Why General Relativity and String Theory Are the Same Admissibility Constraint

General Relativity and perturbative string theory are often related by the statement that Einstein gravity emerges as a consistency condition for string propagation. We show that this familiar result reflects a deeper equivalence. Within Modal Triplet Theory, both the infrared spacetime field equations of General Relativity and the vanishing of worldsheet sigma-model beta functions in string theory arise as distinct effective shadows of a single admissibility constraint enforced by coherent-sector projection and stability. Starting from the same coherent projector, spectral gap, and overlap data, we construct two dual encodings: a four-dimensional infrared effective theory and a two-dimensional worldsheet renormalization-group flow. We prove that coherent admissibility is equivalent, up to controlled truncation error, to the existence of a worldsheet RG fixed point, and that this condition is equivalent to satisfaction of the Einstein equations with suppressed higher-curvature corrections. General Relativity and string theory are therefore not hierarchically related but are technically equivalent encodings of the same admissible coherent fixed-point structure, and they break down simultaneously when admissibility fails.