MacMahon Convolutions and a Divisor-Gap Reformulation of the Twin Prime Conjecture

This preprint presents an exact algebraic and structural reformulation of the twin prime conjecture, shifting the analytical focus from traditional sieve-theoretic methods to the study of quasimodular divisor-sum convolutions. We introduce a quadratic prime-detecting functional, E(n) = (n2 - n + 1)σ1(n) - σ3(n), which acts as an exact interaction detector for semiprimes. By applying a quadratic coefficient-restriction (n = m2) to MacMahon's classical convolution identity for σ3(n), we establish the Square Coupling Theorem. This theorem rigidly links the internal divisor-sum convolution over the hyper-dense square m2 to the boundary divisor sum of its symmetric neighbor m2-1. To quantify the arithmetic distance from a twin prime state, we define the Divisor Gap δ(m) = σ1(m2-1) - m(m+2). Using the AM-GM inequality, we prove a strict discrete separation (the Forbidden Zone): δ(m) vanishes if and only if m-1 and m+1 are twin primes, and is bounded below by a macroscopic margin of m3/2 for all composite failure states. Restricting the analysis to the natural geometric class of twin prime centers m = 6u, we reveal a strict modular conservation law modulo 12, explaining the insufficiency of elementary congruence methods. We then decompose the boundary condition using Ramanujan Fourier expansions, formally reducing the twin prime conjecture to an open spectral threshold problem. Finally, we translate this structural gap into a Dirichlet-type indicator series Z6(s), framing the infinitude of twin primes as an exact convergence dichotomy. Note: The results presented in this paper are equivalence theorems and exact algebraic identities. No claim is made that the infinitude of twin prime centers follows unconditionally from the present methods; rather, the conjecture is rigorously translated into the continuous analytic domains of quasimodular forms and harmonic phase interference.