The Two-Mode Envelope: A Lukewarm Counterexample and a Conjectural Phase Picture for Bradley–Terry Sigmoid-Tilt Models

For a ±1 tournament direction τ, the supremum sup_u Φ_τ(u) of the Bradley–Terry (BT) sigmoid-tilt free energy over score vectors u governs the KL cost of fitting a BT reward model to intransitive preferences. A natural maximum principle — that this supremum never exceeds the linear-ordering ground state h_LO(τ) = max_σ τ·v_σ — is true for small n but is refuted for spectrally balanced tournaments at large n (companion paper B; first deterministic witness n = 811). This paper develops the positive structure the refutation points to. We conjecture that the supremum is a two-mode envelope, sup_u Φ_τ(u) = max(Φ_τ(0), h_LO(τ)): the larger of an annealed free energy Φ_τ(0) = γ·C(n,2) + ½·Στ (order n², the maximally disordered "tied" tournament) and a ground-state energy h_LO(τ) (the "ranked" linear-order face, of order n^(3/2) in the Paley/quasirandom regime that motivates the conjecture), with a crossover at the critical structural label r(τ) = 2γ = log₂9 − 3, where γ = log₂3 − 3/2. For general τ we DISPROVE the envelope. A cyclic blow-up of three copies of Paley(811) is an explicit regular tournament on n = 2433 vertices whose optimum is strictly LUKEWARM (ranked between blocks, tied within), exceeding the tied face by (½ − 3γ)·811² ≈ 1.6×10⁵ and the ranked face by at least 8.2×10², certified end-to-end by the von Neumann bound of B. The envelope conjecture is accordingly restricted to doubly regular directions; the restriction is necessary, not prudential. We give rigorous, n-uniform structural support: a Hessian computation pinning the tied configuration as a strict local maximum with curvature −L_{K_n}/(36·ln2) for every regular tournament; an exact proof that every two-block face is dead for regular tournaments; and a quotient spectral certificate (a clean operator–nuclear duality bound) that all ≤4-block faces are dead at Paley(67). We report tied-side computational evidence at Paley(83) — a best-found ordering with g = 543, below the threshold 578.25 — which is evidence, not a theorem, since a lower bound cannot certify the required upper bound g < 578.25. Finally we develop the practical reading for reinforcement learning from human feedback (RLHF): the response-scale diagnostic r_R = 2·acc_MAS − 1 is the in-sample accuracy ceiling of any score-based reward model, measures ≈ 1.0 on UltraFeedback and PRISM, and is therefore exact about the rankability of systems but silent about reward-model fit quality, whose binding constraint (a measured 0.395 held-out gap of label noise and feature error) lives off the preference graph. The envelope for doubly regular τ and the Paley(83) tied-side conclusion are conjectural; the disproof for general τ, the structural lemmas, the Hessian anchor, and the Θ(n²) triangle-LP upper bound are rigorous. Ancillary pure-NumPy scripts reproduce the two theorem-critical certificates: the lukewarm counterexample C₃[Paley(811)] and the ≤4-block quotient certificate at Paley(67).