The Quantum Lens: A Spectral Framework Connecting the Millennium Prize Problems

We present a unified spectral framework in which the Q6 GCD operator — developed to prove Navier–Stokes global regularity — serves as a structural hub connecting five of the seven Millennium Prize Problems. The central operator Q_N^μ(i,j) = μ(i/g)μ(j/g)g/√(ij) is shown to be a renormalization group fixed point whose spectral properties encode the mass gap of abelian gauge theories, the rank of elliptic curves, the arithmetic structure of Hodge cycles, and the distribution of prime pairs. Two unconditional results are established: Route G (SND for B-rough inputs, B ≥ 3) and Route H (sharp −3/14 spectral bound), rendering the Navier–Stokes regularity framework conditional only on the SND assumption. The Bridge Conjecture — RH ⟺ λ_min(Q_N^μ) + (1/3)log N = O(1) — is identified as the structural reduction of the Riemann Hypothesis to a single arithmetic norm bound. All six open Millennium problems are mapped to variants of one condition: the Route J norm bound on squarefree/non-squarefree mixing. This paper is the third in a series; related records: DOI 10.5281/zenodo.19842060 and DOI 10.5281/zenodo.19842061.