The Genetic Code as an Ordered Quaternion Path: The Three Base Symmetries as the Three Units, the Reading Rule Modulo $\mathbf j$,\\ and the Codons Nature Reassigns

With the author's assignment of the four bases to the quaternion units, A = 1, C = i, G = j, U = k, and codons read as ordered (Path B) products, two exact results follow for the standard genetic code, and a registered prediction on the variant codes passes. The three base symmetries are the three units. Watson–Crick complementation is multiplication by k, the transitions are multiplication by j, and Rumer's transformation U ↔ G, C ↔ A, which carries the eight four-fold ("strong") doublets onto the eight split ("weak") doublets, is multiplication by i, with ±q counted as one destination. The third base is read modulo j. Multiplication by j exchanges A ↔ G and C ↔ U, so the purines form the coset {1, j} and the pyrimidines the coset {i, k} = i{1, j}. In every weak box the meaning depends only on the coset of the third base, with exactly two exceptions (AUA/AUG and UGA/UGG). In every strong box the third base is not read. Nature reassigns the two exceptions. In a test registered before counting, the 65 codon reassignments in the 25 variant codes listed by NCBI fall on AUA or UGA 20 times (30.8%), 9.8 times the uniform rate (p = 6 × 10^-15); UGA is the most reassigned codon of all. The vertebrate and ascidian mitochondrial codes, which reassign both, obey the modulo-j rule with no exception. The companion prediction, that each reassignment moves a code toward the rule, failed as registered. In the standard code the multiplicities that follow are {1, 2, 3, 4, 6}, the set of n for which cos(360°/n) is rational; this set holds in 14 of 26 codes and is not required. The translated chain itself is the ordered backbone product of Path B, which requires one sense: each mirror residue costs three backbone hydrogen bonds, a count that matches seventeen measured peptides with no adjusted parameter.