Worldsheet and Spacetime Consistency as a Conditional Diagnostic Square: Sigma-model beta functions, target-space equations, and the MTT source contract

Abstract

Perturbative string theory supplies a precise relation between two-dimensional Weyl consistency and target-space field equations. At leading order the metric beta function contains the Ricci tensor, but the complete statement also involves the antisymmetric tensor, dilaton, renormalization scheme, higher powers of alpha-prime, string loops, and the domain of the worldsheet theory. It is therefore inaccurate to identify General Relativity and string theory as theories, or to infer their equivalence merely because their consistency equations meet in a low-energy corner. This paper gives the correct technical bridge and states what Modal Triplet Theory (MTT) would have to add. We define a worldsheet diagnostic, a target-space Euler-Lagrange diagnostic, two source encodings, and an explicit comparison operator. A controlled diagnostic-square theorem proves exact and approximate transfer of solutions when these maps intertwine and the comparison operator has the required inverse estimate. A counterexample shows that coincident fixed points alone do not establish this bridge. The standard sigma-model result supplies a perturbative instance after field redefinitions; it does not supply an upstream MTT source. We finish with a twelve-row same-source contract and a separate error ledger. Current q79 work provides five rows, partially provides two, and leaves five open. Accordingly, the paper establishes a rigorous conditional comparison framework, not a derivation of string theory, pure Einstein gravity, or their ontological identity from MTT.