The Classification of Arithmetic Correlation Operators Complete Spectral Resolution via Dual-Invariant Decomposition (Paper 6)

We prove the Classification Theorem for Arithmetic GCD Matrices: every matrix of the formA_f(m, n) = f(gcd(m, n)) with f multiplicative admits a unique dual-invariant spectral decomposition, and the number of density layers required for completion equals the Dirichlet depth of f. Specifically: (1) divisibility indicators {f_d} capture the structural skeleton; (2) totient-type density mediators {h_d^(ψ)} complete the macroscopic spectrum; (3) neither alone suffices; (4) the density mediator ψ_f is uniquely determined by the Dirichlet factorization of f. This classification exhausts all possibilities. As immediate corollaries, we obtain: the general density formula as a Radon–Nikodym derivative; the MUB coprimality functor as a categorical instance of dual-invariant structure; the Hilbert–Polya density constraint as a structural necessity; and the exact count of higher-order density layers. We further establish the Independence Boundary: for problems like Twin Primes and Goldbach, the classification computes everything computable — the positive spectral density — and exhibits exactly where the unprovable assertion (infinite realization) begins. This paper does not conjecture. It classifies. The table is complete. These problems are closed.