Part II: Local Domains: Cylinders as Prefix Predicates

Part Overview

Part II constructs the local domains and proves the central inversion: holomorphic implies continuous. Four chapters build the topological foundation from ABCD coordinates.

the relevant chapter defines cylinders as clopen sets determined by ABCD prefix constraints. The D-ray yields ultrametric stage-k cylinders (depth-based); A/B/C-rays yield angular peel-token constraints. Cylinders form the canonical clopen basis.

the relevant chapter introduces the first disagreement depth δ(t, t’) between NF expansions and the ultrametric distance d(t,t’) = 2^{-δ(t,t’)}. Cylinders are precisely ultrametric balls; the ultrametric triangle inequality holds in its strong form.

the relevant chapter proves Theorem II.T06: Every ω-germ transformer is continuous. This is the core inversion: classical complex analysis moves continuous ⇒ holomorphic; τ moves holomorphic ⇒ continuous, forced by probe naturality and cylinder compatibility.

the relevant chapter notes that Parts I–II work purely topologically: split-complex structure H_τ is not yet needed. The topological skeleton stands alone; split-complex becomes load-bearing in Part VI (Local Hartogs).

Together, these four chapters invert the classical dependency chain and establish the boundary-first paradigm: topology emerges from holomorphy.

Chapters

• Chapter 9: Cylinder Domains from ABCD Refinement
• Chapter 10: The Ultrametric and First Disagreement Depth
• Chapter 11: Holomorphic Implies Continuous (The Inverted Dependency)
• Chapter 12: Split-Complex Structure Not Yet Load-Bearing