Axiom of Topological Degrees of Freedom and Observer-Dependent Descriptions

This paper formulates structural admissibility as a condition under which a candidateconfiguration may be accepted as a structurally describable configuration within a givendescriptive regime. The framework, developed under the name Dimensional-Structural Describability, distinguishes the structural organization available within a regime from theadmissibility of a particular configuration. A candidate configuration is structurally admissible when its active elements, recorded active relations, and boundary records satisfy localcoherence and when every relation exposed across its active carrier receives a treatmentpermitted by the boundary rule of the regime.The resulting axiom system defines induced restriction, boundary-compatible gluing,structural isomorphism, and admissibility-preserving transformations. It also defines an admissible structural evolution as an ordered family of configurations connected by admissibilitypreserving transformations. This temporal extension does not introduce an equation of motion, a physical time law, a conservation law, or a propagation bound. Its role is limitedto specifying the condition under which a later dynamical construction may be treated as astructurally admissible continuation.The formal results show that structural admissibility is invariant under structural isomorphism and is preserved under regime-defined restriction, boundary-compatible gluing,and finite chains of admissibility-preserving transformations. Further results establish corerecord consistency for nested restriction and for the relation between restriction and gluing.The basic gluing operation is conservative at the level of relation records: it preserves relations already recorded as active by union and does not automatically reconcile a relationmerely because that relation becomes internally supported after gluing. A finite 3 × 3 D2lattice realization is constructed to provide a nonempty, directly reconstructible model ofthe system. Finite failure witnesses clarify the roles of relation-incidence conditions, explicit boundary treatment, and the separation of connectedness from the basic admissibilitycriterion.