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 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 Q6 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 shuffle null (n=10,000), the standard code is significantly low-cost across four established, code-independent physicochemical distance metrics with partially overlapping content (Grant ham 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 Q6→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 Q6 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 Q6 as a sensitivity. Event-level conditional-logit modelling shows that topology avoidance and local physicochemical cost provide complementary, only weakly correlated signal (rs=0.15), and that topology adds explanatory value beyond physicochemistry under both Q6 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.