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. The four DNA bases are the four stable combinations of two chemical bits (ring type, group type), so each codon is a vertex of the hypercube {0,1}^6 and Watson–Crick complementation is the antipode w → −w. Rumer's 1966 partition is identified exactly (64/64) with the pure/divided split of the 16 codon boxes, and is proved NOT to be an F2-affine functional (exhaustive search over all 63 linear and 126 affine functionals reaches at most 48/64). The amino acid is a function of 5 of the 6 bits (93.75% purity); the only two impure blocks contain the code's punctuation (start ATG, stop TGA), the two degeneracy-1 amino acids (Met, Trp) and one Ile codon. Exhaustive enumeration of the 46,080 elements of W(B6) shows the code has no exact spinor symmetry (strict orbit conjecture refuted) but preserves a Z2 quasi-symmetry (60/64) broken only by punctuation. The 32 five-bit blocks split 16+16 and Watson–Crick acts on them exactly as the charge conjugation 16 ↔ 16bar of the SO(10) spinor; conjugate blocks carry amino acids of opposite hydrophobicity (r = −0.75, p = 0.0016 against 2×10^4 degeneracy-preserving random codes). The Walsh–Hadamard spectrum shows hydrophobicity is the only physicochemical property that is both ~linear over the weights (R² = 74.9%) and ~odd under the antipode (85.0%), with a chemical mechanism: keto groups add oxygen (r = 0.59, p = 6×10⁻⁷), Watson–Crick inverts codon oxygen content exactly (O + O_WC = 6, r = −1.000), and hydrophobic amino acids prefer less polar anticodons (r = 0.52), consistent with Yarus et al. Honest no-gos are reported (no exact symmetry; SU(4)³ fails by rank; position 3 does not reproduce SU(3) color; the 10+10bar piece was not found). Version 4 — correction notice. The central correction: the "Rumer theorem" of earlier versions (Rumer = ring bit of position 1) was false — that agreement is 32/64, i.e. chance; the correct identification is the box-purity split (64/64), and no F2-functional reproduces it. Also corrected: 5bar is NOT antimatter (10, 5bar and 1 are components of one left-handed generation; matter vs antimatter is 16 vs 16bar); a single bit flip changes chirality and cannot exchange 10 ↔ 5bar (two flips needed — position 2 correlates with the split, it does not implement it); r(O content, hydrophobicity) = 0.59 (p = 6×10⁻⁷), not 0.74; the Crick tRNA minimum is 31 for the 61 sense codons (32 only with the separate initiator tRNA-Met); Walsh energies harmonized (74.9%/85.0%); z = −2.71; the (p = 0.022, OR = 16) pair was not realizable and is replaced by the corrected Fisher p = 0.089; the amino→keto substitution adds oxygen but removes nitrogen only in the pyrimidine pair.