Moment Recurrences, Exponential-Polynomial Spaces, and the Classication Theorem

We develop the analytic and algebraic foundations underlying the moment structure of the reduced kernel Γ̃ in the TCFQ (Twisted Clifford Functional Quantization) model. Starting from the structural analysis of the moment sequence {J_d} (R81), we identify the universal recurrence operator Δ = 1 − 2E⁻² + E⁻⁴ (R82), establish the maximality of the generating space 𝒷₀ (R83), and prove a general mechanism for exponential-polynomial functions (R84). The central result, proved in R85, is a complete classification theorem: a finite-dimensional subspace V ⊂ C^∞[0,π] produces a finite moment recurrence if and only if V is contained in a space generated by functions of the form xᵏeˡˣ. The specific recurrence Δ corresponds exactly to the space𝒷₀ = span{1, x, cos x, x cos x, sin x, x sin x}, recovering and structurally explaining the maximality result of R83. The proof rests on three pillars: an algebraic equivalence between recurrences and rational generating series (R85.2), an operator correspondence t ↔ −d/dx via integration by parts (R85.3), and the classical structure theorem for linear ODEs with constant coefficients (R85.4). Together, these convert an empirical observation about Γ̃ into a complete structural classification of all finite-dimensional spaces admitting finite moment recurrences over [0,π].