Chapter 21: Pasch and the Parallel Postulate

Chapters and earned betweenness (II.D19, II.T15) and congruence (II.D20, II.T16) from the ultrametric structure on τ³. This chapter tackles the two hardest Tarski axioms: the Pasch axiom (a line entering a triangle through one side must exit through another) and the parallel postulate (through a point not on a line, exactly one parallel exists). Both are earned, not assumed. Pasch follows from ultrametric triangle structure; the parallel postulate follows from cylinder separability at each stage of the primorial tower. A critical structural point: the parallel postulate survives τ’s native compactification because the algebraic base is hyperbolic (j² = +1), not elliptic (i² = -1). With this chapter, all Tarski axioms for Euclidean geometry are verified within τ, without importing ℝ, Dedekind completeness, or any continuity axiom.