4 Shadow Symmetry = Time Reversal
4.1 The Three-Step Proof
The map \(\Delta \mapsto 2-\Delta \) has order \(2\).
The orthogonal complement \(\Lambda \mapsto \Lambda ^\perp \) on \(\mathrm{Gr}(2,4)\) acts as the antipodal map on \(S^2\) via the Penrose correspondence.
Under \(\Lambda \mapsto \Lambda ^\perp \), the symplectic normalisation forces \(\omega \mapsto \omega ^{-1}\).
On the principal series \(\Delta = 1+i\lambda \), time reversal \(T\) sends \(\Delta \mapsto 2-\Delta \): the shadow transform. (Most-cited result in ONON52: 16 cross-references.)
Under \(\Delta = 2s\), shadow \(\Delta \leftrightarrow 2-\Delta \) is the functional equation \(s \leftrightarrow 1-s\).
\(\mathrm{Re}(\Delta ) = 1\) under \(\Delta = 2s\) gives \(\mathrm{Re}(s) = \tfrac {1}{2}\).