IV.T146 — Majorana Theorem: zero-U(1) modes are Majorana

Every mode psi with zero U(1)-holonomy charge satisfies C_tau(psi) = pm psi. Proof: sigma maps Q=0 subspace to itself (since -0=0); sigma^2=id (involution), so eigenvalues +/-1. Case sigma(psi)=+psi: C_tau(psi)=psi (Majorana). Case sigma(psi)=-psi: field redefinition psi_tilde=i*psi gives C_tau(psi_tilde)=-i*C_tau(psi)=-i*(-psi)=i*psi=psi_tilde (Majorana via antilinearity). Both cases Majorana. Lab verification (50-digit mpmath): all three neutrino eigenstates have sigma-parities [+1,-1,+1]; sigma-odd eigenvalue = a = iota_tau^p exactly.