Chapter 46: D-Holomorphy and the Cauchy–Riemann Analog

the relevant chapter earned the split-complex scalar ring ℤ_τ[j] (the relevant definition, I.D20), and the relevant chapter constructed the bipolar spectral algebra H_τ = ℤ_τ[j] (the relevant definition, I.D27) with j² = +1 forced by the bipolar structure (the relevant theorem, I.T10). This chapter develops the first layer of analysis on H_τ: D-differentiability (the relevant definition, I.D42), the split-complex analog of complex differentiability. The role played by the Cauchy–Riemann equations in classical complex analysis is taken here by the split Cauchy–Riemann equations (the relevant definition, I.D43), whose characteristic feature is a plus sign where the classical equations carry a minus sign — reflecting the hyperbolic signature j² = +1. In sector coordinates (u, v) = (a + b, a - b), the split-CR equations reduce to ∂ F_+ / ∂ v = 0 and ∂ F_- / ∂ u = 0: each sector component depends on only one variable. The sector independence theorem (Proposition [prop:sector-independence], I.P22) formalizes this: every D-holomorphic function decomposes as f(u, v) = (g(u), h(v)) for independent functions g and h. This is the wave equation ∂^2 f / ∂ u ∂ v = 0. In the classical (orthodox) setting over ℝ, sector independence makes D-holomorphy too flexible — the theory lacks the identity theorem, Liouville’s theorem, and virtually all the rigidity that powers complex analysis. The zero divisors e_+ · e_- = 0 are the root cause. The chapter closes by diagnosing this pathology precisely, setting the stage for the τ-framework’s rescue of D-holomorphy via tower coherence and diagonal discipline .