One Edge Is Enough: Error-Preserving Repair for Stochastic Fixed-Point Iteration

Accurate estimates of a nonexpansive map need not produce stable iterates, because estimation can destroy consistency between successive queries. We show that repairing one adjacent query/reply constraint suffices to preserve estimation accuracy throughout Halpern iteration. On a cube, our rule clips each fresh absolute map-value estimate against the previous reply and preserves the largest input error with constant one. This invariant yields residual $4D/(k+2)+M$ with $O(d)$ recurrence state, where $D$ is the domain diameter and $M$ the largest input error. Conditional robust estimation controls $M$, giving fixed-budget guarantees without knowledge of the noise scale or tail exponent. The coordinate geometry is essential: we prove that universal lossless repair on norm balls is possible exactly in maximum-norm spaces, even with randomization. For contractions, the same clipping argument links inherited-error decay to query travel, yielding a minimum-motion extension with geometric recovery and no additional oracle calls. Matched-cost experiments isolate stabilization across 36 settings with two estimators, while Bellman studies demonstrate recovery from corrupted replies and value-accuracy gains under variance bursts. One stored edge thus provides a low-state interface through which statistical accuracy supports stable iteration and recovery.