Chapter 14: The Primorial Ladder

Part II closed with the No Knobs Principle , asserting that every sector coupling is a rational function of ι_τ evaluated at a specific primorial depth. The word primorial appeared throughout Parts I and II as a background assumption: the stages of the tower, the levels at which τ-effective statements are verified, the index set of the primorial presheaf (I.D83). This chapter elevates the primorial tower from background infrastructure to the central organising object of Part III. We define the primorial ladder, prove that it is cofinal among all modular towers—so that checking a property at primorial levels is sufficient—and introduce the primorial filter, the mechanism by which infinity is tamed. The chapter closes with an export table: every millennium problem’s τ-effective statement reduces to verification at primorial levels.