Chapter 11: Holomorphic Implies Continuous (The Inverted Dependency)

In classical complex analysis, the logical chain runs: topology ⇒ limits ⇒ differentiability ⇒ holomorphy. Continuity is a prerequisite; holomorphy is a strengthening condition imposed on a substrate that is already continuous. In Category τ, this chain is inverted. Holomorphy — defined combinatorially in Book I via ω-germ transformer naturality — is the primitive concept. Continuity is a consequence. This chapter proves the core inversion: every ω-germ transformer on τ³ is continuous with respect to the cylinder topology. The proof is short — a three-line lemma does all the work — because naturality is a stronger condition than continuity. The inverted dependency is not cosmetic; it reflects the fact that τ’s holomorphic structure is logically prior to its topological structure.