Cousin Primes as a Persistence Phenomenon in a Canonically Indexed Sieve

We reformulate the cousin prime problem within a fully deterministic and indexed framework, where divisibility and primality are encoded exactly by congruence conditions on a single index axis. Using the canonical decomposition $n=6t+r$ with $r\in\{1,5\}$, cousin prime candidates are represented as ordered indexed pairs at equal height, corresponding to the integer pairs $(6t+1,6t+5)$. For each prime $p\ge5$, primality induces exactly two canonically anchored congruence obstructions acting on the same index variable. These obstructions are rigid, admit no translational freedom, and never coincide. We analyze their exact periodic structure and their interaction 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 natural index scales. The decisive step occurs at the Archimedean level. Extinction of cousin primes would require complete coverage of finite index intervals by canonically anchored obstruction sets. We show that such coverage is structurally impossible: the cousin obstruction system satisfies bounded local multiplicity and therefore falls within the scope of a general Archimedean non--coverage principle. As a consequence, admissible indices persist beyond every finite scale, and infinitely many cousin prime pairs exist. All arguments are exact, finite, and deterministic. No analytic estimates, density heuristics, or probabilistic methods are used.