The Infinitude of Twin Primes – A Constructive and Harmonic Double Proof

This document provides a double elementary proof of the infinitude of twin primes, developed through two independent approaches: (1) a modular proof using admissible patterns and residue classes in ℤ/nℤ, and (2) a recursive construction based on the sequence of twin centres. Both proofs avoid the use of complex analysis and rely solely on elementary number-theoretic arguments. The results contribute to the understanding of structural properties of twin prime configurations and support their infinitude.