Arithmetic Spectral Multiplier and Resonance Geometry of Logarithmic Translations

The preceding article [AX-135968] introduced a metric structure on the positive rationals induced by the translation geometry of a functional state on a finite logarithmic window. It was shown that exact invariance of a nontrivial L2L^2-state under a nonzero translation is impossible, while the corresponding small-translation defect has a quadratic leading scale. The present article develops the spectral counterpart of this construction. The central question is whether the arithmetic structure itself can enforce an additional, collective form of approximate translation invariance. For a finite logarithmic window Ia=[−a,a]I_a=[-a,a], we derive an exact spectral representation of the aggregate arithmetic translation defect in terms of a nonnegative multiplier Ma(t)M_a(t). Its low-frequency behavior is established and independently verified from the coordinate-space translation expansion. We then analyze the arithmetic phase alignment responsible for deep spectral minima and derive a variance identity in which logarithms of rational ratios naturally reappear. Finally, the arithmetic phase dynamics is represented as a linear flow on a finite-dimensional torus. Recurrence of this flow implies that a global pointwise lower bound for the arithmetic multiplier is impossible for any finite prime system. The remaining problem concerns the interaction between these recurrent arithmetic resonances and the spectral restrictions imposed by compact support of the underlying functional state.