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 GΞ(w) = Ξ(√w)/Ξ(0) and its logarithmic moments σn, to positive semidefiniteness of the single Hankel tower Hr = [σi+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 = εeiθ is a rank-two update of the base Hankel matrix, and Hr > 0 if and only if |1 + λKr(λ,λ)| > |λ| Kr(λ̅,λ), where Kr 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, pn(0)2 = 2(4n+3)/π2 and Kr(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.