On an Invariant Ratio Involving Prime Gaps and Odd Components of Twin Primes

We introduce an algebraic sequence transformation based on the 2-adic factorization of prime gaps (g_n) into their power-of-two and odd components, anchoring our formulation on the classical, universally accepted OEIS integer sequences A007814 (binary valuation, exponent k) and A000265 (odd part, m_n). We formally define a primary sequence A(n) = g_n and a dual sequence B(n) = m_n / 2. We prove that for any pair of twin primes (characterized by a prime gap g_n = 2), the ratio R(n) = A(n)/B(n) is invariant and identically equal to 4. Furthermore, we generalize this relation to prove Bressan's Invariant Ratio Theorem, establishing that for any arbitrary prime gap g_n = 2^k * m_n, the ratio R(n) scales exactly as 2ᵏ⁺¹. Computational verification up to p_n < 100 via Python algorithms is included to empirically support the mathematical proofs.