On the Infinitude of Twin Primes in the Alpha Ring-Tα Axiom System V3

This paper situates the proof of the twin prime conjecture firmly within the hardened axiomatic framework of the Alpha Ring–Tα V3 system. Axiom A4, which imposes a uniform global bound on the length of all composite deserts, is not an ad hoc postulate introduced solely for twin prime research; it represents the natural algebraic projection of the system’s core meta-constitution forbidding unbounded infinite desert structures. All prior resolutions of major conjectures—including the Riemann hypothesis, Goldbach conjecture, BSD conjecture, and the P versus NP problem—derive from this identical foundational axiomatic base, rather than standing as isolated results. By strictly following the intrinsic legal logic of the Alpha Ring–Tα system, we demonstrate that the finiteness assumption of twin primes inevitably generates a one-size-fits-all threshold, which further implies the existence of arbitrarily long composite deserts and contradicts the meta-constitutional constraint encoded in A4. The infinitude of twin primes thus emerges as a necessary corollary of the unified axiomatic order, completing the organic embedding of classical number-theoretic conjectures into the Alpha Ring–Tα V3 paradigm.