Chapter 9: Cylinder Domains from ABCD Refinement

The point set τ³ was defined in the relevant chapter as the collection of τ-admissible ABCD quadruples, completed by the profinite inverse limit. This chapter builds local domains on τ³ using the CRT reduction maps π_k : τ³ → ℤ/P_k that Book I’s Chinese Remainder Theorem provides. The key construction: a stage-k cylinder is the set of all points that agree with a given point modulo the kth primorial P_k. This is a “prefix predicate”: two points lie in the same cylinder if and only if their first k prime reductions coincide. Cylinders decompose along the four ABCD coordinates, are simultaneously open and closed, and form a basis for a topology on τ³. This topology is not chosen — it is the unique topology making all reduction maps continuous. No metric is used; no holomorphy is invoked.