This public preprint / technical research note introduces a path-based refinement of static equivalence on admissible domains. The central idea is that static equivalence between states does not by itself determine whether one state can be coherently realized from another through admissible paths with controlled or vanishing realization cost. The note defines admissible states, coherent paths, path cost functionals, infimal realization cost, and strong admissible identity. In particular, strong admissible identity requires static equivalence together with zero infimal realization cost in both directions on the admissible domain. The resulting structure is related to an extended directed quasimetric / Lawvere-type distance. Gabriel’s horn is used as the main worked example. Its classical finite-volume and infinite-surface-area structure motivates a distinction between static description, admissible lifting, realization cost, topology-sensitive path existence, and access constraints. Supplement A provides a measure-theoretic instantiation of the framework, while Supplement B develops Gabriel-horn realization-cost scenarios. This release is not peer reviewed and is not a journal submission final. It does not claim a physical law, a Banach–Tarski solution, or a universal impossibility theorem. It is intended as a first public, citable, time-stamped technical release of the KFE framework layer developed here.
Csongor Petkes (Sun,) studied this question.