Predictability is dynamically constructed by topological collective modes in deterministic systems

Abstract: Deterministic many-body systems governed by simple interactions can self-organize into macroscopic patterns, and the determinants of long-time behavior are assumed to be encoded in the initial configuration. Here we show that predictability can instead be constructed dynamically rather than being accessible in the initial configuration. We study a generalized cellular automaton of secrete-and-sense cells that self-organizes from disorder into static configurations, rectilinear waves, or spiral waves. Although dynamics are deterministic, the final outcome cannot be reliably inferred from the initial state alone. Treating cell states as a discrete phase field, we uncover emergent topological modes - charged vortices connected by strings that form non-contractible loops. Tracking their dynamics reveals that predictive signatures of macroscopic fate appear only late in the trajectory: vortex annihilation becomes readable through loop loss, whereas vortex persistence remains unreadable until spiral waves form abruptly. These results show how predictability can be dynamically constructed in deterministic nonequilibrium systems.

• The paper demonstrates that predictive information in high-dimensional deterministic automata emerges only through late-stage topological modes like vortex pairs and non-contractible loops.
• It employs extensive cellular automaton simulations showing that initial states lack predictive signals, with machine-learning models performing no better than random guessing.
• The study reveals that dynamic trajectory clustering until the final timesteps marks the onset of macroscopic fate, challenging conventional deterministic predictability.

Dynamical Construction of Predictability via Topological Collective Modes in Deterministic Systems

Summary of Key Findings

This study rigorously interrogates the foundational assumption that the macroscopic fate of deterministic, many-body systems is legible from the initial configuration. Employing a generalized cellular automaton modeling secrete-and-sense cellular circuits on a two-dimensional lattice, the authors demonstrate that the ultimate organizational outcome—static configuration, rectilinear waves, or spiral waves—cannot be reliably predicted from the initial condition. Neither human inspection nor machine learning, including deep neural networks, achieves accuracy beyond random guessing in predicting final pattern types from initial states, even with datasets comprising up to one million independent simulations. This failure is not attributable to insufficient data or model expressivity, but rather to the fact that the dynamical structures carrying predictive information are absent at initialization and constructed only late in the trajectory via emergent topological collective modes, specifically vortex pairs and non-contractible loop strings.

System Architecture and Dynamical Regimes

The underlying automaton comprises cells interacting via finite-range, nonlocal couplings, governed by a minimal gene-regulatory circuit with two diffusible molecular species. Each cell cycles almost deterministically through four gene-expression states, discretized as phase directions, and updates synchronously based on sensed concentrations from neighbors. Initial conditions are maximally disordered, with negligible spatial correlation and balanced state representation. Despite deterministic rules, exhaustive perturbation analysis (single-cell state flips) reveals extreme sensitivity: ~50% of all one-cell changes redirect the system to a qualitatively different macroscopic outcome, with substantial variation in pattern formation timescales.

Three classes of outcomes are observed depending solely on precise initial configuration: static uniformity, propagating rectilinear waves, and spiral waves. Importantly, statistical and informational analysis (mutual information) confirms the absence of extractable predictive signal in the initial configuration, both at the local and global level.

Topological Modes, Emergence, and Temporal Construction of Predictability

The authors recast cell states as a discrete phase field, revealing spontaneously nucleated vortex-like defects characterized by quantized winding numbers (+1, -1, 0). Vortices are topologically charged excitations invisible in the non-phase representation. The dynamics universally proceed through a three-stage process: rapid vortex creation from disorder, diffusive motion and pairwise annihilation (constrained by global topological charge neutrality imposed by periodic boundaries), and eventual outcome manifestation. The emergence and annihilation of vortex pairs are synchronized, driven by lattice topology (torus equivalence), precluding isolated creation or destruction of topological charge.

Tracking vortex interactions, the study establishes that vortex cores behave as Brownian particles, with diffusive motion and annihilation upon proximity. An analytical Monte Carlo model recapitulates the temporal "staircase" of pairwise annihilations and termination times across lattice sizes, with quantitative agreement (Pearson $p=0.95$) between analytical and automaton trajectories.

Critically, predictability is shown to crystallize only late: the fate-defining microstructures (vortex core size, non-contractible loop strings threading vortex pairs) remain indistinguishable across runs destined for different outcomes until the final tens of timesteps. Static and rectilinear outcomes are preceded by topological simplification (declining number of NCL strings), whereas spiral formation is abrupt and remains unreadable until occurrence.

Machine Learning Limitations and Trajectory-based Signatures

An extensive array of supervised learning models (logistic regression, tree ensembles, boosted trees, deep MLPs, CNNs) trained on up to 850,000 examples yield balanced accuracy and ROC-AUC indistinguishable from chance ($\approx0.5$), regardless of training set size or model class. Neither spatial inductive bias nor permutation invariance rescues predictability. Model-agnostic mutual information analysis reveals near-zero dependence between initial state and outcome labels, collapsing to shuffle baselines with increasing sample size.

Late-stage trajectory clustering analysis (graph-based, unsupervised) reveals that fate-aligned clusters only consolidate in the final $\sim20$ timesteps, with up to 97% reduction in trajectory diversity as the system approaches its terminal state. The dynamics segregate into fate-specific archetypes only at this terminal phase.

Topological Constraints and Non-contractible Loop Structures

Charge neutrality and pairwise annihilation of vortices are enforced by the global topology (periodic boundaries). Strings of same-state cells connecting vortex pairs can form non-contractible loops (NCLs) that are topologically non-shrinkable on the torus. The dynamics of NCL strings encode the late-stage forward-time signature: progressive decline in NCL string abundance reliably predicts impending vortex annihilation and outcomes (static or rectilinear wave), whereas spiral outcomes retain persistent NCL strings until abrupt spiral consolidation.

Implications and Theoretical Significance

This work makes a strong, technically substantiated claim: determinism is insufficient for practical predictability in high-dimensional, discrete nonequilibrium systems when the features governing macroscopic fate are not accessible at initialization but are organized collectively and dynamically via topological modes. The results directly challenge deterministic predictability paradigms and define a sharp separation between formal computational determinism and operational accessibility of predictive structure.

Practically, the findings imply that naive application of data-driven pre-commitment prediction strategies in deterministic systems (biological, physical, or engineered) may fail unless the relevant structures have already emerged within the system's trajectory. Theoretically, the identification of topological collective modes—vortex pairs, charge conservation, and non-contractible loop dynamics—as carriers of predictive information provides concrete physical mechanisms underlying computational irreducibility and temporal emergence of accessibility.

The study delineates minimal structural ingredients—cyclic internal states, finite-range nonlocal interactions, and global topological constraints—that demarcate classes of discrete systems in which late and asymmetric predictability is generic. This mechanism is expected to generalize to coupled oscillator networks, excitable media, active nematics, and discrete lattice models with rotational symmetry.

Prospects for Future Developments

Future work should investigate:

• Generalization to larger classes of cellular automata—including higher phase-dimensionality, more complicated topological defects, and additional regulatory motifs—and determine universality of late-stage predictive accessibility.
• Analytical characterization of topological mode dynamics and identification of predictors in other nonequilibrium systems (e.g., excitable biological tissues, coupled synthetic cellular consortia, synthetic oscillator networks).
• Development of machine learning paradigms that leverage trajectory information and topological observables, rather than static initial configurations, possibly integrating online or recurrent approaches, or integrating physical topological constraints.

Conclusion

This study establishes that predictability in deterministic systems, specifically generalized cellular automata with finite-range nonlocal couplings and global topological constraints, is not a static property of the initial configuration but is dynamically constructed by emergent topological collective modes. Predictive information is carried by late-stage dynamical structures—vortex pairs and non-contractible loops—that crystallize only through evolution, with practical prediction impossible from initialization alone, even for expressive machine-learning models. This highlights a fundamental gap between computational determinism and accessible prediction, delineating new directions in the study of complex self-organizing systems and the development of dynamical prediction frameworks.