Operator Package 3: Coherence-Filtered Spectral Selection on Finite Log-Scale Windows in APM/QMU

This paper formalizes Operator Package 3 as the computational “selection engine” of the operator-package series. Operator Packages 1–2 provide (i) an infinite-line log-scale backbone and (ii) a physically realizable finite-window discretization with holonomy-controlled boundary conditions and a compact $\Xi$-kernel filter. Package 3 adds the missing operational step: a coherence-filtered spectral selection operator that turns these ingredients into a finite-dimensional, numerically stable prediction workflow. Core construction.On a finite log-scale window $I=[q_{\min},q_{\max}]$ with length $L_q=q_{\max}-q_{\min}$, define a self-adjoint discretized Hamiltonian$$H_{\mathrm{APM}}=-\frac{d^2}{dq^2}+V_{\mathrm{APM}}(q)$$with quasi-periodic (holonomy) boundary conditions$$f(q_{\max})=e^{i\vartheta}f(q_{\min}),\qquad f'(q_{\max})=e^{i\vartheta}f'(q_{\min}),$$and define the compact self-adjoint $\Xi$-kernel filter$$(W_I f)(q)=\int_{q_{\min}}^{q_{\max}}\Phi(q-u)\,f(u)\,du,$$where $\Phi$ is the real, even kernel whose Fourier transform gives the Riemann $\Xi$ function:$$\Xi(t)=\int_{-\infty}^{\infty}\Phi(v)\,e^{-itv}\,dv.$$For an energy cutoff $E_c$, let$$\Pi_{E_c}=\mathbf{1}_{(-\infty,E_c]}(H_{\mathrm{APM}})$$be the spectral projector onto the finite-dimensional subspace spanned by eigenfunctions with eigenvalues $\lambda\le E_c$. The \emph{selection operator} is$$\mathcal{S}_{E_c}:=\Pi_{E_c}\,W_I\,\Pi_{E_c}.$$The paper proves the basic operator-theoretic properties needed for robust computation: $\mathcal{S}_{E_c}$ is self-adjoint and finite-rank; it admits Rayleigh–Ritz variational characterizations; and its leading eigenvalues obey stability bounds under filter perturbations. Coherence-baseline calibration (exact Fejer smoothing).In the baseline limit $V_{\mathrm{APM}}\equiv 0$, the Hamiltonian eigenbasis coincides with the finite-window scale-harmonic basis (the Package 2 basis). In this limit, the diagonal response of the filter becomes an exact Fejer-smoothed sampling of $\Xi$:$$\langle e_n, W_I e_n\rangle=\int_{-\infty}^{\infty}\Xi(\tau)\,F_{L_q}(t_n-\tau)\,d\tau,\qquadF_{L_q}(x)=\frac{1}{2\pi}L_q\,\mathrm{sinc}^2\!\Big(\frac{L_q x}{2}\Big),$$with $\mathrm{sinc}(y)=\sin(y)/y$. This makes the finite-resolution effect explicit: finite windows do not access pointwise $\Xi(t)$, but a scale-dependent average whose bandwidth scales like $L_q^{-1}$. Outputs and intended use.Package 3 provides: a standardized definition block for $\mathcal{S}_{E_c}$ that downstream papers can cite verbatim; explicit matrix representations in both the Hamiltonian eigenbasis and the scale-harmonic basis; two practical scalar diagnostics for “coherence survival” within an energy band, including $\Lambda_{\max}(E_c)=\max \sigma(\mathcal{S}_{E_c})$ and the band-acceptance trace $\mathrm{Tr}(\mathcal{S}_{E_c})$; a ready-to-implement computational protocol: solve $H_{\mathrm{APM}}\psi_k=\lambda_k\psi_k$ up to $\lambda_k\le E_c$, assemble the finite matrix $(\langle \psi_j,W_I\psi_k\rangle)$, diagonalize, and scan holonomy/window/potential parameters.