Chapter 33: Split-Complex Scalars

the relevant chapter proved that bipolar polarization forces j² = +1 (the relevant theorem, I.T10). This chapter formalizes the extension ℤ_τ[j], the split-complex scalar ring, where every element has the form a + bj with a, b ∈ ℤ_τ. The fundamental idempotents e_+ = (1+j)/2 and e_- = (1-j)/2 split the ring into two sectors, corresponding to the B and C components of the lemniscate boundary. This ring has zero divisors (e_+ · e_- = 0), but the diagonal-free discipline ensures that zero divisors do not cause collapse. ℤ_τ[j] is the natural scalar ring for all subsequent analysis on τ.