Robust error-minimization in the genetic code across physicochemical metrics and variant codes: a graph-theoretic analysis in GF(2) ⁶

The standard genetic code reduces the impact of point mutations, but the robust-ness of this property across physicochemical metrics, naturally variant codes, and codon-reassignment mechanisms remains incompletely quantified. We embed the 64 codons in GF(2) ⁶ , representing the hypercube Q ₆ as a coordinate-dependent subgraph of the encoding-independent single-nucleotide mutation graph H (3, 4), which supports continuous ρ -interpolation between the two and enables joint analysis of physicochemical error minimization and codon-family topology. Under a block-preserving null ( n =10,000), the standard code is significantly low-cost across four distinct amino-acid distance metrics (Grantham p = 0.006; Miyata p < 0.001; Woese polar requirement p = 0.003; Kyte–Doolittle hydropathy p = 0.001), addressing the concern that prior optimality results could be metric-specific; the signal strengthens monotonically as ρ moves Q ₆ → H (3, 4). Across the 27 NCBI translation tables, near-optimality is broadly preserved: 11 of the 12 infor-mative-distance variants retain top-5% placement after BH–FDR correction (yeast mitochondrial is the sole marginal exception). Natural codon reassignments rarely break codon-family connectivity: under H (3, 4), only 6 of 28 observed events are topology-breaking versus 66% of 1,280 candidate moves (RR 0.32, permutation p ≤ 10 ⁻⁴ ). This depletion is robust to alternative topology definitions, clade exclusions, and base-to-bit encodings, although the small breaker subset (4 of 6 from a single yeast-mitochondrial lineage; denominator effect in clade-exclusion robustness) is underpowered for strong cross-clade inference. Conditional-logit decomposition shows that topology avoidance and local physicochemical cost provide complementary, only weakly correlated signal ( r _s = 0.15); a heuristic tRNA-distance proxy does not improve fit, and several variant-code lineages show suggestive tRNA-gene enrichment for reassigned amino acids. Retrospective reanalysis of nine genome-recoding datasets is consistent with—but does not establish—a working hypothesis in which codon-family topology operates at a different biological layer from acute cellular fitness: Syn61 tolerated 18,218 boundary-crossing serine swaps as a class (genome-wide, not per-codon-position viability), yet the same move type is 3.1-fold depleted across natural code evolution. The contribution is the second axis: code evolution is jointly constrained by physicochemical smoothness and codon-family topological integrity, and these two constraints are partly independent.