Finite-core Volterra reductions for a Weyl-positive Riemann phase kernel

We record a Weyl-positive reduction and certificate framework for the Riemann phase kernel associated with the even Riemann kernel $\mathrm{\Phi }$. The manuscript does not present a complete proof of the Riemann hypothesis. Its immediate analytic target is a concrete positivity theorem for a Weyl kernel whose quantum characteristic function satisfies the Kastler–Loupias–Miracle-Sole condition in all numerical tests performed so far. Several natural factorizations are ruled out. In particular, the positive anti-Wick density route is obstructed by a local heat-deconvolution test, and several natural finite-core reductions are excluded by explicit counterexamples. The surviving structure is a finite-core Volterra program upgraded to a closed-trace quotient certificate for the full kernel. We derive exact same-sign finite-core formulae, the second-order theta-mode identity ${\phi }_n(t)=({\mathrm{\partial }}_t^2-1\mathrm{/}4)(e^{t\mathrm{/}2}{e}^{-\pi n^2{e}^{2t}})$ for $n\ge 1$, a Volterra boundary-plus-tail representation, and a quotient Schur factorization for the normalized full-$\mathrm{\Phi }$ source/Volterra model. The latest certificate closes the active trace-range condition, the full-continuum source-inactive domination, and the Douglas/Moore–Penrose Schur hypotheses in the normalized model. What remains outside that certificate is explicitly separated: the quotient-to-original Weyl lift, uniform $\omega$-coverage for $\mathrm{\mid }\omega \mathrm{\mid }<1\mathrm{/}2$, and the final bridge from Weyl/KLM positivity to the intended de Branges or RH-side formulation.