The Resonance Fourier Transform: A Golden-Ratio Spectral Basis with Diophantine Concentration Guarantees

We introduce the Resonance Fourier Transform (RFT), a unitary discrete transform constructed from a golden-ratio frequency grid f_k = frac((k+1)φ), where φ = (1+√(5))/2. The canonical RFT basis U = Φ(Φ^†Φ)^{-1/2} is the unique unitary polar factor of the associated Vandermonde matrix. We prove that U is not equivalent to any permuted or rephased DFT. The central result is a Diophantine spectral concentration theorem for signals drawn from the Golden-Harmonic Subspace Ensemble: Hurwitz's irrationality bound [1] forces nonzero DFT bin offsets for every golden harmonic, producing Dirichlet-kernel leakage (sinc^2(ε) < 1), while the RFT grid contains these harmonics by construction (ε = 0). This structural mismatch is proven via a six-lemma chain grounded in Hurwitz [1], Steinhaus–Sós [2], Weyl [3], and Erdős–Turán [4]. The resulting quantitative gap Δ K_{0.99} is computationally verified across all tested sizes (N ∈ [32, 512]) with bootstrap confidence intervals excluding zero. We provide an analytical per-tone leakage-to-spread inequality as a partial bridge toward a fully deductive ensemble-level bound. All claims are accompanied by 117 deterministic falsifiable tests and an FPGA implementation path. Proven and conjectured results are explicitly delineated throughout.