Chapter 40: Regularity as Positive Structure

What does it mean for a function to be regular at a point? In classical complex analysis, the answer is negative: a function is regular where it is not singular. Regularity is defined by the absence of pathology — the absence of poles, branch points, or essential singularities. One first defines what can go wrong, then declares regularity to be the complement of wrongness.

In Category τ, regularity is a positive condition: the existence of a stabilized ω-germ. A point p ∈ τ³ is τ-regular for a function f if the sequence of ω-germ restrictions stabilizes at a finite primorial stage — if finitely much boundary data suffices to determine f at p. This chapter introduces τ-Regularity (the relevant definition, II.D49), proves the Regularity Criterion (the relevant theorem, II.T34), which establishes a sharp dichotomy — every point is either τ-regular (stabilization at finite stage) or genuinely singular (the germ sequence diverges) — and explains in Remark [rem:positive-negative-regularity] (II.R11) why the classical notion of “removable singularity” has no τ-analogue. The ultrametric topology admits no epsilon-delta wiggle room: cylinders are clopen, so singularities cannot be “approached” and then “removed.” They are either present at a given stage or absent.

The key dependencies are: the Global Hartogs Theorem (I.T31, the Global Hartogs Extension (Book I, Chapter 62)) establishes that boundary data determines interior data; the Idempotent Decomposition Lemma (II.L07, the relevant chapter) provides the canonical e_±-splitting; the Holomorphic ⇔ Idempotent-Supported equivalence (II.T33, the relevant chapter) characterizes holomorphic maps; and the evolution operator (II.D37, the relevant chapter) provides the propagation mechanism through which stabilization occurs.