Modal Triplet Theory and Asymptotic Safety

This paper states the mathematically controlled relation currently available between Modal Triplet Theory and functional-renormalization-group asymptotic safety. An integrable bound on an effective-action flow gives a Banach-space ultraviolet endpoint with an explicit tail estimate, but the bound must be proved for the selected four-dimensional flow and is not implied by an internal finite projector. Exact conjugacy between an MTT coarse-graining map and an FRG step map transports fixed points; approximate conjugacy transports only a residual unless a separate fixed-point theorem is supplied. Scheme changes are controlled by an endpoint-difference estimate, not by unrestricted scheme independence. Finally, a finite-dimensional unstable subspace follows when the linearized RG step is quasi-compact with essential spectral radius below one. These results define a rigorous conditional bridge. The selected MTT-to-FRG chart, physical regulator, four-dimensional damping estimate, and quasi-compactness certificate remain open source obligations.