Neural variational framework for random Young-diagram limit shapes

We develop a structure-preserving neural variational framework for random Young-diagram ensembles, with representations adapted to the structure and scaling of each measure. The method is validated on the Plancherel, uniform, minimal-difference, and fixed-$q$ $q$-Plancherel ensembles, using known asymptotic profiles only for post-training comparison. We then study a quartically deformed hook-length ensemble without assuming an analytical saddle shape. Large-$n$ neural profiles are compared with finite-size MAP profiles obtained from exact-action searches and with mean profiles obtained from corner-transfer Metropolis--Hastings sampling. Increasing the deformation suppresses the leading rows and broadens the support, while the neural, discrete, and sampled mean profiles agree at the percent level. These results provide numerical evidence for a deformation-dependent macroscopic saddle family.