Chapter 25: Spectral Purity and the Critical Line

We harvest the consequences of self-adjointness. The spectral correspondence of Chapter 24 maps zeros to eigenvalues; self-adjointness forces eigenvalues to be real; reality forces the imaginary part of ρ(1-ρ) - ¼ to vanish; and for non-trivial zeros γ ≠ 0, this annihilation implies σ = ½. The Riemann Hypothesis emerges as a spectral purity theorem: K5 diagonal discipline forbids off-diagonal mixing in H_L, and this discipline propagates through the spectral correspondence to constrain the location of zeros. The critical line is the locus where B-sector and C-sector contributions balance exactly, a balance enforced by the bipolar symmetry encoded in Book I’s bi-square characterization.