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In a deterministic system, the future is fixed by the starting state and the rules, but that does not mean an observer can read it from the start. In a non-chaotic cellular automaton studied by Lars Koopmans, Elinor M. Kay and Hyun Youk, each disordered starting layout has one fate: a static configuration, a rectilinear wave or a spiral wave. Yet machine-learning models asked to predict that fate from the starting layout did no better than random guessing. Predictability appears as the pattern evolves, because topological structures build up during the run and make some outcomes readable.
The model in plain terms
The system is a generalized cellular automaton: a lattice of cells, each in one of several states, updated by fixed rules. The authors start it from disordered lattices and track where it ends up. Every starting layout leads to one of three outcomes:
- Static configuration: the pattern stops changing in a fixed arrangement.
- Rectilinear wave: the pattern settles into a wave travelling in a straight line.
- Spiral wave: the pattern settles into a rotating spiral.
Because the rules contain no randomness, the same starting layout always produces the same outcome. The open question the paper addresses is how much of that outcome is visible in advance.
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Outbyte Driver Updater FREEFix the driver behind crashes, sound loss and screen glitchesFind Drivers →Outbyte PC Repair FREEClear out junk files and repair common Windows errorsFree Scan →Determinism is not the same as predictability
The paper separates two claims that are often run together. Determinism means the outcome is fixed by the initial state and the rules. Practical predictability means an observer or model can infer that outcome from the information it has. A system can have the first property without the second.
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The authors use an operational definition: a human observer or machine-learning model predicts the fate better than chance. Hyun Youk put the limits of that definition plainly in institutional coverage:
“So far, we haven’t come up with a deep answer to why topology matters so much in our simulations. And while we have an operational definition of predictability based on the ability of a human observer or machine-learning model to predict fate better than chance, this definition hasn’t been mathematically formalized yet, so rigorously defining predictability and examining its properties are our next goals.”
The result is therefore a model result, not a general theorem about deterministic systems.
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How each outcome becomes predictable
The starting layouts do not reveal their fate at the outset. What changes is how the outcome classes become legible as the simulation runs. The table summarises what the sources establish for each class.
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| Outcome | Predictability at the start | How predictability changes during evolution | Limits and notes |
|---|---|---|---|
| Static configuration | Machine-learning models did no better than random guessing (overall result; per-outcome figure not stated) | Becomes progressively more predictable as the winding field self-organizes | No exact accuracy percentage is given in the sources reviewed |
| Rectilinear wave | Same overall result; per-outcome figure not stated | Becomes progressively more predictable as the winding field self-organizes | No exact accuracy percentage is given in the sources reviewed |
| Spiral wave | Same overall result; per-outcome figure not stated | Becomes accurately predictable only near the moment the wave forms | Accurate prediction is not established before wave formation |
Institutional coverage describes the strongest convolutional neural network as moving from chance-level accuracy at the start to almost perfect accuracy late in the simulation. That is a qualitative description; it is not a reported percentage and should not be read as one.
The topological structures that make the future legible
The authors recode the cell states geometrically and track topological features that do not exist in the random starting layout. These features develop during the simulation, and they are what makes the outcome progressively readable.
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Vortices
Vortices are one class of topological feature the authors identify in the evolving pattern. They are part of the geometric description that lets the authors connect the evolving lattice to its eventual outcome.
Non-contractible-loop strings
A non-contractible loop is a closed path of same-state cells that cannot be shrunk to a point without leaving the lattice, because it wraps around the periodic boundaries. The paper groups such loops into strings, which the authors use as another structural signal in the pattern.
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The winding field
The winding field is the central construct. It describes how connected regions of same-state cells wrap around the lattice. As the field self-organizes during the run, the outcome becomes more predictable for static and rectilinear-wave fates. For spiral waves, the field does not make the outcome accurately predictable until the wave is close to forming.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.What the study does not establish
- No forecasting tool. The sources reviewed describe a computational model. They do not establish a practical forecasting application.
- No biological validation. The model is described as inspired by cell-like communication, but that inspiration does not make it evidence about living tissue. The sources reviewed do not show validation in living systems.
- No claim about chaotic systems in general. The point is that this model is non-chaotic yet initially hard to predict. It does not show how other deterministic systems behave.
- No population statistics or precise accuracy figures. The sources reviewed give no sample sizes, percentages or effect sizes for the machine-learning results.
Source, funding and status
The primary source is the open-access Nature Communications article “Predictability can be dynamically constructed in deterministic systems” by Lars Koopmans, Elinor M. Kay and Hyun Youk, published 11 September 2026. The publisher labels the shared article an early version subject to further edits and replacement by the final Version of Record. The authors are affiliated with the University of Illinois Urbana-Champaign. The publisher lists NIH-NIGMS grant GM147508 and NSF Science and Technology Center for Quantitative Cell Biology grant DBI 2243257, and reports no competing interests.
The headline here is the title of University of Illinois Grainger College of Engineering coverage distributed through Phys.org on 8 October 2026, which includes attributed comments from the authors. Elinor Kay, a physics graduate student and co-author, said: “Despite the simplicity of our system, the cells self-organized in a way that no human or machine could initially predict.” The coverage also quotes her on the broader implication: “This shows that information is always present but slowly becomes accessible, which is very exciting because it implies that there’s a greater order just below our grasp.”
For readers, the practical takeaway is narrow but clear. A fixed future and an accessible future are different things, and in this model the structure that makes the outcome accessible is built by the system itself as it evolves.
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