Constrained Padé ensembles for thermal $$\mathcal {N}{=}~4$$ SYM with the exact $$\mathcal {O}(\lambda ^{5/2})$$ coefficient

Abstract We revisit the constrained log-subtracted two-point Padé (LSTP) ensemble for thermal $$\mathcal {N}{=}4$$ N = 4 supersymmetric Yang-Mills (SYM) thermodynamics in four spacetime dimensions after upgrading the weak-coupling truncation from $$\mathcal {O}(\lambda ^2)$$ O ( λ 2 ) to the exact $$\mathcal {O}(\lambda ^{5/2})$$ O ( λ 5 / 2 ) coefficient. We keep the LSTP interpolation ansatz unchanged and shift the weak-side matching points to the regime where the new term is numerically significant. Within the same parameter scan 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 updated weak-side matching removes the LSTP scan uncertainty but not the difference between the two routes. The minimal HP extension that also carries the exact coefficient is uniquely determined and develops a spurious pole-zero pair at small coupling. Away from that pair the HP and LSTP curves still differ, so this difference persists at the same weak-coupling order.