Chapter 9: Boundary Functoriality (Langlands₀

The Canonical Ladder Theorem

established four enrichment layers E₀ ⊂neq E₁ ⊂neq E₂ ⊂neq E₃ and a universal layer template. This chapter puts that template to work for the first time. The lemniscate 𝕃 = S¹ ∨ S¹ carries a natural space of characters, and these characters are indexed by a lattice ℤ^2 whose two axes encode the multiplicative and additive structure of the series. A canonical functor Φ maps boundary characters on 𝕃 to holomorphic functions in the fibered product τ³. The central result of this chapter is that Φ preserves the bipolar decomposition: the χ_+-sector of the boundary maps into one holomorphic sector, the χ_–sector into another, and the mixed characters into the ω-coupling sector. This functorial preservation is what we call Langlands_0—boundary functoriality—and it is the mechanism that induces the 4+1 decomposition developed in the next chapter.