How often can migration switch a population above or below its total carrying capacity? Five crossings in three logistic patches, and bounds for any number of patches

Unrefereed research candidate. Habitat patches linked by migration: does migration raise or lower the total equilibrium population B relative to the sum S of the patches' carrying capacities? In the standard logistic patch model with an arbitrary irreducible, conservative (possibly asymmetric) migration network, B can switch between above and below S several times as the overall migration intensity grows. Elbetch, Benzekri, Massart and Sari (2022) found three such crossings with three patches and asked for a proof that three is the maximum, and for bounds with more patches. This package answers both requests: three patches allow exactly five crossings (an explicit rational example, certified in exact arithmetic, and a direct algebraic proof that six are impossible); for n patches there are at most 2n−3 crossings unless B is identically S, with no genericity assumption; and for every n ≥ 3 there are n-patch networks with at least n+2 crossings, with an exact four-patch certificate giving six. The five-crossing example has at least three separate peaks of total biomass. Effect sizes: tiny. In the three-patch example B differs from S by at most about 3.3 parts per million between crossings (numerical), approaching a 249 ppm shortfall at high migration; one migration rate is about 1/430 of the largest and the example is finely tuned. The results describe what the model allows, not observations of real populations. The archive contains the 13-page manuscript (PDF and LaTeX), exact standard-library checkers that do not rely on assert, numerical figure tools with pinned dependencies, 12 semantic negative controls, CI and SHA-256 sums. Assurance: written proofs and exact rational certificates; internal AI-assisted review only (not peer review, independent reproduction or formal verification); priority bounded to the recorded search. Related release: doi:10.5281/zenodo.23687852. Creator: Anonymous. Text and data CC0-1.0; code MIT.