The delta orientation is the unique entropy minimizer for self-calibrated ±1 Walsh tilts on the Boolean cube

For each choice of signs ε_a ∈ {±1}, one per nonzero Walsh character χ_a on the Boolean cube G = {±1}^n (N = 2^n points), consider the exponential-family law P_ε(s) ∝ exp(Σa≠0 h_a ε_a χ_a(s)) whose parameters are self-calibrated by the fixed-point conditions EP_ε[ε_a χ_a] = e^(−h_a) for every a ≠ 0: each tilted correlation must equal the exponential of minus its own tilt parameter. We prove that this system has exactly one solution for every sign choice, and that the relative entropy D(P_ε ‖ U) from the uniform law is minimized, strictly and uniquely up to the natural N-element symmetry orbit, by the delta orientations ε_a = −χ_a(s⋆) — the sign patterns whose calibrated law is uniform on N−1 points and nearly extinguishes the single point s⋆. The proof is elementary and quantitative. Its core is a deep-dip trichotomy: any non-delta calibrated law with D ≤ 1/60 must extinguish at least three points of the cube to depth e^(−5), forcing N·D ≥ 3ψ(e^(−5)) > 2.878716 where ψ(x) = x log x − x + 1, while the delta value satisfies N·D_δ < N/(N−1). For n ≥ 6 this closes the theorem analytically; for n ≤ 5 we verify it by certified computation (exhaustive over all 2^(N−1) orientations for n ≤ 4; via a provably complete 176-orbit symmetry transversal of the 2^31 orientations at n = 5), with a posteriori Newton–Kantorovich error certificates throughout. The minimizing value is not a photo-finish: the best non-delta family we exhibit satisfies N·D → 4 log 4 = 5.545…, and we conjecture this is the sharp asymptotic runner-up constant. The self-calibration fixed point appears to be new; its nearest relatives are the extremal theory of Littlewood (±1-coefficient) polynomials and the self-consistent field equations of statistical mechanics, and we state our literature position explicitly in Section 1.4. Reproducibility: all computer-assisted claims ship with certification code (receipts w1–w5) and canonical outputs in the linked repository; the n ≤ 5 verification is exhaustive/transversal with a posteriori Newton–Kantorovich error certificates.