorem implies zero total plaquette circulation
    Supp Note 5 proof of topological charge neutrality; standard lattice edge-cancellation, independent of the cellular-automaton dynamics.
  • domain assumption Maximally disordered initialization (p1=p2~0.5, Moran I~0) is the relevant ensemble
    Methods and Supp Note 1; the machine-learning negative result, fate statistics, and trajectory-clustering claims are conditional on this ensemble definition.
  • domain assumption The tested machine-learning model classes plus marginal per-cell NMI suffice to probe practically extractable predictive structure at t=0
    Supp Note 2.8 admits untested architectures (graph/permutation-invariant); the central absent-at-t=0 claim depends on this coverage assumption.
invented entities (3)
  • Discrete vortices with topological charge +1/-1/0 (phase-recoded CA)
    purpose: Effective carriers of late-time dynamics; their annihilation staircase determines when fate becomes legible
    Constructed from the dynamics via the phase recoding; internally validated by exact charge conservation (Supp Fig. 17) and Brownian recapitulation, but there is no falsifiable handle outside this model.
  • Same-state strings and non-contractible loops (NCL strings) linking vortex pairs
    purpose: Predictive signatures separating static and linear outcomes from spiral outcomes; signal of imminent annihilation
    Descriptive topological observables; the predictive separation (Fig. 4e-f) is empirical but defined with hand-chosen windows and restricted to one-vortex-pair regimes.
  • Brownian particles as effective vortices
    purpose: Minimal coarse-grained model for vortex motion and annihilation timescales
    Effective model; sigma and r_threshold fitted from the same cellular-automaton data (Supp Note 4.5), so its agreement is not an independent confirmation.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Predictability can be dynamically constructed in deterministic systems." pith.science (2026). https://pith.science/paper/2JIM5ZRT

@misc{pith2026260401088,
  author       = {Pith},
  title        = {Pith review of: Predictability can be dynamically constructed in deterministic systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2JIM5ZRT}},
  note         = {Machine review of arXiv:2604.01088}
}
read the original abstract

A deterministic system's future can be fixed from the start yet remain unpredictable because of chaos or computational irreducibility. Here we reveal another source of unpredictability for non-chaotic deterministic systems: collective entities making the future legible are initially absent but dynamically constructed. In a generalized cellular automaton, disordered lattices evolved into static configurations, rectilinear waves, or spiral waves. Although initial configurations fixed each fate, machine-learning models did no better than randomly guessing fates from them. Recoding cell states geometrically revealed self-organization of prediction-enabling topological entities: vortices, non-contractible-loop strings, and, most generally, a winding field describing how same-state regions wrap the lattice. As the winding field self-organized, initially unpredictable fates became increasingly legible. Static and rectilinear-wave fates became progressively predictable, whereas spiral-wave fate became accurately predictable only near wave formation. These results establish that self-organization can build not merely order but entities that make certain futures visible, revealing a gap between determinism and practical predictability.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

26 extracted references

  1. [1]

    & Othmer, H

    A lbert, R. & Othmer, H. G. The topology of the regulatory inter- actions predicts the expression pattern of the segment polarity genes in Drosophila melanogaster. Journal of Theoretical Biology 223, 1–18 (2003). 2. A lbert, R. & Barabási, A.-L. Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47–97 (2002). 3. P athak, J., Hunt, B., Girvan, ...

    2003

  2. [2]

    at least one self-activation must be present

    the off-diagonal elements must contain both activation and inhibition (+1 and −1), and 2. at least one self-activation must be present. Because swapping molecule labels (1 ↔ 2) leaves the dynamics unchanged, we fix M12 = 1 and M21 = 55 −1 without loss of generality. Throughout this work we therefore study the cellular automaton defined by the interaction ...

  3. [3]

    Make the prediction task as easy as possible. We reduced the problem to binary classification: predicting whether a random initial configuration evolves into a static configuration (label = 1) or into any dynamic pattern (rectilinear or spiral waves; label = 0). This task is strictly easier than predicting the full pattern class

  4. [4]

    We generated 1,000,000 independent sim- ulations and evaluated learning curves with up to 850,000 training examples, spanning nearly three orders of magnitude in training set size

    Eliminate data limitations as a confounding factor. We generated 1,000,000 independent sim- ulations and evaluated learning curves with up to 850,000 training examples, spanning nearly three orders of magnitude in training set size

  5. [5]

    dynamic,

    Use a frozen, large test set. All reported performance metrics were computed on a fixed test set of 100,000 previously unseen initial configurations, ensuring that apparent improvements could not arise from test leakage or overfitting. As shown below, even under these deliberately favorable conditions, prediction performance saturates at chance-level bala...

  6. [6]

    An ensemble of randomized decision trees capable of capturing nonlinear, high-order interactions among lattice sites

    Extremely Randomized Trees (ExtraTrees). An ensemble of randomized decision trees capable of capturing nonlinear, high-order interactions among lattice sites

  7. [7]

    A modern boosted-tree method designed for large structured datasets

    Histogram-based Gradient Boosting (HGB). A modern boosted-tree method designed for large structured datasets

  8. [8]

    An aggressively tuned gradient-boosted decision-tree model applied to one-hot en- coded initial configurations

    XGBoost. An aggressively tuned gradient-boosted decision-tree model applied to one-hot en- coded initial configurations

Show all 26 references
  1. [9]

    A high-capacity neural network trained on one-hot encoded config- urations, with early stopping based on validation ROC–AUC

    Multilayer perceptron (MLP). A high-capacity neural network trained on one-hot encoded config- urations, with early stopping based on validation ROC–AUC

  2. [10]

    A spatially structured neural network trained on a two- dimensional, four-channel representation of the lattice

    Convolutional neural network (CNN). A spatially structured neural network trained on a two- dimensional, four-channel representation of the lattice. No manual feature engineering was applied beyond the raw encoding of cell states. 2.6 Learning curves and performance saturation...

  3. [11]

    simultaneous creation of one+1and one −1 vortex,

  4. [12]

    simultaneous annihilation of one +1 and one −1 vortex, or

  5. [13]

    conversion of a (+1)–(−1) pair into a 0-vortex. Proof. By the Theorem, total plaquette circulation vanishes at every timestep t for any instantaneous configuration {ϕi(t)}, independently of the dynamics. By the contour-based winding assignment used in our CA (Remark 2), the to...

  6. [14]

    Because different runs have different durations, we aligned all trajecto- ries by indexing time backward from the terminal event

    Trajectory representation and backward-time alignment Each cellular automaton (CA) run terminates upon reaching a final configuration, which is either static, rectilinear-wave, or spiral-wave. Because different runs have different durations, we aligned all trajecto- ries by in...

  7. [15]

    Unique-trajectory analysis 2.1. Exact identity within a backward-time window To ask how many distinct dynamical routes the system follows as it approaches its terminal state, we selected a backward-time window [1, τ] for varying values of τ ≤ 12,000. For any given τ, we extrac...

  8. [16]

    Motivation The unique-trajectory analysis demanded exact agreement and therefore treated even very small devi- ations between two trajectories as distinct

    Graph-based clustering 3.1. Motivation The unique-trajectory analysis demanded exact agreement and therefore treated even very small devi- ations between two trajectories as distinct. We therefore asked whether a broader version of the same collapse appears when near-identical...

  9. [17]

    The unique-trajectory analysis counted the number of exactly distinct scalar time series in a given backward-time window

    Relationship between the two analyses The two methods quantified related but distinct aspects. The unique-trajectory analysis counted the number of exactly distinct scalar time series in a given backward-time window. The graph-based method counted the number of nearby groups u...

  10. [18]

    Interpretation These analyses establish that one cannot determine the pattern fate of a given CA trajectory from this scalar observable for most of its duration. For thousands or tens of thousands of timesteps, trajectories leading to different final outcomes remained too frag...

  11. [19]

    Molecular-level tuning of cellular autonomy controls the collective behaviors of cell populations.Cell Systems 1:349-360 (2015)

    Maire, T & Y ouk, H. Molecular-level tuning of cellular autonomy controls the collective behaviors of cell populations.Cell Systems 1:349-360 (2015)

    2015

  12. [20]

    P ., Dang, Y ., Y ouk, H

    Olimpio, E. P ., Dang, Y ., Y ouk, H. Statistical dynamics of spatial-order formation by communicating cells. iScience 2:27-40 (2018)

    2018

  13. [21]

    Dang, Y ., Grundel, D. A. J., Y ouk, H. Cellular dialogues: cell-cell communication through diffusible molecules yields dynamic spatial patterns. Cell Systems 10:1-17 (2020)

    2020

  14. [22]

    Moran, P . A. P . Notes on continuous stochastic phenomena.Biometrika 37:17-23 (1950)

    1950

  15. [23]

    Statistical hydrodynamics

    Onsager, L. Statistical hydrodynamics. Nuovo Cimento 6 (Suppl 2):279–287 (1949)

    1949

  16. [24]

    Feynman, R. P . Application of quantum mechanics to liquid helium. Progress in Low Temperature Physics 1:17–53 (1955)

    1955

  17. [25]

    Abrikosov, A. A. On the magnetic properties of superconductors of the second group. Sov. Phys. JETP 5: 1174–1182 (1957)

    1957

  18. [26]

    Kosterlitz, J. M. & Thouless, D. J. Ordering, metastability and phase transitions in two-dimensional systems. J. Phys. C: Solid State Phys. 6:1181–1203 (1973). 82

    1973

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.