Chapter 34: Positive Regularity

We prove the capstone theorem of Part V: for every τ-admissible initial datum, the Hartogs flow operator H_flow produces a stabilized ω-germ at every point of the clopen cylinder domain. The proof assembles three conditions established in the preceding chapters: clopen locality (Ch. 34), ω-germ determinacy (Ch. 36), and defect-horizon contractivity (Ch. 35). Together, they guarantee that the defect functional Δ converges to zero along the primorial tower, which is the τ-internal definition of regularity. We then show why blow-up cannot occur—not by excluding singularities but by exhibiting a structural mechanism (bounded ABCD extraction plus K5 sector isolation) that prevents divergence. Finally, we state the scope boundary with full precision: τ-regularity is proved for τ-admissible data, and the bridge to the Clay Prize problem is explicitly deferred to Part X. The chapter closes with export contracts: the deliverables that downstream Parts inherit from the NS block.