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

We present a clean conditional Hilbert--P\'{o}lya framework on a compact Riemannian spin manifold. For $\widehat H=\slashed D_E+\Phi$ with real scalar $\Phi$, we establish: (I) essential self-adjointness and discrete spectrum; (II) a corrected first heat coefficient $A_1$ for $\widehat H^2$; (III) a centered zeta--determinant $\mathcal D(s)=\det_{\zeta}((s-\frac{1}{2})^2+\widehat H^2)$ which is entire, even, and has no linear Hadamard factor. A Trace--Prime Hypothesis (TP) injects the Dirichlet block $\sum\Lambda(n)n^{-s}$ from the centered heat trace, producing an odd entire remainder $H$. Under sectorial analyticity of a relative centered trace (two concrete SA frameworks: cylindrical ends and short-range on $\mathbb R^d$) we obtain uniform bounds $H(\frac{1}{2}+it)=O(1)$ and, via a Phragm\'{e}n--Lindel"{o}f uniqueness principle, conclude $\mathcal D\equiv\xi$ 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 $\nu_\varepsilon$ and $\tau$-trace limit), and a Birman--Kre\u{\i}n ``implant'' of (TP) in a scattering avatar. The program stays consistent with compact Weyl asymptotics and places all arithmetic content solely in (TP).