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

We formalize a conditional and auditable route to certifying that perfect-play chess is fair up to an explicit tolerance. The method introduces a small positive blur operator $K_t$ on the finite chess state graph, defines a simple draw-oriented imbalance potential $\Phi \in[0,1]$ from local features (king box, pawn shield, loose pieces, safe mobility), and then replaces $\Phi$ by a shifted Lyapunov observable

$$
\Psi_\delta=\frac{\delta+\Phi}{1+\delta}
$$

which avoids the degeneracy at $\Phi=0$ and is the actual control quantity in the proofs.
We impose a Top-n Good-Move Gap (GMG) condition: after a short opening prefix, every reachable non-absorbing position admits at least one of its Top- $n$ moves whose worst-case two-ply blurred expectation of $\Psi_\delta$ contracts by a fixed factor $1-\gamma$. From this we derive a weighted sup-norm spectral gap for the non-absorbing two-ply kernel, a discounted Bellman contraction, and an exponential finite-horizon decay for the residual blurred risk

$$
\operatorname{Risk}_{N, t} \sim(1-\gamma)^N
$$

This finite-horizon object is deliberately phrased in terms of blurred expectation rather than exact sharp-chess mate probability.

To make the hypothesis auditable at scale, we add a tightening framework based on: (a) $\varepsilon$-nets in feature space, (b) certified interval upper envelopes for reply classes, (c) dominance pruning at the level of these envelopes, and (d) adaptive refinement near the threshold. Under standard deblurring and discount-removal consistency assumptions, an audited discounted blurred value transfers to an explicit tolerance bound for the sharp value.