Reversibility and frequency reduction in periodic and random positive systems

Unrefereed research candidate. A population spread over several patches, linked by fixed migration, grows at a long-run rate that depends on how fast its environment changes. This paper asks when making periodic environmental change faster can only lower that growth rate. For the linear patch model x' = [Q + diag r(ωt)]x with an irreducible migration generator Q, the answer is exactly detailed balance (reversible migration: no net circulation around any loop of patches; every two-patch system qualifies). Reversible migration gives a growth rate that never increases with frequency, strictly decreasing unless ri(t) = ai + b(t); every irreversible Q admits weak rotating environments on two patches whose growth rate has an interior frequency maximum, located by an explicit certificate computed from Q alone, with a quantitative gain bound. This settles the periodic monotonicity conjecture of Benaïm, Lobry, Sari and Strickler for a fixed migration matrix (it holds exactly for diagonally symmetrisable migration). For reversible migration the quenched Lyapunov exponent under an irreducible finite-state Markov environment strictly decreases with switching speed unless the frozen matrices share a Perron vector, proving Conjecture 1 of Monmarché, Schreiber and Strickler (2025); the annealed exponent is convex and strictly decreasing when the environmental chain is also reversible. Time-symmetric two-state environments give witnesses according to the sign of an explicit quadratic form, and the classification extends to elliptic diffusions on closed manifolds. Limits: the guaranteed effects are small at weak amplitude (order ε2); the classification is existential and does not compute the frequency curve of a given environment; the random results concern quenched (and, where stated, annealed) exponents. The archive contains the manuscript (PDF and LaTeX sources), bibliography, the checking script verify.py with its retained output (81 exact rational checks and 193 numerical illustrations) and figure, negative controls, a replay comparer, a rating-free review summary, a prior-art summary, CI and SHA-256 sums. Assurance: written proofs and computational checks; internal AI-assisted review only (not peer review, independent reproduction or formal verification). Creator: Anonymous. Text, data and figures CC0-1.0; code MIT.