An Exact Periodic Continued Fraction Representation Linking Dimensionless Physical Coupling Constants

This research note reports a striking mathematical coincidence discovered within the regular continued fraction expansions of the reciprocal fine-structure constant alpha ^{-1} and the electroweak coupling constant at the Z-boson mass scale. We demonstrate that both fundamental physical parameters can be modeled using an identical 9-period infinite tail, \([1; 3, 1, 1, 18, 1, 7, 1, 3]\), combined with distinct integer prefixes [137; 27]and [128; 1, 17]. The resulting algebraic closed-form solutions yield numerical values A approx. 137.03599917705 and B approx. 128.946747361 that fall precisely within the current 2022 CODATA recommended experimental standard uncertainties and high-energy physics literature thresholds. Furthermore, by eliminating the shared periodic variable, we derive a direct, integer-coefficient Möbius transformation under the modular group GL2{Z} that maps the low-energy scale directly onto the high-energy scale. This explicit fractional linear relation proves that the scaling distance between these two critical coupling states can be represented via a rigid algebraic trajectory. Whether this tight mathematical coherence reflects an underlying discrete structure of the vacuum energy scale (renormalization group path) or represents an extraordinary numerical coincidence remains an open question for digital physics and experimental number theory.