IMM Companion: Two Spectral Fingerprints of the Riemann Zero Measure — Unifying the Turán/Jensen and CMO/det2 Programs

We present a structural unification of two historically independent approaches to the Riemann Hypothesis (RH): the Turán/Jensen programme and the CMO/det2 programme. We show that both are dual analytic representations of the same underlying object: the Riemann zero measure μ = Σk δγk. The Turán programme reads μ through its reciprocal power moments p2n = Σk γk-2n via the Stieltjes transform; the CMO/det2 programme reads μ through its Poisson harmonic extension Pλ(τ). The explicit bridge between these two lenses is the Cauchy–Stieltjes identity Pλ(τ) = (λ/π) Im(S(τ + iλ)), where S(z) = Σn p2n z-2n-1 is the Stieltjes transform of μ. We prove that this identity is explicit, bijective, and has not previously been written connecting these two programmes. We further prove that RH is equivalent to the complete Stieltjes moment condition: the Hankel matrices Hk = [[p2(i+j+1)]]i,j=0..k-1 being positive definite for all k. Numerical verification confirms Hk ≻ 0 for k = 1, …, 5, with minimum eigenvalue decaying as ~10-6k, consistent with the infinite-order total positivity required by RH. The remaining analytical challenge — proving Hk ≻ 0 for all k — is identified as an instance of the Stieltjes moment problem applied to the zero measure.This version: Reproducibility bundle added: recovered session scripts (Hermite-Hankel/Jensen suite incl. the scale-normalization control behind the v2 retraction) plus an independent recomputation script (2026-07-12, labeled as such) that reproduces all Section 5 claims: Turan inequalities (13/13), Newton-Hadamard moment recovery vs direct zero sums (ratio 1.0000064 at n=3, ~15 s.f. for n>=4), and Table 1 Stieltjes-Hankel positive-definiteness with all four published lambda_min values reproduced to <0.1% (documented convention note: published Table 1 magnitudes correspond to the pre-descaling 4^n moment normalization; positive-definiteness is convention-invariant). Original Section-5 generating scripts were not preserved; see bundle README for the honest scope statement.