How wide must an eavesdropper be? Leakage of a toric-code logical bit to a strip and the Nishimori point

arXiv:2611.13174v1 Announce Type: new Abstract: Fault-tolerant quantum computers hand pieces of each logical qubit to separate decoder modules and shared readout lines. How much does a piece reveal? We prove that for any region and any Pauli noise, including correlated noise, an observer's optimal advantage in learning a logical bit is exactly 1 - 2 P_fail, where P_fail is the failure probability of maximum-likelihood decoding confined to that region. Two lessons follow. Geometry, not noise, provides the protection: a region that misses a single column of the code learns exactly nothing at any noise level, while a region that spans the code is protected by noise only at impractical distances (d > 350 for one row at a 10^-3 flip rate). And width is a poor guide to risk: under i.i.d.