On the Structural Necessity of Logarithmic Description

No ontology is asserted. A boundary theorem is proved in an explicit admissibility class C.

(A) In a generative ratio sector modeled by G = (R +, ×), any composable ratio increment representation that is continuous at the identity with respect to an admissible refinement filter factors uniquely through ln.

(B) Under admissible finite-resolution dynamics (translation-covariant strongly continuous semigroups with strict damping and spectral separation), any translation-invariant and resolution-invariant identity predicate that robustly separates carriers can be supported only on the pure-point translation spectrum. Hence robust carriers are discrete.

Fossil sectors (preferred rulers without a ratio calculus) lie outside applicability; β-silence is admissible there.