A Scattering Reformulation and Conditional Arithmetic Reduction of the Riemann Hypothesis

This note develops a conditional reduction of the Riemann Hypothesis (RH) within a prime-scattering framework. No proof of RH is claimed. The work separates into three proved components and one explicitly open problem. First, for any finite prime set P and Re (s) > 0, an exact Euler-scattering identity is established: the ratio ZP (s) /ZP (¯s) of truncated Euler factors is unimodular. Second, the regularised Euler product Z^ (3) (s), obtained by removing the first two logarithmic prime contributions, is shown to converge absolutely and to be holomorphic for Re (s) > 1/3 and unimodular on the critical line. Third, the low-order prime contribution is isolated in a finite-scale, PNT-subtracted, smoothed form that is welldefined at every cutoff. All three components combine into a total finite-scale boundary factor Sᵗot_ X (E) that is unimodular at every finite X. The genuine remaining obstacle — uniform analytic completion to a Schur/Herglotz function and a cutoff-independent Carleson bound — is then formulated as an explicit open problem, rather than being asserted as a theorem. The note thereby provides a mathematically clean conditional reduction without overclaiming: RH would follow from a specific, identifiable arithmetic variance bound on the primary (k = 1) prime oscillatory block.