A Spectral Barrier for Paley Tournament Discrepancy, with an Application to a Maximum-Principle Refutation in Bradley–Terry Sigmoid-Tilt Models

We study the maximum principle for Bradley–Terry sigmoid-tilt models: the assertion that the base-2 log-moment-generating functional supᵤ Φₛ (u) of an independent Bradley–Terry product never exceeds the support function hLO (s) of the linear-ordering polytope on the tilt box ‖s‖_∞ ≤ 1. A single-point reduction shows that for a ±1 tournament direction τ the base point u = 0 gives an exact violation test: it violates the principle precisely when r (τ): = g (τ) / (n (n−1) /2) 550. 45 in g-units), the near miss at p = 787, the cotangent singular-value formula, the class-number cross-check Στ = p·h (−p), exact Held–Karp values for small Paley tournaments, and the first-moment crossover n = 787. It runs in seconds and requires only numpy and mpmath.