Twin Primes as a Persistence Phenomenon in a Canonically Indexed Sieve

We reformulate the twin prime problem within a fully deterministic and indexedframework, where divisibility and primality are encoded exactly by congruenceconditions on a single index axis. Using the canonical decomposition n=6t+r with r\1, 5\, twin primecandidates are represented as ordered indexed pairs with fixed relativedisplacement. For each prime p5, primality induces exactly two canonically anchoredcongruence obstructions on the index variable, one for each component of thepair. We analyze the exact periodic structure of these obstructions and theirinteraction at finite levels via Chinese Remainder Theorem assembly. Composite moduli introduce no additional freedom beyond prime intersections, and higher prime powers do not generate new constraints below their naturalindex scales. This yields a sharp separation between the structural congruential layer, whereinfinite compatibility always persists, and the Archimedean layer, whererealizability by actual integers is decided. We show that extinction of twin primes would require complete coverage of finiteintervals of the index axis by canonically anchored obstructions. Such coverage is ruled out by rigidity alone: the twin obstruction systemsatisfies bounded local multiplicity and therefore falls within the scope of ageneral Archimedean non--coverage principle. As a consequence, admissible indices persist beyond every finite scale, andinfinitely many twin prime pairs exist. All arguments are exact, finite, and deterministic. No analytic estimates, density heuristics, or probabilistic methods are used.