Structural Foundations of Identity Quasipotentials: Equivalence Classes, Renormalization Stability, and Invariant Structure

Structural Foundations of Identity QuasipotentialsEquivalence Classes, Renormalization Stability, and Invariant Structure This work develops a rigorous, purely structural theory of identity for systems admitting a quasipotential description. Assuming only the existence of metastable identity dynamics governed by a quasipotential, the paper asks a logically independent question: which features of identity persist under admissible transformations and coarse-graining? Identity is defined here not as a state, trajectory, or potential, but as a structural equivalence class determined by global metastable organization. Specifically, identity is encoded by (i) the adjacency graph of metastable basins, (ii) the ordering of minimal barriers governing transitions between them, and (iii) the stationarity of this structure under renormalization in the quotient space of identity classes. The central result establishes that identity is preserved if and only if the renormalization-group (RG) orbit is stationary in the quotient space of identity equivalence classes. This criterion is both necessary and sufficient and does not require convergence of the quasipotential itself. Identity persistence is thus characterized as a fixed-point property of coarse-graining acting on structural invariants, placing identity within the logic of universality rather than microscopic mechanism. The paper develops: a non-circular definition of identity equivalence, a precise notion of admissible transformations, a structural stability lemma guaranteeing robustness under bounded perturbations, spectral representations of identity (reversible and irreversible cases), a functorial formulation of identity-preserving transformations, and extensions to spatially extended and field-theoretic systems. No biological, cognitive, or ontological interpretation is assumed or required. The results are independent of the specific law generating the quasipotential and apply to any system in which metastable structure is well defined. This work forms the structural counterpart to a companion paper that introduces a law-level variational principle for identity dynamics. Together, the two papers provide a closed, non-circular framework in which dynamics generate identity structure, and structure constrains identity persistence. Keywords: identity, quasipotential, metastability, renormalization group, structural invariants, large deviations, coarse-graining, universality, nonequilibrium systems.