Part IV: Geometry: The Tarski Program

Part Overview

Part IV executes the Tarski program: deriving Euclidean geometry from ultrametric foundations. The two-readout principle (II.D18a, the relevant chapter) established that geometry is the coarse-grain readout of the coherence kernel, parallel to (not dependent on) the fine-grain topological readout of Part III. Five chapters earn betweenness, congruence, Pasch, and the parallel postulate as theorems—not axioms.

the relevant chapter defines the betweenness relation B(x,y,z) from ultrametric ordering on NF prefixes. Theorem II.T15 verifies that betweenness satisfies Tarski axioms T1–T3.

the relevant chapter defines congruence ≅ from the canonical ultrametric distance d(x,y) = 2^{-δ(x,y)}. Theorem II.T16 verifies Tarski congruence axioms C1–C6. Euclidean congruence emerges from a non-Archimedean base.

the relevant chapter proves both the Pasch axiom and the parallel postulate as Theorems II.T17–II.T18. Pasch follows from ultrametric triangle structure; the parallel postulate follows from cylinder separability.

the relevant chapter explains that split-complex holomorphy generates wave-type PDEs (not Laplacian). Characteristic curves define a causal structure. Euclidean geometry emerges as the static limit (wave speed → ∞).

the relevant chapter outlines the denotation map from τ-geometry to classical Euclidean geometry. ℝ^4 appears as a limit of τ-approximations, not as an ambient space. This bridges to the calibration dictionary of Part V.

Together, these five chapters show that Euclidean geometry is a theorem in τ, earned from the axioms.

Chapters

• Chapter 19: Betweenness from Ultrametric
• Chapter 20: Congruence from Canonical Distance
• Chapter 21: Pasch and the Parallel Postulate
• Chapter 22: Wave-Type Causal Structure
• Chapter 23: Orthodox vs.\ τ Denotation Bridge