Constrained Padé Ensembles for Thermal $\mathcal{N}=4$ SYM with the Exact $\mathcal{O}({\lambda }^{5\mathrm{/}2})$ Coefficient

We revisit the constrained log-subtracted two-point Padé (LSTP) ensemble for thermal $\mathcal{N}=4$ supersymmetric Yang–Mills (SYM) thermodynamics in four spacetime dimensions after upgrading the weak-coupling truncation from $\mathcal{O}({\lambda }^2)$ to the exact $\mathcal{O}({\lambda }^{5\mathrm{/}2})$ coefficient. We keep the interpolation ansatz unchanged and shift the weak-side matching points to the regime where the new term is numerically significant. The admissible set collapses from $9$ nominal survivors ($3$ distinct curves) to a single distinct curve, the crossover range shrinks to a unique value, and the pointwise band width drops to zero within numerical resolution. The Hermite-Padé (HP) central curve does not coincide with the unique LSTP survivor, so the exact weak-coupling coefficient removes the LSTP scan uncertainty but not the difference between the two routes. The next step is to compute the unknown $\mathcal{O}({\lambda }^{-3})$ strong-coupling coefficient.