The Twin Prime Formula and the Infinitude of Twin Primes

I introduce the Twin prime detector formula D(p), a closed-form expression involving ratios of Gamma function products that evaluates to 1 if and only if (p,p+2) is a twin prime pair. By applying the Gauss Multiplication Formula to simplify the MAF to 〖(2π)〗^((p+2 -(σ(p)+σ(p+2))/2)), I establish an exact equivalence between the twin prime condition and the vanishing of an arithmetic exponent involving the sum-of-divisors function σ. Building on this characterisation, I construct a GMF-weighted Dirichlet series L(s) whose double pole of order 2 at s = 1 encodes the twin prime distribution. I derive the nonvanishing of the Hardy-Littlewood constant C₂ directly from the GMF local correction factors, establish the sieve dimension κ = 2 as a structural consequence of the two-product GMF architecture, and obtain a Bombieri-Vinogradov-type remainder bound through a GMF-derived zero-free region. Assembling these components, I prove π_2 (x)∼2C_2 Li_2 (x)∼(2C_2 x)/ln^2⁡x →∞, establishing the infinitude of twin primes.