Kolmogorov Cycles in the String Landscape: A Rank-Two Constraint on Irreversibility

Preprint of a manuscript submitted to Physical Review D (2026). This is the version that is under review; the text and figures have not changed since submission. We show that irreversibility in black-hole-mediated landscape transitions is structurally rigid: the full N(N-1)/2-dimensional cycle space of an N-vacuum hub collapses to only N-1 independent affinities, each black-hole species contributing one rank-two block, and the resulting affinities obey a cocycle identity that a future directed rate-weighted network would confirm or exclude. This structure follows from the hub geometry once the dS-dS links are of potential type, and is independent of the hub's internal physics. The framework is the edge-space Hodge decomposition of the vacuum-transition rates, x = (1/2) B^T phi + h, used as a diagnostic of the measure. For standard Coleman-de Luccia rates, in both Hartle-Hawking and Vilenkin conventions, the cycle component vanishes identically and every Kolmogorov cycle residual equals unity: the measure information does not live in the cycle sector. The gradient potential phi then makes the reversible sector readable from the rates alone - detailed balance in edge-space form, center(ln pi) = phi - with pi proportional to exp(S) (HH) and pi proportional to exp(-S) (Vil); outside it the deviation D = center(ln pi) - phi has the first-order response D = 2 L_a^+ B (a . h) + O(|h|^3). A read-off protocol uses the cycle fraction rho = |h|/|x| to triage networks: rho = 0 is exactly readable for a conservative generator and 0 < rho <~ 0.2 is readable to first order. We verify these identities on random networks up to N = 1000 and on an explicit enumeration of the Carifio edge-tree network (N = 82, m = 199), and locate the narrow mass window, from the smallest per-edge threshold M_c(H_B) ~ 0.370 M_Pl^2/H_A to the smallest per-edge upper bound M_max(H_A) ~ 0.46 M_Pl^2/H_A, in which detailed balance is broken. Volume growth, by contrast, is invisible to both phi and h and requires a separate KL comparison with exp(S). Every figure and every tabulated or quoted number in the manuscript is generated by the archived analysis code. The code, its cached numerical outputs and the eight figures are archived separately at doi:10.5281/zenodo.23749448, together with a SHA-256 manifest (CHECKSUMS.sha256) covering every file in that archive; no numerical result in the paper is entered by hand.