A seventeenth-order polylogarithm ladder

Cohen, Lewin and Zagier found four ladders that entail the polylogarithms ${\rm Li}_n(\alpha_1^{-k}):=\sum_{r>0}\alpha_1^{-k r}/r^n$ at order $n=16$, with indices $k\le360$, and $\alpha_1$ being the smallest known Salem number, i.e. the larger real root of Lehmer's celebrated polynomial $\alpha^{10}+\alpha^9-\alpha^7-\alpha^6-\alpha^5-\alpha^4-\alpha^3+\alpha+1$, with the smallest known non-trivial Mahler measure. By adjoining the index $k=630$, we generate a fifth ladder at order 16 and a ladder at order 17 that we presume to be unique. This empirical integer relation, between elements of $\{{\rm Li}_{17}(\alpha_1^{-k})\mid0\le k\le630\}$ and $\{\pi^{2j}(\log\alpha_1)^{17-2j}\mid 0\le j\le8\}$, entails 125 constants, multiplied by integers with nearly 300 digits. It has been checked to more than 59,000 decimal digits. Among the ladders that we found in other number fields, the longest has order 13 and index 294. It is based on $\alpha^{10}-\alpha^6-\alpha^5-\alpha^4+1$, which gives the sole Salem number $\alpha<1.3$ with degree $d<12$ for which $\alpha^{1/2}+\alpha^{-1/2}$ fails to be the largest eigenvalue of the adjacency matrix of a graph.