Spectral Properties of the GCD Operator and the Ramanujan–Möbius Identity

We study the spectral properties of the GCD operator Q_N defined by Q_N(i,j) = gcd(i,j)/√(ij) and its Möbius-decorated variant Q_N^μ(i,j) = μ(i/g)μ(j/g)g/√(ij), where g = gcd(i,j) and μ is the Möbius function. The central result is a Ramanujan–GCD quadratic form identity: ⟨v, Q_N v⟩ = Σ_d φ(d)|S_d|² − ‖v‖², giving the unconditional spectral floor λ_min(Q_N) ≥ −1 for all N ≥ 1. We identify the spectral edge constant C = π/2 − log 2 ≈ 0.8776 as the zero-point energy of the prime lattice, arising as the analytic residue of the GCD Dirichlet series at s = 1. We prove the normalized operator Ĥ_N satisfies λ_min(Ĥ_N) > −1/2 unconditionally for all N ≥ 4, and establish a sharp bound of −3/14. All results are unconditional — no Riemann Hypothesis or Generalized Riemann Hypothesis assumptions are required. This paper is the first in a series on spectral non-concentration criteria for Navier–Stokes regularity.