A Sieve-Theoretic Reformulation of the Twin Prime Conjecture

This paper is a request for collaboration:The previous paper presented a sieve-theoretic reformation of the Goldbach Conjecture. The same methodology is now applied to the Twin Prime Conjecture. Polignac's Conjecture asserts that for every integer m, there exist infinitely many pairs of primes (p, q) such that |p − q| = 2m. This is trivially satisfied for m = 0, since infinitely many primes exist. The technique introduced in the prior work can be extended to any fixed m, though additional effort is required to cover all cases. This paper focuses specifically on the case m = 1, corresponding to the Twin Prime Conjecture. The problem is reformulated as the identification of semiprimes of the form pq = m² − n² = (m − n)(m + n), thereby reducing a prime pair to a single semiprime and enabling the application of sieve-theoretic analysis. A domain is selected in which the von Mangoldt function Λ(pq) vanishes for all odd composite values, thus avoiding unbounded analytic errors. Residue class elimination modulo small primes is employed to construct bounding equations describing the expected range and density of valid prime pairs. These equations are bounded and then numerically evaluated. All twin prime pairs are explicitly verified below 10¹², and it is sieve-theoretically proven that for all values of n > 10¹², the number of twin primes in the interval [31n/32, 33n/32] exceeds (1.3201 ± 0.0002)·n ⁄ (16 log² n). Consequently, the asymptotic relation limₙ→∞ π₂(n) = ∞ follows, confirming the infinitude of twin primes. This result constitutes a sieve-theoretic proof of the Twin Prime Conjecture and illustrates the generalizability of the semiprime-based sieve framework.