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 sup_u Φ_s(u) of an independent Bradley–Terry product never exceeds the support function h_LO(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) < 2γ = log₂9 − 3 = 0.169925…, where g(τ) = 2·MAS(τ) − n(n−1)/2 is the maximum acyclic-subgraph advantage; thus r(τ) < 2γ refutes the principle outright (the criterion is a base-point test, not a characterisation of failure). We exhibit an explicit, deterministic, machine-checkable counterexample at the Paley tournament on n = 811 vertices, certified by a von Neumann trace bound yielding r(Paley(811)) ≤ 0.16825 < 2γ; a first-moment argument with a padding lemma refutes the principle non-constructively for every n ≥ 787. Our main theorem is a matching negative result on the certification method itself. Writing 2g(Paley(p)) = max_P ⟨PQPᵀ, Σ⟩ with Q the Paley skew matrix and Σ_ij = sgn(j−i), we prove that every bound on g(Paley(p)) obtained through a unitarily-invariant matrix-norm (symmetric-gauge) dual pairing is at least (1/π)·p^(3/2)·ln p·(1+o(1)), with the leading constant 1/π obtained from the exact singular values σ_k(Σ) = |cot((2k−1)π/2p)| and the evaluation ‖Σ‖_* = (2/π)·p·ln p + O(p). The bound is the minimum of the Schatten–Hölder family — exactly, at every finite p, by a power-mean argument — so no such bound is log-free: the log p separating deterministic upper bounds from the conjectured Θ(p^(3/2)) truth is intrinsic to the spectral category. The barrier applies verbatim to the wider family of skew Hadamard difference-set tournaments. The ancillary script verify-certificate.py reproduces all certified numerical claims: the Gram identity QQᵀ = pI − J, the p = 811 certificate verified in interval arithmetic (bound 55262.26 < threshold 55812.72, rigorous gap > 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.