CALLIOPE · BCOM

We use Conway's Game of Life as a fully computable laboratory for the Kolmogorov-Theory (KT) account of persistent patterns developed in WP0162. In this toy world the dynamics, the agent/world boundary, the perturbations, the identity conditions, and the interventions are all explicit, so KT's claims can be operationalized and measured. We argue, and demonstrate, that persistence is not an intrinsic yes/no property of a pattern but is ensemble-relative: a pattern persists within a perturbation ensemble if its local rule keeps its identity inside a viability (reach) set; death is leaving that set; regulation is a compressive rule that enlarges reach by acting on the world. We (i)~separate the two coupling channels of a hand-crafted eater by directed ablation; (ii)~measure persistence signatures directly from the automaton using a finite compression proxy, showing that a spaceship's persistence is frame-relative; (iii)~evolve a local learned eater whose reach, with the same identity and region as a passive block, rises from $0.000$ (passive block) and $0.018$ (hand-crafted eater) to $0.892$ ; (iv)~show death outside reach, the learned eater being destroyed by a spaceship absent from its training ensemble; (v)~give a finite behavioral witness that the rule's competence is glider-shaped (selectivity $I(S;G)=0.46$ bits); and (vi)~find that the effective regulator compresses to a minimal sufficient rule --- ``a cell lives iff it has $3$ live neighbours'' --- a single tuned threshold that turns a glider-propagating substrate into a glider-eating regulator. Reach is set by the program, not the substrate; and the cleanest model of the niche is the shortest rule that preserves reach. We situate the construction relative to Growing Neural Cellular Automata.

Conway's Game of Life as a fully controllable laboratory for testing when and why patterns survive — and the answer turns out to be: it depends entirely on the program, not the stuff.

The core idea is that "persistence" isn't a yes/no property baked into a pattern. It's relative to a threat ensemble — the specific set of perturbations a pattern might face. A Game of Life eater survives gliders hitting it from one direction but dies to gliders from another. A blinker survives almost nothing. The paper formalizes this with a quantity called reach: the fraction of a perturbation ensemble from which a pattern successfully recovers its identity. Death is simply leaving that set.

The most striking result is what happens when you hold the identity (a 2×2 block) and the region fixed, and only change the local update rule inside that region. A passive block has reach 0.000 — any glider destroys it. A hand-crafted eater has reach 0.018. An evolved rule, found by an evolutionary strategy over a 177-parameter neural network, reaches 0.892 — absorbing gliders from all four directions it was never explicitly trained on, thanks to a built-in rotational symmetry. Same shape, same location, wildly different survival. The rule is what matters.

Then comes the compression punchline. That 177-parameter network is massively over-parameterized. When you shrink the rule down, reach stays flat at 0.892 all the way to a two-parameter rule. And that minimal rule is interpretable: "a cell lives if and only if it has ≥ 3 live neighbors." One threshold. Conway's original rule (B3/S23) births cells at exactly 3 neighbors and keeps them alive at 2 or 3. The evolved rule just drops the state-dependence and thresholds at 3 — which happens to eat gliders instead of propagating them. The paper frames this via algorithmic information theory: among all rules that achieve the same reach, the universal prior selects the shortest one. The minimal rule wins not because it's clever but because it encodes almost nothing except the niche model itself.

The paper also demonstrates death cleanly: the evolved eater, trained only on gliders, is destroyed by a lightweight spaceship it never encountered. And it gives a behavioral measure of how "glider-shaped" the rule's competence is — observing whether the rule succeeds tells you 0.46 bits about whether the incoming object was a glider. This isn't the full uncomputable algorithmic mutual information between rule and world, but it's a concrete, finite witness of niche-specific knowledge encoded in the program. The whole construction is fully reproducible: fixed seeds, pure Python, every figure regenerable from listed scripts.