Operator Package 2: Discretized Log-Scale Spectral Mechanics for APM Boundary Closure

This paper standardizes Operator Package 2: an APM-ready discretization layer that converts the continuous log-scale backbone of Operator Package 1 into a controlled, self-adjoint, finite-window module with discrete mode families. The organizing principle is that physical APM systems occupy a finite scale window and that Aether-unit packing/holonomy induces boundary closure. Finite QMU-normalized scale window. The log-scale coordinate is taken as$$q=\log(\ell/\lambda_C),\qquad q\in[q_-,q_+],$$with window length$$L_q=q_+-q_-=\log(\ell_{\max}/\ell_{\min}).$$ Ledger-One normalization anchor. The module is ledger-anchored through the standardized closure$$A_u\cdot \mathrm{curl}={F_q}^{2}{\lambda_C}^{2}.$$Define the propagation constant$$\kappa_0:=F_q\lambda_C,$$so $A_u\cdot \mathrm{curl}={\kappa_0}^2$, providing the canonical scaling used to normalize discretized generators and Hamiltonians. Deliverables on the interval Hilbert space. On $\mathcal{H}_I=L^{2}([q_-,q_+],dq)$ the package provides three reusable operator components: (1) Discretized log-translation generator (quasi-periodic closure).Define$$P_{\vartheta}f:=-i\,\frac{df}{dq},$$with domain $f\in H^{1}([q_-,q_+])$ satisfying the quasi-periodic boundary trace$$f(q_+)=e^{i\vartheta}f(q_-),\qquad \vartheta\in[0,2\pi).$$This yields a discrete scale-harmonic basis with eigenfrequencies$$t_n=\frac{2\pi n+\vartheta}{L_q},\qquad n\in\mathbb{Z},$$and eigenfunctions$$e_n(q)=L_q^{-1/2}\exp\!\big(i\,t_n\,(q-q_-)\big).$$The operator is self-adjoint and diagonal in this basis, producing a controlled discrete scale spectrum parametrized by $(L_q,\vartheta)$. (2) Ledger-normalized Sturm--Liouville scale Hamiltonian (discrete spectrum).A self-adjoint APM Hamiltonian template is defined by$$H_{\mathrm{APM}}f:=-\frac{d^{2}f}{dq^{2}}+V_{\mathrm{APM}}(q)f,$$with separated Robin boundary conditions at $q_-$ and $q_+$, ensuring a purely discrete spectrum on the compact interval. A ledger-normalized form is obtained via$$H_{\mathrm{L}}:={\kappa_0}^{2}\,H_{\mathrm{APM}},$$preserving the $A_u\cdot\mathrm{curl}={\kappa_0}^2$ anchor at the operator level. (3) Truncated xi-kernel operator (compact, self-adjoint, Fej\'er smoothing).Using the even theta-built kernel $\Phi$ from Operator Package 1, define the truncated integral operator$$(W_I f)(q):=\int_{q_-}^{q_+}\Phi(q-u)\,f(u)\,du.$$On a finite window this operator is Hilbert--Schmidt, hence compact and self-adjoint, with a discrete spectrum accumulating only at $0$. In the discrete scale-harmonic basis $\{e_n\}$, the diagonal matrix elements admit the exact finite-window formula$$\langle W_I e_n,e_n\rangle= \frac{1}{L_q}\int_{-L_q}^{L_q}(L_q-|v|)\,\Phi(v)\,e^{-i t_n v}\,dv,$$equivalently showing that finite-window truncation produces a Fej'er-type smoothing of the full-line multiplier in discrete scale-frequency space. APM interpretation. The boundary phase $\vartheta$ functions as an APM holonomy dial (periodic, anti-periodic, or quasi-periodic closure), converting scale ratios into discrete, ledger-normalized mode families. The package therefore supplies an explicit, reusable discretization mechanism that is compatible with APM practice: finite scale windows, boundary closure, self-adjointness, and a precise connection between discrete log-scale spectra and xi-kernel filtering on bounded domains.