The Leading Root of the Partial Theta Function: Three Coefficient Conjectures

This preprint studies the leading formal root of the partial theta function and addresses three coefficient conjectures of Alan Sokal. It establishes inequalities for the Euler exponents of the root and an adjacent series, together with log-convexity of the root coefficients. It also identifies equality cases and gives lower bounds for selected exponent sequences. The arguments combine Ferrers-enriched trees and free Lie algebra decompositions with exact integer certificates for a finite coefficient range and an analytic contour estimate for the remaining indices. The deposit includes the manuscript PDF, LaTeX source, and computational supplement. The finite computations support the stated proofs; they do not by themselves verify the general theorems.