The Illusion of Scale: Approximation, Identifiability, and the Entropy Tax

Predictive error and structural error are quantities on different spaces, joined only through the inverse of an observation operator. This note makes that joint explicit. It defines a query-relative inverse modulus and shows that predictive convergence transfers to structural convergence, uniformly over an admissible class, exactly when that modulus vanishes in the limit as the observational tolerance tends to zero. Three regimes follow: strict non-identifiability, unstable inversion, and practical non-identifiability. They separate the cases in which additional scale cannot help from those in which it can. A two-point argument shows that no estimator, and therefore no quantity of computation, distinguishes hypotheses with identical observational laws; a compactness argument shows that, under the stated conditions, restricting the admissible class can convert query-level uniqueness into uniform stability over that class. The resulting picture is an accounting identity: identification is paid for in data, interventions, assumptions, or proofs, and computation searches only whatever those payments leave behind. None of the individual mathematical facts is new. The contribution is a single object that makes the scaling claim precise, and that states the conditions under which the claim is false. Contents: illusion_of_scale_note.pdf is the full note (10 pages, version 1.0). visual_abstract.pdf is a one-page visual abstract of the same material. The LaTeX sources for both documents are included.