A Conditional Scalar Hilbert–Pólya Program: Self-Adjointness, Centered Determinants, and a Trace–Prime Hypothesis

We present a clean conditional Hilbert–Pólya framework on a compact Riemannian spin manifold. For H^=\slashedDE+Φ\widehat H=\slashed D_E+\PhiH=\slashedDE+Φ with real scalar Φ\PhiΦ, we establish: (I) essential self-adjointness and discrete spectrum; (II) a corrected first heat coefficient A1A_1A1 for H^2\widehat H^2H2; (III) a centered zeta–determinant D(s)=det⁡ζ((s−12)2+H^2)\mathcal D(s)=\det_\zeta((s-\tfrac12)^2+\widehat H^2)D(s)=detζ((s−21)2+H2) which is entire, even, and has no linear Hadamard factor. A Trace–Prime Hypothesis (TP) injects the Dirichlet block ∑Λ(n)n−s\sum\Lambda(n)n^{-s}∑Λ(n)n−s from the centered heat trace, producing an odd entire remainder HHH. Under sectorial analyticity of a relative centered trace (two concrete SA frameworks: cylindrical ends and short-range on Rd\mathbb R^dRd) we obtain uniform bounds H(12+it)=O(1)H(\tfrac12+it)=O(1)H(21+it)=O(1) and, via a Phragmén–Lindelöf uniqueness principle, conclude D≡ξ\mathcal D\equiv\xiD≡ξ after central normalization. We include: Schatten control, Seeley complex powers, a Gaussian Tauberian toolkit, a toy realization of the Trace–Prime identity as a trace-per-unit-length on the multiplicative line (with regularized comb νε\nu_\varepsilonνε and τ\tauτ-trace limit), and a Birman–Kreĭn “implant” of (TP) in a scattering avatar. The program stays consistent with compact Weyl asymptotics and places all arithmetic content solely in (TP).