Proof of the Infinitude of Twin Primes via Trigonometric Primality Test: A Period-6 Geometric Structure

This paper presents two major results:
1. Proof of the infinitude of twin primes through geometric-structural analysis of the period-6 framework.
2. A novel trigonometric primality test based on elementary formulas with connections to the Riemann Hypothesis.
 
Two elementary formulas are introduced: 
Formula 1 identifies prime-eligible positions using cos(πx/3) = 1/2, arising naturally from the period-6 structure after eliminating multiples of 2 and 3; 
Formula 2 tests divisibility by primes p ≥ 5 using cos(3πx/p) = −1 to detect odd multiples. 
Together they provide a complete deterministic primality test using only elementary trigonometry.

The period-6 structure creates exactly two prime-eligible rows — numbers ≡ 1 and 5 (mod 6) — separated by a fixed distance of 2. Since both rows provably contain infinitely many primes with equal asymptotic density, and the geometric relationship is fixed by periodicity, infinitely many twin prime pairs must exist.

Remarkably, the critical value 1/2 — identical to the conjectured real part of Riemann zeta function zeros — emerges naturally from the geometric structure cos(π/3) = 1/2, suggesting deeper connections to the Riemann Hypothesis.

The discovery has interdisciplinary origins, combining pattern design intuition with mathematical training in electronics, illustrating how fresh perspectives can illuminate classical problems. A complete C implementation is provided for verification. The author welcomes collaboration from the mathematical community.