2604.00603 Spectral Invariance in International Football: A Multi-Scale Markov Analysis of Match Outcomes, 1902–2024
We model international football match outcomes (win, draw, loss) as a first-order Markov chain and investigate the spectral properties of the resulting transition matrices across 122 years of data (1902–2024; 47,914 matches, 332 teams). Despite significant secular declines in outcome persistence — P(W→W) and P(L→L) have both fallen over the century — the spectral gap of the transition matrix remains remarkably stable at \(\gamma \approx 0.