Structural Origin of the Logarithmic Response Registry: A Conditional Derivation from Complete Response Modules

This paper develops a structural derivation of the logarithmic response registry within a declared minimal complete-response branch of the broader admissibility-first programme. The result is explicitly conditional on the declared response grammar, common scalar phase action, independent frame covariance, and determinant-character scalarization. It is therefore presented as a falsifiable structural theorem on a specified branch, rather than as an unconditional theorem applying to every conceivable physical description.