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

Abstract The standard genetic code reduces the impact of point mutations, but the robustness of this property across physicochemical metrics, naturally occurring variant codes, and codon-reassignment mechanisms remains incompletely quantified. Embedding the 64 codons in GF(2) 6 represents the hypercube Q 6 as a coordinate-dependent subgraph of the encoding-independent single-nucleotide mutation graph H (3, 4), and enables continuous ρ -interpolation between the two. Under a quartet-pattern shuffie null ( n =10,000), the standard code is significantly low-cost across four established, code-independent physicochemical distance metrics with partially overlapping content (Grantham p = 0.0062; Miyata p < 0.001; Woese polar requirement p = 0.003; Kyte–Doolittle hydropathy p = 0.001), and the signal strengthens monotonically as ρ moves Q 6 → H (3, 4). A structure-aware sensitivity analysis under the alignment-derived ProtSub matrix (Jia & Jernigan 2021) yields the most extreme percentile of any measure tested ( p = 0.0004; all five p-values pass Bonferroni at α = 0.05). Across the 27 NCBI translation tables, near-optimality is preserved: 11 of 12 informative-distance variants retain top-5% placement after BH–FDR correction. Natural codon reassignments avoid disrupting codon-family connectivity: under the encoding-independent H (3, 4) adjacency, observed events are topology-breaking at relative risk 0.32 versus the candidate landscape (permutation p ≤ 10 −4 ). The H (3, 4) result is stable by construction; the Q 6 decomposition is representation-specific and fails to show depletion under 8 of 24 base-to-bit encodings, so we report H (3, 4) as the primary test and Q 6 as a sensitivity. Event-level conditional-logit modelling shows that topology avoidance and local physicochemical cost provide complementary, only weakly correlated signal ( r s = 0.15), and that topology adds explanatory value beyond physicochemistry under both Q 6 and encoding-independent H (3, 4) adjacency. Retrospective reanalysis of nine genome-recoding datasets is consistent with codon-family topology operating as an evolutionary-trajectory constraint distinct from acute engineering fitness. 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. Highlights Codon-space geometry links genetic-code robustness and reassignment paths Standard and variant codes preserve broad physicochemical error minimization Reassignments are depleted for codon-family topology-breaking moves Conditional-logit models separate topology from physicochemical similarity Synthetic recoding shows boundary conditions for natural-code constraints