One Operator, Four Faculties: Value, Time, Self and Contradiction as the Hodge Components of an Agent's Flow, with Belnap's Four Values as Its Regimes

Abstract

Claim. We propose a topological account of an embodied AI agent’s memory, in which four faculties, the value of a goal, time,the self as efference, and contradiction, are the component types of a single combinatorial Hodge decomposition of the agent’sempirical flows, read with no learned parameters. Using frozen adversarial synthetic worlds and the 1977 text adventure ColossalCave, we find that value is the gradient, time is the circulation, the self is the exact part, and contradiction is the curl. Belnap’s four belief values are the topological regimes of the same flow, dispatched with no hand-tuned threshold. We claim nothing about real perception. Results. Each identification survives the control built to destroy it, at ≥10 seeds with 95% confidence intervals, so the decomposition is measured rather than analogical. The four components then form a single diagnostic surface, on which a reward that cycles, a belief base that cannot be ordered and a memory key that aliases distinct situations appear as one defect at one component, and the operator that detects that defect also names its repair. Potential-based shaping is known to leave optimal policies unchanged, and we measure the converse: a non-gradient perturbation of matched magnitude captures the greedy policy and drops optimality from 1.000 to 0.327. Belief revision presumes that its entrenchment order is well-founded, and curl-freeness decides that question instead of assuming it, telling a resolvable conflict from a money-pump at 1.00 where counting contradictions reaches 0.49. Whether a state representation is Markov enough for value becomes a quantity the agent reads off its own flow, and removing the curl it exhibits restores planning to oracle level. The geometry stays informative because nothing optimises against it, and training toward it proves null-to-harmful in all three forms we tried.