Variational Order Stability for Young Diagram Extremizers

This note is concerned with stability phenomena in Young diagrams, with an emphasis on their behavior under natural sequential growth. In the literature on representation stability, stability properties of Young diagrams are typically explained via representation-theoretic structures, categorical frameworks, or probabilistic limit-shape arguments. The present notes take a complementary perspective, aiming to reorganize the underlying proof logic from a variational and order-evolution viewpoint. More specifically, without introducing additional representation-theoretic assumptions, we examine how near-optimality propagates along the growth of Young diagrams and how such propagation enforces stability of macroscopic observables. From this viewpoint, the decisive mechanism appears to be a form of rigidity arising from the interaction between order evolution and variational structure, rather than representation-theoretic symmetry per se. As part of this reorganization, the proof structure is abstracted into a tentative framework, referred to here as Variational Order Stability (VOS). At this stage, this abstraction primarily serves as a conceptual tool for understanding Young diagram stability, and its general scope and limitations remain to be tested in further concrete settings. These notes are intended as a preliminary research record. The results and formulations should be regarded as exploratory, with the goal of isolating structural mechanisms underlying stability phenomena rather than presenting a finalized theoretical framework.