An exact congruential reformulation of the failure condition for twin prime pairs

We associate to each integer n ≥ 1 the candidate twin pair (6n−1, 6n+1), whose product is 36n²−1. We prove that this pair fails to be a twin prime pair if and only if the index n lies in one of the congruence classes n ≡ ±6⁻¹ (mod p) determined by a prime p ≥ 5 satisfying p ≤ √(6n∓1). This provides an exact reformulation, in the language of congruences, of the failure condition for twin pairs, recovering the classical twin-prime sieve framework and making explicit the bound needed to exclude the case p = 6n∓1. Counterexamples show that this bound is indispensable.