Allen's Universal Equation of Admissible Structure

This record is version 2.0 of Allen’s Universal Equation of Admissible Structure.Version 2 introduces formal notation corrections and typographic refinements without altering the underlying mathematical content or theorems. This work introduces a domain-independent mathematical law governing admissible updates of structures. The universal equation states that an update is applied if and only if the resulting structure is coherent; otherwise, the structure remains unchanged. The formulation is intentionally minimal, requiring only the primitives of structure, update, and coherence, without assuming any physical, logical, or computational semantics. Three equivalent realizations of coherence are presented: a predicate form, a scalar functional form, and a certificate (verifier) form. Four core theorems are proved: (i) coherence preservation under evolution, (ii) equivalence of the three coherence formulations, (iii) minimality of the primitive assumptions, and (iv) universality with respect to arbitrary domains admitting a notion of acceptable outcomes. The result establishes a foundational, stand-alone update law applicable wherever coherent and incoherent outcomes can be distinguished, independent of domain interpretation.