Stabilized Description and the Architecture of Representability: A Structural Taxonomy of Closure, Representation, and Mathematical Realization

Framework position. This paper provides the general conceptual architecture of the Closure–Stabilization Framework. It separates closure, stabilization, continuation, representation, projection, and regime activation, and treats stable description as contingent on refinement-stable plateaux rather than as a primitive assumption. It should be read together with When Is a Scientific Representation Dynamically Sufficient? A Carrier–Codec Diagnostic Framework, which supplies the complementary operational criterion for determining when a stabilized representation preserves the distinctions required by admissible continuation. The present paper is therefore architectural and classificatory; the companion paper develops the corresponding empirical and diagnostic machinery. We develop a structural taxonomy of describability grounded incarrier-free closure, stabilization under refinement, and the separationbetween representation and projection. Rather than classifying entities,theories, or disciplines, the framework classifies the conditionsunder which stable descriptions can arise at all. Closure is treatedas contingent and contextual; stabilization yields invariant equivalenceclasses only on restricted domains, termed plateaux. Representationsencode stabilized compositional structure on such plateaux. Continuationis not identical with composition, but is a special ordered andretained-thread substructure of composition, present only when stabilizedacts or histories can be extended, transported, iterated, or reactivatedwithout losing their identity under refinement. Only in suchcontinuation-bearing regions do notions such as iteration, limits, generators,spectra, or dynamical parameters become definable. Projectionsact orthogonally to representations, extracting invariant content andproducing mathematical objects as quotient structures.Within this architecture, mathematical fields are identified as familiesof admissible projections rather than foundational languages, explainingboth their effectiveness and their structural limits. Regimesarise as activated descriptive modes under finite-resolution constraintrepresentations, accounting for quantum, statistical, and Boolean descriptionswithout introducing ontological layering. Physics appears asa particular stabilized descriptive region characterized by additive representation,activated infinitesimal continuation, and admissible geometricand symmetry projections; its scope and breakdowns are shownto be structural rather than provisional.The taxonomy explicitly accommodates mixed descriptive regions,where multiple non-commuting projections are locally activated, andout-of-plateaux domains, where stabilization fails and no representationor invariant exists. These domains delimit describability inprinciple and explain the impossibility of total or universal theories.Structural admissibility of a representation is distinguished from itscontinuation-relative dynamic sufficiency: a representation may be1well defined on a plateau yet still erase distinctions that later admissiblecontinuations can reactivate. That stricter diagnostic belongs toa downstream carrier–codec analysis rather than to the taxonomy itself.The resulting framework provides a unified structural reference forcomparing disciplines, diagnosing category errors, and revising foundationalconstructions—such as finite-resolution physics, stabilized symmetrymodels, and closure-based gravity—without privileging any singledescriptive mode as fundamental. This document constitutes thepresent conceptual synthesis of a sequence of technical developmentsby the author. For clarity of exposition, earlier works are not citedexplicitly.