The genetic code as a broken spinor: binary structure, Watson-Crick charge conjugation, Walsh spectrum and SU(5) decomposition

Abstract

We show that the standard genetic code is a precise algebraic object, not a contingent table. The 64 codons are the weights of the spinor of so(13); Watson-Crick complementation is the antipode w→-w; the SO(10)→SU(5) step splits the spinor 16 into 10 (matter, hydrophilic) + 5̄ (antimatter, hydrophobic) + 1. The Walsh-Hadamard spectrum reveals that hydrophobicity is the only physicochemical property that is simultaneously ~linear over the weights (R²=77.2%) and ~odd under the antipode (84.4%). The chemical mechanism is exact: 5̄ has more keto groups (C=O) and thus more oxygen; Watson-Crick inverts O/N (r=-1.000); hydrophobic amino acids prefer less polar anticodons (r=0.52, p=5×10⁻⁵), consistent with the RNA-amino acid binding experiments of Yarus et al. (2009). Two levels are distinguished: the algebraic-chemical scaffolding is an exact derivation (Lie theorems + chemistry), while the coupling to hydrophobicity is statistical with a physical mechanism.