Chapter 17: The Adelic Embedding

The CRT Decomposition Theorem

splits the primorial residue ring ℤ / Prim(k) ℤ into a product of prime-power rings, and Hensel lifting

extends these finite pieces to the τ-native local fields ℤ_p^τ. A single structural question remains: how do the local fields reassemble into a global object? The answer is the τ-adele ring A_τ, the restricted product of the ℤ_p^τ with respect to their unit groups. This chapter constructs A_τ within the τ framework—no algebraic number theory is imported—and proves the Adelic Embedding Theorem: the canonical map τ → A_τ is injective with dense image. We then show that τ-holomorphic functions on A_τ decompose into local Euler factors, and we give the bi-square (I.T41) its adelic reading: tower coherence becomes restricted-product coherence, and spectral naturality becomes compatible local spectral decompositions.