A reduction: no finite Hankel prefix certifies the Riemann Hypothesis without Xi-specific input, and the exact rank at which a hidden pair appears

A reduction, displayed in the Frey–Ribet–Wiles style. The Riemann Hypothesis is equivalent, through the square-variable function G_Ξ(w) = Ξ(√w)/Ξ(0) and its logarithmic moments σ_n, to positive semidefiniteness of the single Hankel tower H_r = [σ_i+j+1] at every rank r (self-weighted Hamburger rigidity). The note shows that this equivalence cannot be truncated: any finite rank R proposed as a certificate for the genus-zero class is defeated by an explicit real entire function of order 1/2 with a nonreal zero pair whose first R Hankel blocks are positive definite.

New ingredient. An exact criterion for the first failing rank r_* of such a hidden pair: the inserted pair λ = εe^iθ is a rank-two update of the base Hankel matrix, and H_r > 0 if and only if |1 + λK_r(λ,λ)| > |λ| K_r(λ̅,λ), where K_r is the bilinear Christoffel–Darboux kernel of the base. This costs one three-term recurrence per rank and admits ball-arithmetic certificates. For the sine base, p_n(0)² = 2(4n+3)/π² and K_r(0,0) = 2r(2r+1)/π².

Computed rank. For the sine base with the pair ε = T^-2, θ = T^-1: r_* = 1.5283707 T + O(1), certified with python-flint/arb through T = 10⁵ (rank 152,838) and computed with PARI/GP to T = 10⁷. At T = 3·10¹² this places the first failing block near rank 4.6·10¹², about 7·10⁵ times the previously proved Lambert-W lower bound of 6,421,928. The linear law is empirical and stated as a conjecture; the criterion, the closed forms and the certified rows are proved.

Claim boundary. This is a reduction inside the genus-zero class and a computation for one abstract countermodel. It proves nothing about the literal Xi moments, supplies no Xi-specific detection rank, and does not prove or disprove RH. The attached package contains the full addendum (proofs), a fail-closed standard-library verifier (807 assertions, prints RH_OPEN), fourteen arb certificates with ball radii, and the GP evaluator.