A Finite Lattice and Modular–Composite Framework for Twin Prime Admissibility

This paper presents a structural proof of the infinitude of twin primes based on a finite deterministic model of composite elimination. Restricting attention to the arithmetic progressions 6k − 1 and 6k + 1, we encode divisibility by primes p ≥ 5 as ordered elimination events acting on a lattice of twin slots. When projected onto modular phase systems, these elimination schedules collapse into a finite family of cyclic pathways with deterministic rail alternation.The interaction of these pathways defines a finite-state dynamical system with guaranteed recurrence and structural overdetermination. Collisions among elimination events produce irreversible loss of effective elimination capacity. A Hall-type capacity obstruction shows that permanent coverage of all admissible phases is impossible. As a result, uncovered twin slotsrecur infinitely often, yielding infinitely many twin primes.The argument is finite, combinatorial, and unconditional. It does not rely on analytic density estimates, probabilistic models, or hypotheses regarding prime distribution. Infinitude follows from structural impossibility of complete elimination rather than from quantitative abundance.Please note that ChatGPT was utilized for computation and proof drafting purposes.