Dynamical Spectrum of Power-Free Integers in Quadratic Number Fields and Beyond

The paper explicitly proves that for imaginary quadratic fields (Theorem 16) and real quadratic fields (Theorem 19), the Fourier–Bohr spectrum is L^⊛ = (i ψ_δ/√δ) ∑_{p∈P′_O} p^{-κ_p} in the imaginary case and L^⊛ = (ψ_d/(2√d)) θ(∑_{p∈P′_O} p^{-κ_p}) in the real case, with the denominator criterion and the direct-sum quotient L^⊛/O^* described exactly as in the candidate solution . The paper’s proof proceeds via weak model sets/CPS and ideal duality (using a^* = a^{-1}O^* and (∩ a_j)^* = ∑ a_j^*), while the model computes the same spectrum by identifying the Mirsky factor with the compact rotation X = ∏ O/p^{κ_p} and reading off eigencharacters via Pontryagin duality; the paper even remarks that it “bypasses the somewhat tedious computation with the characters,” underscoring the methodological difference . The identification of O^* as a scaled copy of O (imaginary: O^* = (i ψ_δ/√δ) O; real: O^* = (ψ_d/(2√d)) O) used by the model is precisely what the paper proves (via the chosen pairings) . Hence, both reach the same result; the proofs differ in emphasis (rotation + Pontryagin dual vs. CPS + ideal duality).