Geometry-Aware STFO Shadow: Path-Dependent Actions on the Young Graph

This note introduces a geometry-aware extension of the STFO-shadow model defined in Examples for the Coupling Problem: Small-n Sanity Checks. By adding a path-dependent term to the discrete Lagrangian, the resulting action becomes non-conservative and distinguishes between different growth histories on the Young graph. Explicit small-n examples are computed, and the resulting weighted path measure is shown to take the form of a discrete path-integral partition function over growth histories.