Explicit-Formula-Tied Spectral Recovery of Riemann-Zeta and Dirichlet L-Function Zeros from Finite Prime Data

We formulate the correspondence between prime-power data and zeros of zeta and Dirichlet L-functions as a structured line-spectral inverse problem. Under RH or GRH, explicit-formula normalization converts each conjugate zero pair into a sinusoidal component whose amplitude and phase are analytically determined by its frequency, leaving only one free parameter. We show that this explicit-formula tying improves frequency localization across several classical spectral estimators. In matched baseline experiments, the best tied method achieves a frequency MAE of 0.007665, compared with 0.014337 for the best untied baseline. We further analyze finite-window interference, automatic model selection, joint refinement, spectral crowding, and selection-aware inference. Extending the method to a family of Dirichlet L-functions reveals a strong arithmetic spectral fingerprint: own-catalogue enrichment has median 10.16, whereas foreign-catalogue enrichment is 0.982; all 12 own-catalogue comparisons remain significant after Benjamini–Hochberg correction, while none of 132 foreign comparisons do. Negative controls and scaling experiments also identify limitations of FDR calibration and asymptotic CRB behavior. These results are computational and conditional on critical-line models, and do not constitute evidence or proof of RH or GRH.