Operator Package 7: Identification Closure for the APM-Derived Xi Multiplier

This record releases Operator Package 7 in the Riemann Function Operators series. Package 7 closes Module A (Identification) of the RH proof program formalized in Operator Package 6 by proving that the APM-derived xi multiplier coincides with the classical Riemann $\Xi$ function. Starting from the canonical APM theta partition derived in Operator Package 4 (resonance-cycle exchange realized as Poisson summation / modular inversion), this paper adopts the unweighted winding-lattice normalization selected by APM packing closure and defines:\[\theta_{\mathrm{APM}}(x)=\sum_{n\in\mathbb{Z}} e^{-\pi n^2 x},\quad\Theta_{\mathrm{APM}}(q)=\theta_{\mathrm{APM}}(e^{2q}),\quadG(q)=e^{q/2}(\Theta_{\mathrm{APM}}(q)-1),\]\[\Phi_{\mathrm{APM}}(q)=\frac12\left(\frac14-\frac{d^2}{dq^2}\right)G(q),\quad\Xi_{\mathrm{APM}}(t)=\int_{-\infty}^{\infty}\Phi_{\mathrm{APM}}(q)e^{itq}\,dq.\]The main result is an exact identification chain:\[\theta_{\mathrm{APM}}=\theta\quad\Rightarrow\quad\Phi_{\mathrm{APM}}=\Phi_{\Xi}\quad\Rightarrow\quad\Xi_{\mathrm{APM}}(t)\equiv \Xi(t)\]as entire functions on $t\in\mathbb{C}$, where $\Xi(t)=\xi(\tfrac12+it)$ is the classical Riemann $\Xi$. To prevent hidden normalization drift, the paper also introduces an explicit degeneracy-weighted deformation $\theta_{\rho}(x)=\sum \rho(n)e^{-\pi n^2 x}$ and proves that non-canonical sector weights produce an entire discrepancy term in the induced multiplier unless the weights are trivial. This isolates the identification to the canonical APM normalization and provides a negative-control test for implementations. Finally, Package 7 provides operational checkpoints (Maclaurin jet tests via kernel moments, first-zero agreement checks, and deformation sensitivity tests) intended to validate numerical pipelines before proceeding to the remaining RH modules: Module B: large-limit determinant closure (Operator Package 5 conjecture), Module C: positivity/canonical-system mechanism forcing real zeros. Series context Packages 1--3: operator backbone (log-scale generator, finite-window discretization, coherence selection). Package 4: APM-derived theta kernel (de-circularization via resonance-cycle exchange / Poisson summation). Package 5: determinant generator and large-limit closure conjecture. Package 6: formal RH proof program (Modules A--C). Package 7 (this paper): completes Module A by proving $\Xi_{\mathrm{APM}}\equiv\Xi$.