Entropy contraction on complete bipartite graphs via reverse erasure: only the rarest symbol matters

Unrefereed research candidate. In one sentence: for the Bayesian reverse of an erasure channel, the worst-case loss of relative entropy depends on the prior only through its rarest symbol, and this settles a conjecture of Caputo, Chen, Gu and Polyanskiy on the half-lazy random walk on complete bipartite graphs. Caputo, Chen, Gu and Polyanskiy (Electronic Journal of Probability, 2025) conjectured that one step of the half-lazy random walk on the complete bipartite graph Kn,n shrinks relative entropy by at most the factor log n / (2 log 2n), with single vertices as the only worst starting points; they proved the case n = 3 with computer assistance. This paper proves the conjecture for every n, and for Km,n with M = max(m, n) in place of n. The key observation is that one step of this walk splits into two copies of the reverse-erasure channel: keep the current symbol with probability α, otherwise replace it by a fresh draw from a prior ρ. For this channel the relative-entropy contraction coefficient depends on the prior only through its smallest atom q, at every retention probability. Up to the sharp threshold α = qq/(1-q) the coefficient is log(1/q) / log(1/(αq)), a translation of a weighted mean inequality of Gao (credited); above the threshold the worst inputs become full-support exponential tilts and no closed form is given. The uniform binary case is known (Nair and Wang; Zhao). Further consequences: a rank-one weighted bipartite extension, which also settles a numerical observation of Jadhav (2026) on the lazy star; a reduction of any block-reset chain (any laziness, all powers, stars, multipartite graphs) to a chain with at most two states per block, whose values are computed numerically; and the exact one-way communication rate for agreement distillation below the threshold. There is no mixing-time improvement: the spectral gap of the walk is 1/2. The archive contains the manuscript (LaTeX source, bibliography and PDF), the figure and its script, four verification scripts (random-sample suite, minimum-mass diagnostics, ten must-fail negative controls with near-extremiser families, and block-reset extension checks with five must-fail controls) with recorded outputs in normal and optimised Python modes, exact small-kernel fingerprints, the dated prior-art gate and audit, raw registry records for every reference, a rating-free internal review summary, development notes, CI and SHA-256 sums. Assurance: written proofs; numerical diagnostics are evidence, not proofs; point-optimality windows for other laziness values are numerical; internal AI-assisted review only (not peer review, independent reproduction or formal verification); priority rests on a bounded literature search. Creator: Anonymous. Text, data and figures CC0-1.0; code MIT.