Uniform eventual domination of permanents of doubly stochastic matrix powers

For every positive integer n, this preprint proves the existence of an integer K(n) >= 2 such that per(A^k) <= per(A) for every real n x n doubly stochastic matrix A and every integer k >= K(n). Powers use ordinary matrix multiplication. The exponent is uniform over the whole doubly stochastic polytope, including its boundary, and the result implies the nontrivial per-k conjecture attributed to Foregger. The argument is nonconstructive and provides no explicit numerical threshold.

The accompanying archive contains the manuscript sources, 63 Lean 4.19.0 source modules, a portable verification script, source hashes, and saved successful local compilation and axiom-audit records. The pinned mathlib commit is c44e0c8ee63ca166450922a373c7409c5d26b00b.

Status: preprint, not peer reviewed. Independent human review has not been completed. A public-literature audit dated 13 September 2026 did not locate a prior general resolution, but does not certify bibliographic priority or exhaustive coverage. Chang's 1983, 1984 and 1990 full texts remain gaps. The manuscript discloses OpenAI Codex assistance. No empirical dataset is involved.