Chapter 40: Characters on the Algebraic Lemniscate

the relevant chapter earned the algebraic lemniscate 𝕃 = (H_τ, ω_𝕃, σ) as the pre-geometric boundary of τ, and the relevant chapter formalized the split-complex scalar ring ℤ_τ[j] with its idempotent decomposition e_+ = (1+j)/2, e_- = (1-j)/2. This chapter develops the character theory of 𝕃: ring homomorphisms from the bipolar spectral algebra into the split-complex scalars. The fundamental characters χ_+ and χ_- project onto the B-sector and C-sector respectively. The polarity character χ

is recovered as a character in this formal sense. The full character group Char(𝕃) is a group under pointwise multiplication, and every character traces back to the bipolar partition of primes established by the Prime Polarity Theorem . Characters are the spectral probes of 𝕃: they detect sector membership and will drive the spectral decomposition of the relevant chapter and the bipolar Fourier analysis of the relevant chapter.