Temporary workspace for robust reconfiguration of stochastic factorizations

Abstract

We prove an exact correspondence between temporary width in stochastic factorizations and vertex-cover reconfiguration. The task is to correct one response while keeping every protected response within a prescribed total-variation tolerance and returning all added components to an unused state. For normalized incidence responses of connected Δ-regular bipartite graphs without four-cycles, the minimum temporary reserve is r_ε(s_row) = w(G) − N, where w(G) is the minimum peak cover size in a reconfiguration between the two bipartition classes and N is the initial width. The lower bound covers arbitrary continuous changes of both stochastic factors, including simultaneous changes, whenever the per-row tolerance ε and optional smoothing level ζ satisfy ε + ζ ≤ 1/(16Δ²). A closed, set-valued decoder turns every admissible continuous path into a finite cover path, and a matching construction lifts cover moves without an extra component. The correction itself forces the terminal cover; no final factorization is prescribed. Expanding graph families yield two realizations of the same full response matrix, with the same minimum width, for which the same correction needs zero or linearly many temporary components. If permanent growth is allowed, one retained component suffices instead. Reclaiming that component while preserving the corrected response has peak width between w(G) and w(G) + 1. These are statements about the defined stochastic-factorization model, not a general physical law of memory or a claim about trained systems. Preprint; not peer-reviewed. The manuscript contains full proofs. The accompanying reproducibility archive supplies source, deterministic checking code, recorded finite results and hashes. The finite checks verify identities and normalizations, not the general all-path theorem. Text, LaTeX, PDF and documentation: CC BY 4.0. Original verification code and recorded results: MIT. AI systems assisted analytical development, checking, coding and drafting; the human author is responsible for the claims and final text. Internal AI-assisted review is not journal peer review.