The Corrector Gap of the Genetic Code from E8: Structural Connection to Cancer and Testable Predictions

The degeneracies of the standard genetic code, {1, 2, 3, 4, 6}, are derived from the Coxeter number h(E6) = 12 in the breaking chain E8 → E6 → SO(10) → SU(5) → G_SM of the TTOE framework — h = 12 is the only value in [2,100] whose proper divisors reproduce them, and the code is never introduced as input. From this the corrector gap p_syn = 138/576 = 23.96%, the per-position anisotropy of the codon (4.2% / 1.0% / 66.7%) and the neutral count baseline dN/dS = 3.17 follow algebraically. Validation against real tumour data: SynMICdb (2.81M mutations: p_syn = 23.4%, error 2.4%) and TCGA-BRCA (93,326 SNVs: 25.0%, error 4.4%). Driver genes concentrate 93% of their mutations at codon positions 1–2, where the gap is thin, versus 66% in passengers (p = 0.008, Mann–Whitney), and 14/15 known oncogenic hotspots fall at positions 1–2. The mitochondrial-code prediction (PRED-073/074) has been tested on 1,907 somatic mtDNA substitutions (Ju et al. 2014): sign confirmed — p_syn(mito) = 17.6% observed vs. 25.0% nuclear; the mitochondrial code is compatible with the data (p = 0.25) while the nuclear code is rejected (p = 8×10⁻⁶); the observed gap exceeds the code-only prediction, the remainder attributed to selection. The position-1 anisotropy of the mitochondrial code (zero deg-1 amino acids) is confirmed. Declared scope, stated in the paper itself: this work provides no clinical tools, does not improve MutSigCV/PolyPhen-2/CHASM, and does not cure, prevent or diagnose. Its contribution is theoretical — deriving from first principles the algebraic baseline that oncology takes as empirical input — with every claim classified by epistemic status.