Outer Functional Analysis, Volume III: Harmonic Observations, Phase-Space Geometry, and Multiscale Reconstruction

This volume develops an iterated outer-functional framework for harmonic analysis. Native functions, distributions, measures, and kernels form the base level; harmonic observations acting on these objects form the first functional level; and sharp constants, extremizer moduli, concentration profiles, and stability moduli form second-level outer functionals. Higher levels describe sensitivity, curvature, persistence, branching, obstruction, robustness, and certified reconstruction. The theory strictly separates genuine functional level from derivative order, frequency scale, multilinear interaction order, representation label, spectral cutoff, discretization parameter, quotient shadow, and retained-coordinate depth. Within this typed framework, the volume establishes envelope and Schur-complement variation formulas, implicit-function results modulo symmetry, Fredholm reduction for degenerate extremizers, projective blow-up governed by decisive jets, and multiscale reconstruction of outer-functional derivatives and stability remainders. Classical Fourier, real-variable, representation-theoretic, oscillatory, microlocal, time-frequency, restriction, Kakeya, and dispersive methods are retained on their native domains. Finite Fourier, wavelet, wave-packet, Gram, Hessian, and spectral data yield infinite-dimensional conclusions only through explicit compatibility, closed-graph, tail, compactness, and external-faithfulness hypotheses. Quantitative morphisms transport sharp values, extremizer information, concentration profiles, and stability conclusions between admissible harmonic-analysis problems. Open endpoint and compactness questions remain explicitly conditional rather than being encoded into definitions. **Keywords** Outer functional analysis; harmonic observation functionals; sharp constants; extremizers modulo symmetry; concentration compactness; stability inequalities; envelope theorem; Schur complement; Fredholm reduction; projective blow-up; decisive jets; Fourier restriction; oscillatory integrals; microlocal analysis; time-frequency analysis; multiscale reconstruction; inverse limits; rigorous numerics; quantitative transference; external faithfulness.