PRH | Aux | 4.1 • A Finite, Auditable Certificate for Chess Fairness via Blur

This note describes a conditional and auditable method for showing that perfect-play chess is fair up to a stated numerical tolerance. The main idea is to replace exact sharp chess by a slightly smoothed version, called a blurred model, and then prove that a simple imbalance measure tends to shrink after both sides make strong moves.

A central difficulty is that sharp chess is not genuinely smooth. One square, one tempo, one undefended piece, or one hidden forcing resource may change the truth of a position completely. Therefore the framework should not pretend that every sharp position inside a blurred cell behaves identically. This presentation allows exceptional sharp tactics below the blur scale. Such exceptions are not ignored; they are charged to explicit exception budgets.

The additional refinement in this version is that exception budgets may be time-indexed or phase-indexed. An unresolved tragedy near the beginning of the game is not treated as morally identical to an unresolved tragedy after many stabilizing full moves. Early exceptions receive their own prefix accounting; later exceptions are propagated through the remaining contraction estimate. The uniform one-step budget is recovered as a simplified special case.

To make the argument readable, we introduce each ingredient before using it. We define the state space, the blur operator, the two-ply transition kernel, the imbalance potential, the shifted Lyapunov observable, time-indexed exceptional-tactic budgets, and the key good-move assumption with defect. The imbalance potential is built from simple local and tactical features rather than from a full chess engine evaluation.