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

We study the spectral properties of the GCD operator QN defined by QN (i, j) = gcd (i, j) /√ (ij) and its Möbius-decorated variant QN^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, QN v⟩ = Σd φ (d) |Sd|² − ‖v‖², giving the unconditional spectral floor λₘin (QN) ≥ −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 ĤN satisfies λₘin (ĤN) > −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.