The Langlands Transfer Graph

We reframe the Langlands program as a systematic edge-construction effort within the proof category \(\mathbf{Spec}\) introduced in [Nagy, 2026a]. Each proved Langlands-type result — from Artin reciprocity to the modularity theorem — corresponds to a morphism between spectral domains carrying a computable decorrelation gain \(\delta \in [0,1]\). We compute \(\delta\) explicitly for the major known correspondences, finding \(\delta \approx 1.0\) uniformly for problems with algebraic correlative structure. We then identify a structural gap: no known or conjectured Langlands transfer provides \(\delta > 0\) for additive correlative problems (twin primes, binary Goldbach, bounded gaps). This algebraic–additive gap is the quantitative reason why the Langlands program — despite being the most powerful source of new spectral edges in mathematics — has not yet contributed to additive number theory. We characterize what an "additive Langlands transfer" would require, connect the analysis to specific open proof circuits, and propose a research program for identifying the missing morphisms.