Twisted Bracelets for Sorting by Transpositions: the Transposition Diameter of $S_{16}$
Sorting By Transpositions (SBT) seeks the minimum number of transpositions required to sort a permutation $π$ on $n$ symbols into the identity $ι$. Let $N=n+1$. A cyclic-target pair $(ω,β)$ consists of an even permutation $ω$ and an $N$-cycle $β$ for which $ρ=ωβ$ is an $N$-cycle. An SBT instance is the special case $(\barι{\barπ}^{-1},\barπ)$, where $\barπ$ and $\barι$ encode $π$ and $ι$, and $\barι{\barπ}^{-1}\barπ=\barι$. For a prescribed fixed-point-free cycle type, fixed-content words encode $ω$, with colors distinguishing cycles and ranks recording their orientations relative to $β$. A word is realizable exactly when $ρ=ωβ$ is an $N$-cycle. Permutations of equal-part colors and shifts of rank origins form auxiliary symmetries that, together with word rotation and position reflection coupled to rank inversion, define a twisted dihedral action. Its orbits are twisted bracelets, and its realizable orbits correspond bijectively to extended-toric equivalence classes of cyclic-target pairs, where reflection is adjoined to classical toric equivalence. The correspondence yields a Burnside identity, and our method generates one encoding word for each such class. The transposition diameter $TD(n)$ is the largest transposition distance in $S_n$. Combining fixed-point contraction and reductions of the ambient instance space based on cycle structure with exhaustive verification of every remaining twisted bracelet, we prove $TD(16)=9$, closing a twenty-five-year gap. This result also yields $TD(19)=11$ and, for every $n\equiv1\pmod{3}$ with $n\geq16$, $TD(n)\leq\left\lfloor(2n-2)/3\right\rfloor-1$, improving the previous general upper bound by one for these $n$.