Chapter 41: Spectral Decomposition

the relevant chapter introduced the lemniscate characters χ_+ and χ_- (the relevant definition, I.D37) as ring homomorphisms from the algebraic lemniscate 𝕃 into the split-complex scalars ℤ_τ[j]. This chapter proves the spectral decomposition theorem: every element of 𝕃 decomposes uniquely into B-sector and C-sector components via these characters. The decomposition x = χ_+(x) · e_+ + χ_-(x) · e_- is canonical — it requires no choice of basis, no approximation, and no topology. The master constant ι_τ = 2/(π + e) governs the relative spectral weight of the two sectors, and the profinite topology on ℤ_τ provides the convergence framework for spectral sums.