Evaluating infinite Hilbert space operators $L^2(1, \infty)$ under standard point-wise Dirichlet boundary constraints induces wave collapse or Robin phase-drift[span_13](start_span)[span_13](end_span). To preserve essential self-adjoint integrity ($E_n \in \mathbb{R}$), the framework defines a non-local boundary domain constraint $\psi(1) = \int_1^\infty K(x)\psi(x) dx$[span_14](start_span)[span_14](end_span).
The closed-form regularization kernel $K(x) = \frac{2}{9x} \cos\left(\frac{2\pi \ln x}{3 \ln 3}\right) \sin\left(\frac{\pi \ln x}{4 \ln 6}\right)$ combines Base-3 clock rotations with Base-6 anti-parallel inversions directly into the calculus of $K(x)$[span_15](start_span)[span_15](end_span).
This graph plots the non-local kernel $K(x)$ across spatial expanding coordinates $x \in [1, x_{\text{max}}]$[span_16](start_span)[span_16](end_span). Oscillatory positive and negative wave nodes illustrate how continuous destructive cancellation bounds integral energy, balancing localized boundary energy against infinite asymptotic limits[span_17](start_span)[span_17](end_span).