Part II: Local Domains: Cylinders as Prefix Predicates
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.