Depth, Width, and Minimal Global Coupling between Primes. Reflections on the Corridor between Symmetry and Arithmetic with Commander Sol

This research note studies the expressive corridor between prime-permutation symmetry and full arithmetic. Starting from multiplicative arithmetic, operator lifts, and the rational adeles, it separates prime individualization from genuine cross-prime coupling. A finite-color locality no-go excludes order and successor from relations that factor through finitely many local types. Uniform cyclotomic Frobenius coupling is then shown to recover absolute prime gaps and, more strongly, to shatter arbitrary finite sets of odd primes via the Chinese remainder theorem and Dirichlet's theorem. Combined with multiplicative finite-support coding, this yields a first-order interpretation of full arithmetic (Cyclotomic Coding Collapse). In the opposite direction, finite S-adic congruence towers have unbounded resolution depth but remain homogeneous enough to admit nontrivial automorphisms, so they still do not orient the primes. The paper therefore proposes a new target regime: unbounded resolution depth with bounded independent shattering width, formalized through prime-shattering dimension and connected to VC/NIP ideas. Adjoint Frobenius data from Ramanujan's Delta form are proposed as a next test class, not as a solution.