On the Impossibility of Primitive Space, Time, and Force in Physical Description

A physical theory is a closed system of distinctions. Distinctions require relations;

relations require composition; composition must remain meaningful under finite observational discrimination and refinement. A descriptive language is formalized as a family of quotient-level assignments Φϵ : Oϵ → Iϵ into composable descriptors, modulo gauge. Three requirements are forced by description itself:

(A) composability without external reference, 

(B) refinement coherence, and

(C) non-circular definability.

Requirement B is made explicit in two tiers: (B1) naturality under refinement (semantic coherence without invertibility), and

(B2) gauge coherence (invertible coherence). Under A, C, and B2, any language employing primitive space, primitive time, or primitive force violates admissibility. Stable identity factors through finite-resolution partitions and is therefore discrete at each ϵ; continuous carriers cannot be fundamental identity carriers under refinement coherence. For ratio-like relational observables (scale factors) whose operational concatenation is multiplicative by definition, quotient-level observability forces measurability with respect to the quotient σ-algebra; under this forced regularity, additive homomorphic coordinates are logarithmic uniquely up to affine gauge. An admissibility projector on gauge-classes of languages is defined; its fixed points are characterized and shown unique up to gauge equivalence. The resulting fixed-point vocabulary—logarithmic relational representation for ratio-regimes, discrete spectral identity, ordering only as an observable, and interaction only as refinement-stable constraints—is named the Unified Substrate Theory (UST) admissibility protocol. This is conditional descriptive necessity, not ontology.