Renormalization as a Consistency Requirement in Finite-Power Irreversible Systems

Landauer-type bounds provide local energetic floors for logically irreversible information processing. This paper introduces a minimal formal setting in which a finite physical system repeatedly executes irreversible closure events. Under a finite power constraint, it is shown that when overhead depends on raw residue, sustained nondegenerate operation is impossible: either the closure rate activity collapses or the maintained entropy per closure collapses. An effective renormalization operator is then defined as a coarse-graining map from raw residue to an effective residue scale. The main theorem proves that the effective overhead must remain essentially bounded, forcing any admissible renormalization to suppress extensive growth of the effective residue.