A Log-Scale Operator Package for the Riemann Xi Function: A Foundation for APM-Linked Mathematics

This paper standardizes Operator Package 1: a reusable, ledger-compatible operator module on the Hilbert space $L^{2}(\mathbb{R},dq)$ intended to support future Aether Physics Model (APM) work that touches analytic number theory. What the package contains (Operator Package 1).Log-scale translation generator. A self-adjoint operator with explicit domain:$$P=-i\,\frac{d}{dq},\qquad \mathrm{Dom}(P)=H^{1}(\mathbb{R}).$$With the unitary Fourier transform $\mathcal{F}$ on $L^{2}(\mathbb{R})$, the operator is diagonalized as$$\mathcal{F}P\mathcal{F}^{-1}=M_{t},$$where $M_t$ is multiplication by $t$. Xi-convolution operator. A bounded self-adjoint convolution operator$$(Wf)(q)=(\Phi*f)(q),$$whose kernel $\Phi$ is…