Exact Activation Rank in Multiplicative Excursion Walls: All-Order Boundary Cyclicity, Explicit Finite Caps, and Sharp Low-Order Minors
Tao Lin · Zenodo (CERN European Organization for Nuclear Research) · 2026
We close the structural strict-rank problem left open by the boundary-history factorization for prime-power activation walls in finite-scale Weil operators. At an activation q, the order-k wall jet factors through a k-dimensional boundary-history core. We isolate the correct strict-rank criterion, replace first-extremum histories by raw boundary Krylov vectors through a canonical unitriangular renewal transform, and identify the resulting problem with a positive weighted walk on a multiplicative endpoint graph. The endpoint graph is finite exactly for q = 2, 3, 4, 5. For every prime-power activation q ≥ 7, in both Legendre parities and at every order k ≥ 1, we prove all-order boundary-Krylov independence and hence strict activation rank k in a finite fixed-parity Legendre truncation. Prime activations are separated by valuation, odd prime squares admit a direct separator construction, and all higher prime powers are handled by an auxiliary-prime Beatty-type geodesic family. We also give an explicit finite cap for the truncation dimension in terms of the radius-(k − 1) endpoint quotient ball. The sharper identity Nmin(k, q, ε) = k remains a distinct moment-determinant problem. We prove Nmin(2, q, 0) = 2 for every q ≥ 3 and Nmin(3, 4, 0) = Nmin(3, 4, 1) = 3. For general cubic rank we derive an exact return/through decomposition: the through channel is governed by the Dirichlet convolution (Λ * Λ)(n), while the return channel is a weighted log-ratio autocorrelation. A Fourier–Mellin formula converts the latter into a signed spectral energy of a prime Dirichlet polynomial, explaining why raw total positivity and positivity of spectral energy alone do not establish the universal sharp law.