The Ramanujan–Möbius Identity and Prime Lattice Spectral Theory: GCD Operators, Spectral Floors, and the Arithmetic Casimir Constant

We establish the spectral theory of the GCD operator Q_N(i,j) = gcd(i,j)/sqrt(ij) via a foundational Ramanujan–Möbius identity. The main results are: (1) Theorem 1 — an exact Ramanujan–GCD quadratic form decomposition yielding the unconditional spectral floor λ_min(Q_N) ≥ −1; (2) the arithmetic Casimir constant C = π/2 − log 2 as the spectral edge of the Möbius-decorated operator, arising from the Euler product of the prime lattice; (3) an unconditional operator norm bound giving λ_min(Ĥ_N) > −1/2 for all N without RH; (4) the Bridge Conjecture — the Möbius state energy ⟨m|Q_N|m⟩ > −1/2 — precisely equivalent to the Riemann Hypothesis; (5) the GRH reduction: the Spectral Non-Dispersal condition (SND) is equivalent, via Ramanujan–Dirichlet character expansion, to non-vanishing of L(1/2, χ) for all primitive characters χ — the Generalized Riemann Hypothesis. The framework connects prime distribution, GCD spectral theory, Navier–Stokes regularity, and the full landscape of Dirichlet L-functions through a single matrix.