Spectral Regulation of Scaling Flows on the Adèle Class Space: Intrinsic Regulator Derivation, Coercive Resolvent Reduction, and Dual Trace Factorization

Harmonic analysis on the global Adèle Class Space \mathbb{X} \equiv \mathbb{A}_\mathbb{Q}/\mathbb{Q}^\times occupies a central position at the intersection of automorphic representation theory, analytic number theory, and noncommutative spectral geometry. The infinitesimal generator of multiplicative dilation, H = -i(x\frac{d}{dx} + \frac{1}{2}) \cong -i\frac{d}{du}, possesses a continuous, unbounded spectrum \sigma(H) = \mathbb{R} when acting on the automorphic Hilbert space \mathcal{H}_\mathbb{A} \equiv L^2_0(\mathbb{X}, d\mu). Extending this unbounded scaling flow across the infinite product of non-Archimedean valuation trees produces severe domain instabilities, phase divergence, and ultraviolet singularities.

In this paper, we construct an unconditional, non-perturbative operator-theoretic framework for the two-sided regularized dilation operator: \mathcal{D}_\theta \equiv H \exp\left(-\theta |H|\right) = H \exp\left(-\theta \sqrt{H^2}\right). We establish that the spectral cutoff \theta is an intrinsic mathematical invariant uniquely determined by a Three-Phase fixed-point hierarchy.