Continuous Algebraic Diversity: Unifying Spectral, Wavelet, and Time-Frequency Analysis via Lie Group Actions

We provide a computable criterion for selecting among Fourier, wavelet, and time-frequency analysis by extending the algebraic diversity (AD) framework to Lie groups acting on L^2(\mathbb{R}). To our knowledge, there is no other criterion that provides this selection capability. The group-averaged estimator generalizes from a finite sum over group elements to an integral with respect to Haar measure. A Continuous Replacement Theorem establishes signal-noise separation under equivariance and ergodicity conditions, with a noise operator \mathcal{N}_G = C_ρ^{-2} determined by the Duflo-Moore operator that explains the frequency-dependent noise floor in wavelet analysis as a consequence of the affine group's non-unimodularity. A Unification Theorem shows that classical spectral analysis corresponds to the translation group, wavelet analysis to the affine group, time-frequency analysis to the Heisenberg-Weyl group, and spherical harmonics to SO(3). The commutativity residual δ, extended to Hilbert-Schmidt operator norms, provides a principled selection criterion among these groups. A double-commutator generalized eigenvalue problem solves the blind group matching problem in polynomial time. A Discretization Recovery Theorem establishes that all discrete AD results are sampling approximations to the continuous theory, with \mathbb{Z}_M o (\mathbb{R},+) as M o \infty.