Modular Period Structure and Fractal Laws of the Affine Sequence T_k = 6·2^k + 1

We study the integer sequence T_k = 6·2^k + 1 (OEIS A004119) and establish a complete theory of its modular periodicity. For each positive integer m, we define the OMARWA period P(m) as the minimal period of the sequence T_k mod m, and prove the Core Reduction Theorem: P(m) = ordm'(2) where m' = m/gcd(m,6). We derive fractal period laws, a super-period L = 30 via three independent paths, and observe coincidences with exceptional Lie-theoretic Coxeter numbers. All principal theorems are formally verified in Lean 4 with Mathlib (654 declarations, zero sorry axioms, 25 modules). Software DOI: 10.5281/zenodo.19024222.