Why General Relativity Appears in Perturbative String Theory: A careful MTT interpretation of worldsheet consistency and target-space dynamics

It is well known that the requirement of quantum Weyl invariance in perturbative string theory yields the Einstein field equations for the target-space metric. This result is often summarized by the slogan that General Relativity “falls out” of string theory, suggesting a hierarchical derivation. We show that this interpretation is misleading. Within Modal Triplet Theory, both the infrared Einstein equations and the vanishing of worldsheet sigma-model beta functions arise as distinct effective encodings of a single admissibility constraint imposed on the coherent sector. Using explicit encoding and scheme-equivalence assumptions, we prove that coherent admissibility is equivalent, up to controlled truncation error, to the existence of a worldsheet renormalization-group fixed point, and that this condition is in turn equivalent to infrared Einstein dynamics with suppressed higher-curvature corrections. General Relativity and perturbative string theory are therefore dual diagnostic shadows of the same coherent fixed-point condition, rather than one being derived from the other. The correspondence holds within a shared universality class and breaks down simultaneously when admissibility fails.