Chapter 22: Wave-Type Causal Structure

Chapters – earned Euclidean geometry as a theorem: betweenness, congruence, Pasch, and the parallel postulate all follow from the ultrametric structure of τ³. But Euclidean geometry is static—it has no notion of propagation, no preferred direction, no causal order. This chapter adds dynamics. The split-complex unit j with j² = +1 (I.T10, Book I) gives the codomain H_τ a hyperbolic algebraic structure. The split-complex Cauchy–Riemann equations yield the wave equation ∂^2 f / ∂ x² - ∂^2 f / ∂ y² = 0, not the Laplace equation. This is hyperbolic, not elliptic: it has characteristic curves (null lines) that define a causal structure on τ³. The B/C asymmetry from prime polarity provides a preferred propagation direction. Euclidean geometry reappears as the static limit: when wave-type coupling vanishes, the wave equation degenerates to Laplace, and the causal structure disappears. The forced 1 + 3 split of the relevant chapter is the algebraic seed of spacetime.