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

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 $\mathcal{O}({\lambda }^2)$ and $\mathcal{O}({\lambda }^{-3\mathrm{/}2})$ ($\lambda$ is the 't~Hooft coupling), including the non-analytic ${\lambda }^{3\mathrm{/}2}$ and ${\lambda }^2\log \lambda$ 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.