Normalized Interference and Near-Null Geometry in a Critical Prime Archimedean System

Abstract

This document studies a mathematical structure built from two operator channels on the same finite-dimensional space: an arithmetic "prime channel" p, assembled from a finite sum over prime powers, and an analytic "archimedean channel" a, assembled from an integral over a fixed density. Compared through a common energy H0 = p² + a², the residual R0 = p + a develops an extreme near-null geometry on a distinguished deep spectral subspace D_r: eigenvalue depths of the associated operator T0 reach 10⁻¹⁰⁰ and beyond, produced by two layers of cancellation, a scalar layer and a residual layer, sitting on top of an exact common integral architecture shared by both channels. The document separates what is proved from what is numerically observed from what remains open, using a consistent four-level marker system throughout (exact identity, derived asymptotic result, computational regularity, open problem), and records at comparable length a set of plausible simpler explanations for the near-null geometry that were formulated precisely, tested computationally, and eliminated — including a rank-5 "universal core" construction that is explicitly retracted once its defining measure is shown to diverge. The a priori construction of D_r directly from p and a, without first diagonalizing the comparison operator, remains an open problem. A second, independent result concerns the near-null projector once it exists: fixing a rank and a trajectory of configurations, the response of the projector to smooth trajectory deformations is shown to admit an exact perturbative expansion, R(ε) = 1 + A(𝒢)ε + B(𝒢)ε² + O(ε³). Both coefficients are derived — not fitted — from the exact projector family via its reduced resolvent and a second-order projector-perturbation identity obtained directly from idempotency, and both are confirmed against independent numerical measurement across multiple geometries, including a sign reversal of the second-order curvature B between two configurations that the same formula predicts correctly in both directions.