Part I: Interior Points and the τ³

Part Overview

Part I defines the point set of τ³ and reveals the fibration structure τ³ = τ¹ ×_f T².

The approach is coordinate-first: Book I earned the ABCD chart Φ(x) = (A,B,C,D) as a total, injective address system for every finite object (I.D17, I.T04). This Part extends the chart beyond finite objects by identifying the τ-admissible quadruples—those ABCD tuples satisfying the constraint lattice forced by the normal-form structure—and completing the finite ABCD space profinitely to include limit points at ω.

The key new result is the omega readout: the coordinate limit of the ABCD chart along the primorial tower. In base coordinates (D,A), ω collapses to a single point (Ω,Ω). In fiber coordinates (B,C), ω has one-dimensional structure: the coupled γ/η dominance flip produces the algebraic lemniscate 𝕃 as the fiber readout of the point at infinity. The lemniscate is not imported; it is the coordinate shadow of ω.

This fiber degeneration—from a full two-dimensional (B,C) parameter space at finite stages to the one-dimensional lemniscate at the boundary—is precisely what distinguishes the fibered product τ¹ ×_f T² from a Cartesian product. The asymmetry between base (D,A) and fiber (B,C) is structural, forced by the greedy peel-off order (not by convention).

The Part concludes by extending the boundary bipolar decomposition e_± = (1 ± j)/2 (I.D21) to all τ-admissible points, and by clarifying why the ABCD four-ray structure replaces the quaternionic algebra proposed in the first edition.

Interior points are defined here as coordinate readouts. The holomorphic structure on this point set—the passage from addresses to analysis—is the subject of Parts II through VI.

Chapters

• Chapter 4: τ
• Chapter 5: The Omega Readout and the Lemniscate
• Chapter 6: The τ³ Fibration Emerges
• Chapter 7: Bipolar Decomposition of Interior Points
• Chapter 8: ABCD Structure Replaces Quaternions