Lonely runner for 14 runners

• Slug: lonely-runner-fourteen
• List: P44
• Solver: SuperGrok CLI grok-4.6 --reasoning-effort xhigh
• Status: open (LRC(13)); one dent — the tight tuple is excluded at ten primes, 191–239
• Area: Diophantine approximation / computational combinatorics
• Sources: Sungkawichai and Trakulthongchai, Eleven, twelve, and thirteen lonely runners (arXiv:2604.23906); Trakulthongchai, Nine and Ten Lonely Runners
• Started: 2026-08-17

Statement

Computer-assisted work in 2025–26 established the lonely-runner conjecture through 13 runners. The next case, 14 runners (13 nonzero relative speeds), remains open.

Tonight

A certified finite-reduction or modular-sieve certificate for 14 runners, a new excluded speed configuration with an independently checkable witness, or a documented incomplete search. Isolated floating-point scans are not a new bound. Fetch the 13-runner papers before searching.

Outcome (2026-08-17)

Incomplete: SuperGrok hit quota before a walkthrough. Independently replayed rk_inclusion.py --selftest and two p=191 witnesses in verify_witness.py. Exhaustive leftover salvage not replayed. LRC(13) open. See ATTACK.md.

Outcome (2026-08-22, Opus 5)

Read ST26 in full first. Two corrections to the record, then a result.

Correction: what the 2026-08-17 campaign was searching for is false. The statement "every nonzero $v\in(\mathbb Z/14)^{13}$ with a zero coordinate admits $s,j$ with $sv+r_{13}(j/191)\in\{1,\dots,12\}^{13}$" fails at $v=2\cdot(1,\dots,13)$, which no pair among all 2674 saves. Correction: that vector is entirely even, and ST26 Definition 2.1 already discharges it through its gcd condition — a branch that is nearly empty when $k+1$ is prime and wide when $k+1=14$, and which had been dropped in transcription.

With the branch restored, and checked exhaustively in 240 s:

Every $(u_1,\dots,u_{13})\in\mathbb Z^{13}_{>0}$ with $\gcd(u_1,\dots,u_{13})=1$ and $u_i\equiv i \pmod{191}$ has the lonely runner property. Equivalently $(1,2,\dots,13)\notin J(13,191)$. The same holds at 193, 197, 199, 211, 223, 227, 229, 233 and 239, and fails at every prime up to 41.

ST26 Proposition 1.4 proves the analogue only when $k+1$ and $p$ are both odd primes, so $k=13$ is outside it; their Section 5.2 reports $(1,2,\dots,13)$ as the sole survivor of the $\times2$ lifting ladder, and it is eliminated here with no $\times7$ and no $\times14$ lift.

This is not a bound on the number of runners. LRC(13) is open and the reason is untouched: ST26 Section 7 names computing $I(13,p,1)$, about $10^{17}$ tuples at $p=191$, as the bottleneck.

Verifier compute/q1/cover.c, certificate compute/q1/certs/T2_p191.txt, independent checks in compute/q1/check_unsaved.py and compute/q1/spotcheck.c. See WALKTHROUGH.md and ATTACK.md.