Formal epiplexity-emergence of Conway’s Game of Life

Prove that the discrete-time Conway’s Game of Life update map Φn on an n×n binary grid, together with an appropriate sequence of initial-state distributions {Xn}, is epiplexity-emergent. Specifically, establish the existence of time bounds T1 and T2 with T1(n)=o(T2(n)) and an iteration schedule k(n) such that, as n→∞, (i) the difference in conditional epiplexity between the two observers for one-step prediction, S_{T1}(Φn(Xn) | Xn,n)−S_{T2}(Φn(Xn) | Xn,n), remains Θ(1), while (ii) the difference for k(n)-step prediction, S_{T1}(Φn^{k(n)}(Xn) | Xn,n,k(n))−S_{T2}(Φn^{k(n)}(Xn) | Xn,n,k(n)), grows unbounded (ω(1)).

Background

The paper introduces epiplexity as the structural information extractable by a computationally bounded observer and defines an epiplexity-emergent pair (Φ,X) when two observers, with time bounds T1 and T2 where T1=o(T2), see comparable structural complexity for one-step evolution yet a growing structural complexity gap for multi-step evolution. Conway’s Game of Life is presented as a canonical case of emergence: a simple local update rule generates complex higher-level patterns (e.g., gliders and oscillators) that may need to be learned by compute-limited observers to make accurate multi-step predictions.

While the authors provide empirical evidence consistent with emergence in related cellular automata, they explicitly note that a formal proof that Game of Life satisfies their epiplexity-emergent definition is not established. Resolving this question would rigorously ground the intuitive claim that compute-limited predictors must internalize rich structure to substitute for infeasible brute-force simulation over long horizons.