Chapter 45: Automorphic–Galois Duality

The BSD block (Chapters 45–47) built the E₁ → E₂ bridgehead: proto-codes exist, their count is governed by the BSD Coherence Theorem (the relevant theorem, Ch. 47), and the rank–L-value equality is proved. This chapter opens the Langlands block by identifying the stage on which the Langlands programme plays out in Category T. The character lattice ℤ^2 from the 4+1 sector decomposition (the relevant definition, Ch. 10) carries two natural axes: the m-axis (Galois/arithmetic, decomposing prime by prime via the CRT) and the n-axis (automorphic/spectral, decomposing eigenvalue by eigenvalue via the spectral trichotomy). At E₀, both axes carry identical algebraic structure; at E₁, they differentiate—the m-axis acquires arithmetic content and the n-axis acquires spectral content. We define the Local Langlands Instance at each prime , construct the Euler product as the CRT decomposition of the spectral determinant, and prove that the automorphic–Galois duality in T is an instance of Mutual Determination on ℤ^2 (Proposition [prop:duality-mutual-determination-z2]).