An Analytical Framework for Goldbach Deviation Structure: Spectral Representation, Persistence, and Amplitude Hierarchy

This paper proposes an analytical framework for interpreting the structural properties of Goldbach deviations δ (N) = (G (N) - HL (N) ) /HL (N), where G (N) is the Goldbach representation count and HL (N) the Hardy-Littlewood prediction. Conditional on the Generalized Riemann Hypothesis (GRH), we develop three principal results: (1) Long-range persistence: The observed Hurst exponent H ≈ 0. 84 can be attributed to the quasi-periodic structure of L-function zeros via an explicit spectral representation. (2) Two-tier amplitude hierarchy: The global envelope scaling κglobal ≈ C₂ (twin prime constant) appears to control the maximal deviation, while the local arithmetic fine structure is governed by a coupling constant κₗocal = 1/24 arising from modular regularization of the singular series. (3) Spectral correspondence: The FFT peaks correspond to imaginary parts of low-lying Dirichlet L-function zeros as predicted by the explicit formula. Additionally, the framework accounts for the anomalous negative interaction terms (cₚq < 0) via the inclusion-exclusion principle inherent in sieve weights. We argue that these phenomena are interconnected manifestations of arithmetic structure controlled by L-function zeros, though rigorous proofs remain open problems in several cases. This work is intended as a theoretical companion to the empirical study "Fractal-Spectral Structure of Goldbach Deviations" (Chen, 2026).