Finite-tree series for the quadratic Feigenbaum constants
Year: 2026
Type: preprint
Abstract: We give absolutely convergent series for the two Feigenbaum constants of quadratic period doubling. Each coefficient is a finite sum over labelled plane rooted trees, with weights obtained from elementary Taylor coefficients and the Lambert value W₀(e⁻¹). The coefficient prescription requires no numerically solved fixed point or eigenvector, validation centre, iteration count or projection dimension. A finite binomial grouping places both evaluations at the same interior point of an auxiliary analytic family. A computer-assisted proof in a parameter–spatial Wiener algebra establishes continuation from an affine origin. A comparison in a common analytic norm identifies the endpoint with the classical quadratic fixed point. A uniform Schur-complement estimate identifies its unstable eigenvalue. The contribution is this particular finite coefficient construction together with absolute convergence and identification, rather than the general existence of convergent representations. MSC 2020: 37F25, 37E05, 65G30, 05C05. Supplementary files contain the LaTeX source, executable interval-validation and coefficient-reproduction materials, and the pinned Lean proof of the scalar duality on its real interval.
Author: Na, Sungsoo
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7224906336