A Contradiction-Based Proof Sketch Supporting the Twin Prime Conjecture

We assume the negation of the Twin Prime Conjecture, that there exists a largest twin prime pair, and derive structural consequences that lead to contradiction. The argument leverages parity, divisibility, and recursive shifting to demonstrate that such maximal pair cannot exist. Empirical reinforcement and modular analysis further support the contradiction-based framework, strengthening the case for the infinitude of twin primes.