Operator Package 4: APM-Derived Theta Kernel and De-Circularization of the Xi Filter

This paper is the fourth module in a three-stage operator program (Packages 1--3) for studying the Riemann $\Xi$-function using self-adjoint log-scale mechanics and finite-window spectral selection. Packages 1--3 introduce a compact kernel operator whose Fourier multiplier is $\Xi(t)$, but the kernel is imported by matching classical analytic number theory data (theta-built input), leaving a single circularity if the long-run objective is a Hilbert--Pólya-style closure program. Operator Package 4 removes that circularity by deriving the kernel from Aether Physics Model (APM) first principles using APM lexicon and invariants. The core construction builds an APM-native partition function from: the APM space-resonance cell structure (Euclidean coordinates coupled to two orthogonal cyclic parameters), integer loxodromic winding sectors constrained by the APM Aether geometry constant, and packing/holonomy closure constraints. From the APM partition, the paper defines an induced log-scale kernel $\Phi_{\mathrm{APM}}(q)$ and its Fourier multiplier $\Xi_{\mathrm{APM}}(t)$. The paper proves the necessary analytic properties (real, even, rapidly decaying kernel; entire, even multiplier) and shows that the operator-theoretic machinery of Packages 1--3 carries over verbatim under the substitution $\Phi \mapsto \Phi_{\mathrm{APM}}$ (and $\Xi \mapsto \Xi_{\mathrm{APM}}$). Key result (programmatic)The paper isolates the remaining step required for an RH-resolution program to a single identification target:prove $\Xi_{\mathrm{APM}}(t)\equiv \Xi(t)$ from APM invariants and/or construct a characteristic determinant (secular equation) whose zero set coincides with $\Xi_{\mathrm{APM}}$. This sharply separates (i) the physically derived kernel and operator well-posedness from (ii) the final identification/uniqueness step. What’s inside APM-native definitions for frequency vs. resonance and the space-resonance cell. An APM winding-lattice partition and its modular inversion symmetry (cycle-exchange reciprocity). A derived xi-kernel $\Phi_{\mathrm{APM}}(q)$ and induced multiplier $\Xi_{\mathrm{APM}}(t)$. Proofs of parity/reality, super-exponential decay, and entire extension of $\Xi_{\mathrm{APM}}$. Drop-in replacement rules for Operator Packages 1--3 (infinite line and finite-window operators). A clean statement of the “Identification Target” theorem (the remaining RH program fork).