Anchored Indexed Congruences and Structural Non–Coverage

We introduce an anchored indexed framework for the study of divisibility congruences, based on the canonical decomposition $n=6t+r$ and a geometric interpretation in the $(r,t)$--plane. In contrast with classical covering systems, all congruences are anchored canonically by arithmetic divisibility, and no freedom in the choice of residues is allowed. Within this framework, congruences modulo $2$ and $3$ act vertically and induce a canonical reduction to the residual columns $r=1,5$. For every modulus coprime with $6$, the induced congruence on the index variable determines a unique arithmetic progression with a fixed anchor. We show that congruences associated with composite moduli introduce no new progressions, being exact intersections of those associated with their prime divisors. Our main result is a deterministic non--coverage theorem: the family of indexed progressions canonically induced by divisibility cannot cover the index axis. The proof is finite and exact, relying solely on canonical anchoring and intersection properties, without probabilistic, density, or analytic methods. This anchored perspective clarifies why coverage phenomena disappear once residues are no longer free parameters, and provides a structural explanation for the persistence of infinitely many primes in the residue classes $6t\pm1$.