math.NT — Number Theory
The Willow Transform and the Blind Zone Theorem: A Sharp Information-Theoretic Obstruction in the Turán Hierarchy for the Riemann xi-Function
Let Xi(t) = xi(1/2 + it) be the Riemann Xi-function, let Xi(t) = sum_{k>=0} (-1)^k c_k t^{2k} be its even Taylor expansion at the origin, and let D(n,r) = det[c_{n+i-j}] (i,j = 0..r-1) be the Toeplitz minors whose positivity for all n >= 0, r >= 1 is equivalent to the Riemann Hypothesis (RH). We prove a sharp and quantitative obstruction in the resulting hierarchy of criteria. Parametrising the zeros of G(z) = sum c_k z^k by beta_j = -1/z_j, we establish by the dual Jacobi-Trudi identity the exact identity D(n,r) = s_{(r^n)}(beta), the Schur function of the rectangular partition (r^n) in the variables beta. A zero of zeta off the critical line contributes to beta a conjugate pair rho e^{+-i theta} whose angle theta is exactly the vertical angle of the violating pair of zeros, while zeros on the critical line contribute positive real variables. For two variables we prove the closed form s_{(a,b)}(rho e^{i theta}, rho e^{-i theta}) = rho^{a+b} sin((a+1-b) theta) / sin theta = rho^{a+b} U_{a-b}(cos theta), U_k being the Chebyshev polynomial of the second kind, so that the sign is governed by the classical criterion U_k(cos theta) > 0 if and only if (k+1) theta < pi. Since a Schur function in two variables can be expanded through the coproduct s_lambda(X sqcup Y) = sum_mu s_mu(X) s_{lambda/mu}(Y), and since s_mu(X) > 0 for positive real X, positivity reduces to the two-variable factors, whose index k = a - b is bounded by the number of columns r, not by the shift n. This yields: Blind Zone Theorem. If a violating pair subtends a vertical angle theta < pi/(r+1), then every Toeplitz minor D(n,r') with r' <= r is strictly positive, notwithstanding the presence of non-real zeros. The critical strip together with gamma_1 = 14.134725141... forces theta <= theta_max = 2 arctan(1/(2 gamma_1)) = 0.07071826... rad, whence pi/theta_max = 44.424... and therefore every order r <= 43 is completely non-discriminating. We emphasise that the exponential bound is on the order r and not on the shift n; we exhibit the mechanism and show the contrary n-based bound is false. We also record the refutation of a "universal covering law" previously conjectured by the author, and a floating-point artefact that nearly produced a false refutation of the main theorem. No claim of a proof of RH is made.