Exact Minimum Field Partition for One-Step Prediction in a Finite Five-Agent ABCW Model

Abstract

Which distinctions in a present state must be retained to predict a specified future exactly? We study this question on a finite reachable set of a deterministic five-agent ABCW model in which binary agent actions and strategies coevolve with a weighted directed influence field. The dataset is generated from four initial-field families and all 4,096 action-strategy initial conditions, yielding 56,536 transitions and 2,562 distinct current fields. We retain the current action and seek the coarsest field-only partition that uniquely determines the complete one-step-ahead field on every observed transition. Nine natural feature families and all 511 of their nonempty combinations provide strong approximate predictors but no exact nontrivial field compression. We therefore construct an incompatibility graph whose vertices are current fields and whose edges join fields that produce different next fields under a shared observed action. A proper 692-coloring gives a constructive upper bound. An independent anchor-action decomposition, followed by exact solution of the 50 nontrivial induced subgraphs, gives the matching lower bound. Hence the minimum number of field classes is exactly 692. This number applies only to the specified finite reachable set, field-only compression, retained current actions, one-step target, and exact-prediction requirement; it is not a universal macrostate count for ABCW systems.