Finiteness and completeness of sporadic sextuples

Determine whether the sporadic near-tight sextuples with k=1 form a finite list and whether the nine sporadic sextuples listed in Table \ref{tab:sporadic6} constitute the complete list.

Background

The six-speed theorem leaves a finite but unspecified exceptional set E6E_6. In the exhaustive search up to speed 110, nine primitive near-tight sextuples lie on none of the three classified critical subtori; all nine have k=1k=1 and are listed in Table \ref{tab:sporadic6}.

The paper notes two apparent structural patterns among these examples but does not establish whether further sporadic tuples occur at larger speeds or whether the observed list is complete.

References

We do not know whether the list of sporadic sextuples is finite as a list of tuples with $k=1$ only, or whether Table \ref{tab:sporadic6} is complete; the nine tuples are of two visible kinds, a near-tight quintuple with a further runner of moderate speed (for instance $(1,4,5,6,7)$, whose $\ML$ is $2/11$, with $22$ or $33$ adjoined) and tuples with no apparent structure.

— Odd denominators in the Lonely Runner spectrum for six speeds  (2609.03444 - Cordella, 3 Sep 2026) in Section 4.1, “Six speeds,” immediately following Table \ref{tab:sporadic6}