Modal Triplet Theory and Asymptotic Safety

We establish a precise connection between Modal Triplet Theory (MTT) and functional renormalization group (FRG) approaches to quantum gravity, showing that asymptotic safety arises as a controlled shadow of the coherent-sector ultraviolet structure of MTT. Working within a bounded-geometry time-slab and the coherent universality class, we define a background-covariant FRG effective average action and a discrete renormalization-group step map. Using Gaussian ultraviolet damping induced by the coherent projector, we prove the existence of a well-defined ultraviolet endpoint functional in a Banach space of analytic local actions supplemented by entire form-factor terms. We show that this endpoint is stable under admissible regulator variations and that the linearized renormalization-group map has a finite-dimensional unstable subspace. As a consequence, fixed points observed in asymptotic-safety truncations are obtained as shadows of the ultraviolet endpoint under controlled truncation and canonical rescaling. The results clarify why asymptotic-safety methods exhibit stability despite scheme dependence and place them within a broader universality framework provided by MTT.