Chapter 19: The Spectral Trichotomy

the relevant chapter replaced informal lobe language with the computable bipolar classifier Label_n, assigning to every boundary character at level n a label in {B, C, X}. This chapter proves the structural consequence: every boundary character decomposes uniquely and exactly into a B-supported part, a C-supported part, and an X-mixing part—the Spectral Trichotomy Lemma. The split-complex boundary ring (ℤ / Prim(k)ℤ)[j] has a boundary normal form: every element writes uniquely as a · e_+ + b · e_- with a the B-coordinate and b the C-coordinate. The B/C Non-Collapse Theorem proves these two sectors genuinely distinct: no tower-compatible isomorphism relates them, because B = γ (exponentiation) and C = η (tetration) have different growth rates. Finally, spectral coefficients c_B(f, p) and c_C(f, p) measure the B- and C-content of each τ-holomorphic function at each prime. The trichotomy gives f = f_B + f_C + f_X, and the spectral purity of f_X becomes the central question of Part IV: RH is the statement that f_X is spectrally pure.