A One-Dimensional Renormalization Group Framework for UV Completion of Quantum Field Theory and Gravity

We present a framework for ultraviolet (UV) completion of quantum field theory and gravity based on three foundational principles: timelessness (implemented via the Hamiltonian constraint), a variational principle selecting stationary configurations, and the absence of a preferred scale (resolution independence). These principles naturally give rise to a renormalization group (RG) structure in which scale plays the role of relational evolution. Within this framework, we analyze the coupled RG flow of gauge, Yukawa, scalar, and gravitational sectors. Including gravitational corrections, we demonstrate the existence of a nontrivial UV fixed point. A stability analysis of the RG flow reveals that only a single relevant direction survives, while all other directions are either irrelevant, redundant, or inconsistent with refinement stability. This leads to a one-dimensional critical surface, implying that all couplings are determined functions of a single parameter. We construct the corresponding one-dimensional RG-reduced effective action, in which the Yukawa--scalar sector dominates the relevant deformation, while gauge couplings remain approximately fixed. This results in a highly predictive structure, yielding universal relations among observables. In particular, we derive a correlation between the Higgs and top quark masses and show that deviations from Standard Model couplings are strongly constrained. The same framework naturally extends to cosmology. Nonminimal scalar couplings generate the Planck scale dynamically, while the scalar sector gives rise to Starobinsky-type inflation with predictions consistent with current observations. The cosmological constant emerges as a small residual effect controlled by the RG deformation parameter. Taken together, these results suggest that quantum field theory and gravity may admit a UV completion in which the full dynamics is governed by a single relevant parameter, determined by fundamental consistency principles.