Renormalization and Structural Incompleteness A General Formal Framework Independent of IR/UV Language

We propose a minimal mathematical framework to analyze a common situation in sci-entific modelling: a theory T may yield non-physically interpretable outputs (e.g. diver-gences) for some admissible data, while a normalization procedure N transforms theseoutputs into physically admissible and empirically accurate predictions. Without using anyinfrared/ultraviolet terminology, we prove a structural non-injectivity result: under a mildrichness assumption on the space of admissible theories and a non-injectivity assumption onthe correction map, the corrected predictions cannot identify a unique “raw” model. Thisprovides a rigorous and general notion of structural incompleteness for the uncorrecteddescription.