SSD–HLV Modulation of Hydrogen Strong Lines: Cross–Scale Geometric Signatures in Precision Spectroscopy

We present a unified residual analysis of the NIST “Strong Lines of Hydrogen (HI)” dataset, focusing on structured deviations that persist after the application of state-of-the-art quantum electrodynamics (QED) corrections. By combining the Sigma–Schmidt Duality (SSD) framework with an effective Helix–Light–Vortex (HLV) geometric parametrization, we analyze the residuals using a log–modulated ansatz featuring a discrete helical Z3 phase structure. Our analysis reveals a statistically significant reduction of χ2 relative to null and randomized control models, with a dominant n = 3 Fourier mode consistently emerging across independent Balmer and Paschen sub–series. Model comparison using information criteria (AIC/BIC) confirms the robustness of this geometric pattern against overfitting. These results establish high–precision hydrogen spectroscopy as a sensitive testbed for detecting structured residual signatures and provide a reproducible statistical framework for assessing effective, geometry–organized deviations within precision atomic data.