The Multiscale Field Equation of Persistence: Architecture, Accessibility Geometry, Potential Conversion, and Cascade Support

This working research note states a multiscale field equation of persistence in constraint form, using a primitive ontology of architecture, accessibility, and potential conversion. The formulation synthesizes prior work on viability geometry, structural persistence, resolution-covariant maneuverability, cascade admissibility, and passive margin exhaustion into a single field-coordinate structure. The note introduces a conserved primitive object ε at the lowest resolution and resolution-dependent effective building blocks eℓ. Architecture Aℓ is interpreted as stored lower-level structure, maneuverability Mℓ as accessible effective structure, support flux Λ as transferred effective structure, and accessibility geometry Gℓ as the generated geometry of admissible motion. The local persistence field is written as Ψℓ = (Aℓ, Gℓ, ωℓ), where Aℓ is represented mathematically by a scalar divergence coordinate, Gℓ is generated by effective admissible regions and tangent cones, and ωℓ is the potential-conversion intensity induced by the passive generator. The note states sign-resolved architecture evolution equations governing the rate of change of stored architecture as a function of support surplus. Negative surplus produces collapse, positive surplus produces higher-level formation, and balanced support produces persistence. The pressure functional Πℓ is identified with the minimum sustaining flux of cascade admissibility, while the support-surplus variable Sℓ = Λℓ−1 − Πℓ classifies collapse, persistence, and emergence. Transformation relations among the four relational states — potential, maneuverability, architecture, and support flux — are recorded explicitly, with conservation of the primitive ε as the background constraint on all reorganization. The resulting constraint-form field equation links the local persistence field, the active viability constraint, the support-surplus evolution equations, and the multiscale pressure-support cascade. In this formulation, persistence is stored lower-level structure held at a higher resolution by accessibility geometry and support flux; collapse is release of stored architecture into lower-resolution accessibility; and emergence is stabilization of surplus support into new higher-level architecture. This document is conceptual and interpretive. It does not claim a derivation of thermodynamics, spacetime geometry, or any established physical theory. Its purpose is to record the ontology, constraint-field equation, and field-coordinate interpretation organizing the persistence, geometry, maneuverability, and cascade frameworks.