Chapter 28: Replaces i — Polarity, Not Rotation

Chapters and earned the transcendental constants π and e. This chapter earns the algebraic constant j—the split-complex unit satisfying j² = +1, j ≠ ± 1. The classical imaginary unit i (with i² = -1) never arises in Category τ. The reason is structural: τ carries a bipolar flip (a discrete ℤ/2 symmetry exchanging the B and C channels), not a continuous rotation SO(2). Polarity—the exchange of two sectors—gives j² = +1. Rotation—the smooth cycling of a single circle—would give i² = -1. The fibered product τ³ = τ¹ ×f T² has two fiber coordinates (B and C), and j encodes their exchange. The chapter defines the bipolar idempotents e± = (1 ± j)/2, which are the canonical sector projections: e_+ projects onto the B-channel, e_- onto the C-channel. Every split-complex element decomposes uniquely as z = z_+ e_+ + z_- e_-, and this is the spectral decomposition inherited from the boundary ring (I.D19, Book I).