A Congruence Construction for Infinitely Many Twin Primes

This preprint presents a congruence-based construction aimed at establishing the existence of infinitely many twin primes. The construction begins by restricting integers to residue classes modulo 6 compatible with twin prime pairs and then successively excludes divisibility by larger primes for both members of the pair. At each finite stage, the Chinese remainder theorem provides compatible admissible integers. A finite stage primality criterion, based on the fact that every composite integer has a prime divisor not exceeding its square root, is then used to identify admissible integers that form twin prime pairs within the corresponding quadratic range. The global argument incorporates a quantitative prime selection result showing that, at sufficiently large scales, enough primes remain outside the prescribed forbidden residue classes to sustain the construction. Combining the finite stage construction with the prime selection argument yields admissible twin prime candidates at arbitrarily large stages.