Constrained Pad\'e Ensembles for Thermal N=4 SYM: Quantified Uncertainties and Next-Order Predictions

We quantify the transition between weak and strong coupling in thermal ${\cal N}=4$ supersymmetric Yang--Mills (SYM) theory in four space-time dimensions by constructing an \emph{admissible ensemble} of log-aware Pad\'e approximants that incorporate the weak- and strong-coupling expansions through $\mathcal O(\lambda^2)$ and $\mathcal O(\lambda^{-3/2})$ ($\lambda$ is the 't~Hooft coupling), including the non-analytic $\lambda^{3/2}$ 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.