Transition-pair structure and near-extremal third-position transition robustness in the standard genetic code

Synonymous mutations concentrate at the third codon position of the standard genetic code. We analyse this robustness as an exact combinatorial question and separate two objectives that are easily conflated: the unweighted count of silent third-position substitutions, and the count restricted to transitions, which often occur at higher per-edge mutation rates in vivo. For the unweighted objective, constancy on every four-codon wobble fibre is the unique global maximiser (96 silent edges), and at a fixed number of encoded signals k the sharp maximum is 81 of 96 edges at k=23. Direct enumeration of the standard code-under the 23-signal convention, in which the three termination codons are treated as distinct labels-shows that it does not approach this optimum: it realises 63 of the 96 third-position edges (77.8% of the fixed-k maximum), with eight of sixteen fibres non-constant. The picture inverts for transitions. We prove that the third-position transition graph is a perfect matching of 32 edges whose silent count is maximised exactly when the pyrimidine pair and the purine pair are each constant within every fibre-a strictly weaker condition than fibre constancy-and that the fixed-k transition maximum remains 32 for all k ≤ 32. The standard code attains 29 of 32 (90.6%): every evenly split 2+2 fibre splits along the pyrimidine/purine boundary, and the three lost transition edges are confined to functionally singular loci (initiation, tryptophan, and the stop set). The divergence between the two objectives-far from the unweighted optimum, near the transition optimum-is the central finding, and identifies transition-pair constancy rather than full wobble constancy as the structural principle the standard code approximates. Lean 4 verifies the abstract unweighted bounds, the transition theorems (with the k=23 fixed-signal instance closed by an explicit witness), and the principal standard-code silent-edge counts; the finite transition-weighting frontier and benchmark statistics are supported by independent executable scripts and tabulated outputs. An independent benchmark of amino-acid replacement severity against 50 000 random codes complements these exact results.