Number-Theoretic Modular Symmetries in the Structure of the Fine-Structure Constant α−1 and QED Vacuum Shielding version 2

This paper analyzes the mathematical synthesis of repeat-

ing decimal cyclic groups generated by fractions of primes ending

in 7 (p ∈ {7,17,37,137}). We explicitly map out the 8-digit cyclic

permutation block generated by 1/137, demonstrating its deterministic

numerical breakdown into components that match the physical value of

the inverse fine-structure constant α−1 ≈ 137.035999. Following the

foundational traditions of dual-wave mechanics established by Louis de

Broglie and extended deterministically by Marius Borneas, we show that

the 0.000497 quantum deviation is mathematically traceable to localized

QED vacuum polarization loops.