Finite-Height Verification and Finite-Order Hamburger Positivity for the Riemann ξ-Function

This preprint establishes a quantitative transfer principle from finite-height verification of the Riemann Hypothesis to finite-order positivity in a Hamburger moment tower associated with the transformed nontrivial zeros of the Riemann ξ-function. For λρ=−14(ρ−12)2,pn=∑ρλρn,λ_ρ=-1/4(ρ-12)^2, p_n=∑ρλ_ρ^n, we study the shifted Hankel matrices Hr,n=[pn+i+j]i,j=0r−1.Hr,n=[pₙ₊ᵢ₊ⱼ]i,j=0ʳ⁻¹. Assuming the Riemann Hypothesis has been rigorously verified through height HH, we prove that for every fixed c<12c<12, all matrices Hr,nHr,n are positive definite for 1≤r≤clog⁡H,n≥1,1≤ r≤ clog H, n≥1, once HH is sufficiently large. The proof uses a positive core of verified critical-line zeros, an adversarial bound for all unverified zeros above HH, explicit zero-counting, and a Lagrange interpolation argument for transformed nodes asymptotic to j−2j⁻². The exact ideal-node weights ∣Lj(0)∣=2j2r(r−j)!(r+j)!|L_j(0)|= {2j²ʳ}{(r-j)!(r+j)!} yield exponential rather than superexponential interpolation growth, leading to the logarithmic order law. Using the current rigorous verification height H=3,000,175,332,800,H=3{,}000{,}175{,}332{,}800, we also give an explicit zero-ordinate-free finite certificate proving positivity for all shifts n≥1n≥1 through matrix order r=8r=8. The result does not assume the Riemann Hypothesis above the verified height and does not constitute a proof of the Riemann Hypothesis.