A Third-Corner Shadow Bridge in Modal Triplet Theory

We establish a three-corner equivalence between perturbative string theory, asymptotic safety, and coherent-sector fixed points in Modal Triplet Theory (MTT). In MTT, physically viable effective descriptions are selected by an admissibility constraint enforced by coherent-sector projection, a finite spectral gap, and controlled truncation. We show that this same constraint admits three distinct encodings: a coherent-spine fixed-point structure in the upstairs theory, a worldsheet renormalization-group fixed point in the string corner, and a functional renormalization-group ultraviolet fixed point in the asymptotic-safety corner. Using a notion of controlled conjugacy between scale-step maps, we prove that fixed points and their stability spectra correspond across all three encodings up to truncation error controlled by the spectral gap. The result yields concrete cross-framework predictions, including matching leading irrelevant operators and correlated breakdown surfaces. The apparent plurality of ultraviolet completions of gravity is thus reinterpreted as an artifact of encoding choice rather than of fundamentally distinct physics.