A Cross-Domain Dual-Sector Spectral Fingerprint: Paper, Methods, and Reproducibility Materials

This paper investigates recurring spectral ratio structure across three distinct settings: a 465-state hydrogen dipole construction, Kerr black-hole quasinormal-mode decay rates, and graph spectra extracted from spatial images. It develops a reproducible framework for measuring how spectral ratios populate a fixed library of fifteen mathematical constants, then tests the recurrence, robustness and specificity of the resulting profiles using alternative libraries and structured controls. The finite investigation finds substantial cross-domain profile similarities. At the primary 3% matching tolerance, hydrogen-to-image Spearman correlations range from approximately 0.886 to 0.996 for five of six primary image datasets, with the embryo dataset providing a clear exception. The hydrogen-to-Kerr correlation is approximately 0.521 and remains essentially unchanged under shell-block multiset-preserving controls. In the Kerr study, the observed incidence exceeds all 4,999 holdout uniform controls and all but 22 of 4,999 coarse-density controls. These positive findings coexist with an important specificity limit: the geometry-matched alternative-library bank yields a joint reference fraction of 23.39%, failing the frozen 5% criterion. The results therefore support recurring, reproducible spectral ratio structure without establishing that the selected fifteen constants uniquely capture it. Hydrogen control results also depend on the spectral readout, helping identify which features survive particular transformations and preserved structures. The paper derives an exact connection between pairwise spectral ratios and autocorrelation in logarithmic spectral coordinates. It relates this framework to adjacent-gap ratio statistics and recent work on power-law spectra and random-matrix universality, while distinguishing their different observables, operators and evidential roles. The contribution is a finite empirical study and a reusable diagnostic framework for comparing spectral structure across domains. It does not establish a universal physical-realization law or a uniquely privileged constant library. Image provenance limits and unreproduced historical claims are explicitly documented. The accompanying public verification package provides executable code, retained numerical inputs, control data, generated tables, integrity hashes and instructions for checking the reported results. Reusable image-processing and spectral-analysis functions allow technically competent readers to apply the method to their own images and investigate its generalization beyond the datasets studied here.