On the Deterministic Distribution of Twin Primes in Residue Classes: A Local Hardy–Littlewood Formula

DOI: 10.5281/zenodo.18169229Title: On the Deterministic Distribution of Twin Primes in Residue Classes: A Local Hardy–Littlewood FormulaAuthor: Miles Enoch Tracy contact: milestracy@yahoo.com ORCID: https://orcid.org/0009-0006-7781-8259 **Description/Abstract:** We prove that for any modulus M > 2, twin primes (p, p+2) can only occur in residue classes a(mod M) satisfying gcd(a, M) = 1 and gcd(a+2, M) = 1. The density enhancement factor for such allowed classes relative to a uniform distribution across all reduced residue classes is given explicitly by **R(M) = Π_{p|M, p>2} (p-1)/(p-2)** \[R(M) = \prod_{\substack{p \mid M \\ p > 2}} \frac{p-1}{p-2}.\] For M = 12, this yields R(12) = 2, explaining the observed absence of twin primes in classes 1 and 7 mod 12 and their doubled density in classes 5 and 11. The proof is elementary, relying only on modular arithmetic and the Chinese Remainder Theorem, yet provides a local finite-modulus analogue of the Hardy–Littlewood twin prime constant. Empirical verification up to 10^6 confirms the formula for moduli 6, 12, 30, 60, and 210. This result bridges sieve-theoretic exclusion with probabilistic models, offering a quantitative explanation for residue-class biases in twin prime distributions. **Keywords:** twin primes, residue classes, modular arithmetic, Hardy–Littlewood conjecture, sieve theory, Chinese Remainder Theorem. **MSC Classes:** 11N05, 11N35, 11A41, 11B25.