Chapter 48: Boundary Characters via Idempotent Support

Part VII proved two decisive results: the Idempotent Decomposition Lemma (Lemma [lem:idempotent-decomposition], II.L07), which splits every holomorphic map f = e_+ · f_+ + e_- · f_- into its B-channel and C-channel projections, and the Three-Lemma Chain (the relevant theorem, II.T33), which shows that holomorphy is equivalent to idempotent support. This chapter turns that equivalence inward: we restate Book I’s boundary ring ℤ_τ spectral characters (I.D19, I.D22–I.D23) in the language of idempotent decomposition, proving that every spectral character valued in calibrated H_τ (the relevant definition, II.D35) admits a unique canonical decomposition χ = e_+ · χ_+ + e_- · χ_-. The character algebra A_spec(𝕃) is the algebra of idempotent-supported characters — there are no others. This chapter sets up the input side of the Central Theorem: the boundary characters, fully organized by their idempotent support structure, ready for the Hartogs extension of the relevant chapter.