Beta-Kernel Truncation Bases for the Functional Renormalization Group: Complete Error Analysis and Adaptive Nyström Propagator Approximation

We construct a family of Beta-kernel basis functions uₘ^ (p) (t) = C₌, ₏ tᵐ (1−t) ᵖ for truncating the effective action and performing Nyström approximation of the regularised propagator in the Functional Renormalization Group (FRG). Working in one-dimensional quantum mechanics with a mass regulator Rₖ = k², we derive the closed-form Green's function and establish three main results: a complete integration-by-parts error bound for the monomial truncation basis yielding cumulative truncation error EN = O (N⁻¹) ; a rank-one collapse theorem showing the regularised propagator reduces to a rank-one operator in the large-m limit; and a closed ODE system for propagator values and adaptive Nyström nodes via the Dirac–Frenkel–McLachlan variational principle, with exact self-consistency for a single node at the midpoint. All analytical estimates are confirmed by explicit numerical computations at Mₖ = 1.