15.11 Cellular-automaton theory induction

Cellular automata provide procedurally generated worlds with exact local semantics and potentially unfamiliar global behavior. A benchmark generator can choose a new alphabet, neighborhood, equivariant local rule, and initial condition after the proposal model’s training cutoff. The system observes selected trajectories and interventions without receiving the hidden rule or the evaluator’s macroscopic vocabulary.

The challenge has four levels:

1. Microscopic recovery. Infer a local update rule that reproduces the observed trajectories and survives held-out cell interventions.

2. Structural recovery. Identify translation symmetries, conservation laws, invariant subspaces, attractors, and compositional factorizations.

3. Macroscopic invention. Propose persistent domains, particles, interfaces, collision classes, or other higher-level objects together with maps from cell configurations to those objects.

4. Persistent theory construction. Use the invented vocabulary to predict new interactions, compress families of proofs or simulations, and transfer to related automata under changes of coordinates or local rules.

The third level is the decisive creativity test. Naming a visually striking pattern is insufficient. A macroscopic object must have an operational recognizer, transformation laws, boundary conditions, and consequences that are independently executable. The fourth level asks whether the new theory continues to organize later experiments rather than disappearing after the trajectory that suggested it.

In the AGENTIC realization, the structural component chooses cell, rule, and pattern perturbations that discriminate candidate microscopic and macroscopic explanations. OPTIC constructs detectors, coarse-graining maps, collision tests, and reusable simulation macros. RELIC allocates experiments across initial conditions and time horizons, then decides when an admitted macroscopic object should enter its planning vocabulary. The closed loop is essential: a detector without an experiment policy may never encounter a decisive collision, while an experiment policy without a typed detector cannot recognize the event it sought.

Experiment: Cellular-automaton emergence benchmark. Epistemic status: proposed benchmark design, not a completed result.
Input: blinded trajectories and a registered set of cell, pattern, rule, and neighborhood interventions from a newly generated automaton.
Proposal: a microscopic rule, structural sketch, and candidate macroscopic theory with explicit abstraction and realization maps.
Admission: exact replay, held-out interventions, invariant checks, prediction of collisions and long-horizon behavior, and reuse on later rules.
Creativity label: rule inference is Mathematics–E; an admitted, predictively useful macroscopic ontology is a candidate Mathematics–T result.
Boundary: success is relative to the generated world and does not establish that the same ontology describes a natural system.