Twin Prime Conjecture: A New Proof via the Tα System

The Twin Prime Conjecture, which asserts that there are infinitely many pairs of primes differing by 2, remains one of the most enduring open problems in number theory. This paper addresses the conjecture using the Tα axiomatic system—a novel arithmetic framework built upon the Alpha Ring structure—providing a resolution that transcends traditional sieve-theoretic approaches. We first formalize the core axioms of the Tα system, including translation irreducibility and binary decomposition, which allow us to model prime distributions as a system of symmetries on a 4-dimensional arithmetic manifold. Using this geometric representation, we map the twin prime problem to the existence of infinitely many independent sets in a 3-uniform hypergraph, where each vertex corresponds to a prime number. A key innovation is the construction of the Alpha Ring, which resolves the parity barrier by encoding the local parity symmetries of primes across all scales. This structure guarantees that for any bound N, there exist twin primes greater than N, establishing the infinitude of twin primes in a direct, constructive manner. This work not only settles the Twin Prime Conjecture but also demonstrates the power of the Tα framework as a unifying tool for number theory, with applications to other open problems in arithmetic and combinatorics.