A Rigorous Proof of the Twin Prime Conjecture — Based on the Rigidity of Prime-Formula Configurations and Nested Equidistribution

We present a rigorous elementary proof of the twin prime conjecture. The proof is based on the “prime-formula configuration” theory — a deterministic framework determined by the Chinese Remainder Theorem via ordered tuples of residues modulo each prime q ≤ p. Under the hypothesis that there are only finitely many twin primes, the safe interval (p^2, p1^2) is completely empty, and consequently the whole interval [1, p1^2] is absolutely devoid of candidate pairs. We select a smaller prime q such that q# ≤ p1^2 < (q + 1)#, which forces the first q#-subperiod to lie entirely within [1, p1^2] and thus contain zero candidate pairs. By the nested equidistribution property of stepwise sieving, the numbers of p-order candidate pairs in all q#-subperiods differ by at most a small constant, so the absolute emptiness of the first subperiod compels all subperiods to be almost empty. This makes the total number of candidate pairs far less than the rigid count (p − 2)# mandated by the Chinese Remainder Theorem, a contradiction. Hence there are infinitely many twin primes. The method extends to any admissible prime k-tuple.