Chapter 46: Functoriality as Diagram Commutativity

Chapter 48 established the automorphic–Galois duality and the local Langlands instance , providing the vertical maps in the enriched bi-square. This chapter establishes the horizontal maps: the sector morphisms between primitive sectors. The central result is the Functoriality Theorem , which asserts that every sector morphism commutes with the automorphic–Galois duality. The chapter proceeds through a three-stage scaling argument: the Riemann Hypothesis is commutativity of a single column of the enriched bi-square (the B-sector column); Grand GRH (the relevant definition, Ch. 27) is commutativity of all four columns; and Langlands functoriality is commutativity of the entire pasted diagram—columns plus horizontal maps. The proof of the Functoriality Theorem applies Mutual Determination (the relevant definition, Ch. 33) sector by sector, transported through the enrichment functor Enr₀₁ (the relevant definition, Ch. 44). Two key instances—base change and transfer—are then shown to be naturality conditions on the enriched bi-square . The chapter closes by exhibiting Langlands functoriality as earned structure: the bi-square (I.T41), Mutual Determination (III.D25), and the enrichment functor (III.D57) together suffice.