Proof of Twin Primes Conjecture

This paper presents a self-contained, constructive framework establishing the infinitude of twin primes via factor-shift dual-sieve dynamics on the central 6k ± 1 integer lattice. We demonstrate that every prime modulus p_i ≥ 5 operates as an orthogonal dual siever that eliminates exactly two disjoint residue classes per modulus length, advancing from diagonal square entry points with deterministic alternating strides (d_1i, d_2i). This mechanism rigorously decouples composite generation into Main Arithmetic Progressions (6k + 1) and Intermediate Arithmetic Progressions (6k - 1). By identifying the global synchronization node (commonk) and its complementary reflection node (commonIntermediatek), we prove that for every primorial horizon P_n ≥ 11#, the forward and backward sieve dynamics exhibit exact bilateral mirror invariance across the periodic span. Under the contradiction hypothesis that twin primes terminate at a finite maximum p_ltwin, the macroscopic interval up to a distant primorial p_z# forms an absolute zero-gap dead zone. By extending the modular framework to the square-root cutoff p_y# = pyrm, this dead zone is replicated m times under the closed active prime set A_y. We show that the total elimination capacity of the deep-space primes across all m replicated blocks is upper-bounded by m · l, whereas the exact mirror sector strictly demands the full Chinese Remainder totient ∏ᵢ₌₁^y (p_i - 2) survivors. Because the exponential growth of the lower primorial totient permanently outpaces the deep-space capacity, uneliminated candidate gaps must leak into the dead zone. As all prime divisors up to √(6r+1) are accounted for, these survivors unconditionally constitute genuine twin primes strictly exceeding p_ltwin, completing the proof by contradiction.