Constrained Padé Ensembles for Thermal N=4 SYM: Quantified Uncertainties and Next-Order Predictions

Abstract

We quantify the transition between weak and strong coupling in thermal $\mathcal{N}=4$ supersymmetric Yang--Mills (SYM) theory in four space-time dimensions by constructing an \emph{admissible ensemble} of log-aware Padé approximants that incorporate the weak- and strong-coupling expansions through $\mathcal{O}(\hat{I}{»}^2)$ and $\mathcal{O}(\hat{I}{»}^{-3/2})$ ($\hat{I}»$ is the 't~Hooft coupling), including the non-analytic $\hat{I}{»}^{3/2}$ and $\hat{I}{»}^2\log \hat{I}»$ terms. This replaces single-curve estimates with a reproducible uncertainty band and a well-defined central curve across the intermediate regime. The framework is \emph{predictive}, setting testable benchmarks for forthcoming perturbative and holographic calculations.

v3: Transport section removed (to appear separately). HP coefficients listed explicitly; spurious $\hat{I}{»}^{5/2}\log \hat{I}»$ artifact subtracted, corrected $A_{5/2}^{\mathrm{HP}}=-43.8\pm 0.1$. LSTP leakage derived; $p=3$ motivated. Strong-coupling HP structure proved exactly