Syntax Cannot Exhaust Semantics: A No-Reduction Theorem for Reflexive Systems Paper 53 of the NEMS Suite

Paper 51 proved that no final internal self-theory can exist in a sufficiently expressive diagonally capable reflexive system. A natural escape hatch remains: perhaps the failure is only in the semantic packaging; maybe a purely syntactic formal closure could still exhaust what matters. The present paper closes that gap. We prove that no purely syntactic internal structure can be total and exact for realized semantic truth in a diagonally capable reflexive system. In slogan form: syntax cannot exhaust semantics. We define syntactic theory-objects, semantic adequacy, and semantic exhaustiveness; we prove that syntactic semantic exhaustion would induce a final self-theory and thus yield contradiction. The theorem is not anti-formalism or anti-syntax—it is anti-exhaustion. We include a careful comparison with Tarski-style truth undefinability. The development is mechanized in Lean 4 as the SyntaxSemantics library in reflexive-closure-lean. Primary anchors: semanticExhaustiveᵢnducesfinal, noₛyntacticₛemanticₑxhaustion (; verify in. lean if renamed). Trust boundary. Formal claims refer to the SyntaxSemantics definitions in the pinned artifact; Tarski comparison () is conceptual classification, not a duplicated classical proof.