Chapter 10: The 4+

Five generators. Four primitive sectors. One coupling sector. The ABCD decomposition from Book I assigns to each of the four primitive generators α, π, γ, η a dominant coordinate in the quadruple Φ(x) = (A, B, C, D). Restricting this decomposition to the boundary character space ℂhar(𝕃) yields four pairwise non-redundant sector projections—one per generator. The fifth generator ω does not produce a fifth sector. It mediates coupling between the existing four: the ω-coupling sector consists of characters with non-trivial projection in both the B and C lobes simultaneously. The result is a 4+1 decomposition, not a 5-fold splitting. The decomposition is not postulated but induced by the Langlands_0 functor of the relevant chapter: the boundary-to-interior functor Φ maps the four primitive character families to four independent holomorphic sectors and the mixed characters to the coupling sector. At the first enrichment level E₁, the template acquires physical content—four fundamental interactions plus a mass mechanism—but no physical constants, no coupling values, and no Lagrangians appear. That derivation is strictly deferred to Books IV and V.