Chapter 59: Interface Width and τ

The τ-Tower Machine (Chapter 55) computes by inspecting primorial stages. How many stages does a given computation need? This chapter introduces interface width — the minimal primorial depth at which a computation stabilizes — and defines τ-admissibility as the finiteness of this width. The Interface Width Principle shows that τ-admissible functions factor through a single finite quotient ℤ/Prim(k₀)ℤ: the entire infinite tower collapses to one level. This collapse is earned: the boundary characters χ_+, χ_- are admissible with width 1, and composition preserves admissibility with sub-additive width bounds (Proposition [prop:earned-admissibility]). Interface width is the E₂ analog of the mass gap (E₁): a finite threshold separating tractable structure from infinite regress.