Research — lonely runner, 14 runners

Papers replayed

Sungkawichai and Trakulthongchai, Eleven, twelve, and thirteen lonely runners, arXiv:2604.23906v1 (26 Apr 2026). https://arxiv.org/abs/2604.23906. Fetch with python3 scripts/arxiv_fetch.py 2604.23906; the reading copy used here was pdftotext -layout into compute/refs/st26.txt. compute/refs/ is local only. Read in full on 2026-08-22. The published frontier: LRC(k) for k ≤ 12, i.e. up to 13 runners.

What was actually used, with the section numbers, because the folder had been citing this paper loosely:

• Definition 2.1 — (k,p,l)-proper, including the gcd branch gcd(l, v_1,…,v̂_i,…,v_k) > 1. Footnote 1 replaces lp by l. This is the term the 2026-08-17 transcription dropped.
• Lemma 2.2, Lemma 2.4, Corollary 2.5, Proposition 2.7 — the reduction from LRC(k) to J(k,p) = ∅ over a prime set with ∏ p ≥ B_k.
• Lemma 2.6 — B_k = (C(k+1,2)^{k-1}/k)^k, from Malikiosis–Santos–Schymura. ln B_13 = 13(12 ln 91 − ln 13) = 670.3497….
• Remark 3.2 — if p > k+1 then I(k,p,l) = ∅ only if (k+1) | l. At k=13 this forces 14 | l, so a ×2 ladder alone can never finish.
• §5.2, the k=11 lifting diagram S1 ×2 ×2 ×2 ×2 ×3 ×3 → S7. This is the load-bearing observation for this folder: k+1 = 12 is composite and they simply lift through its prime factors, with no polynomial method anywhere. Compositeness of k+1 is a change of route, not a wall.
• Proposition 4.1 — the polynomial identity in F_{k+1}, odd prime k+1. It exists only to avoid the c = k+1 lift. Its proof compares leading coefficients of two degree-k polynomials agreeing on k+1 points; both steps need a field, so Z/14 is genuinely unavailable.
• Lemma 4.2, Lemma 4.3, Proposition 4.4, Proposition 1.4 — the chain from Prop 4.1 to "every u with gcd(u)=1 and u_i ≡ i (mod p) has the LR property", valid when k+1 and p > k²+k are both odd primes.
• §5.1 — the equivalence reduction (permute, negate, scale by Z_p^×), giving about p^k / (2^k (k−1)!) representatives.
• Table 1 — the prime sets and ln ∏ p vs ln B_k for k = 10, 11, 12.
• §7 — names the k=13 bottleneck explicitly: "the primary bottleneck in extending our results to k = 13 is the efficient computation of I(k,p,1)". At p=191 under §5.1 that is about 1.1e17 representatives. Also Conjecture 7.1 (a universal denominator D for non-tight tuples).

Trakulthongchai, Nine and ten lonely runners, arXiv:2511.22427, E-JC 33 (2026). https://arxiv.org/abs/2511.22427 (python3 scripts/arxiv_fetch.py 2511.22427). Source of the lifting scheme (Definition 31, Lemma 5, Lemma 7) that ST26 generalize. Not re-derived here beyond what ST26 restate.

Cited through ST26, not opened

• Malikiosis, Santos, Schymura, Linearly exponential checking is enough…, Forum of Mathematics Sigma 13 (2025) — the B_k bound, used as quoted.
• Goddyn and Wong, Tight instances of the lonely runner, Integers 6 (2006) A38 — tightness of (1,…,k). Used qualitatively: (1,…,k) has a witness time only at s/(k+1) with gcd(s,k+1)=1, which is why every s=0 constraint in compute/q1/cover.c is hit. The code asserts this at run time rather than relying on the citation.
• Rosenfeld, arXiv:2509.14111 (eight runners) and arXiv:2512.01912 (nine).
• Bedert, arXiv:2511.16636 — the 1/(2k) + k^{-5/3+o(1)} gap. Different regime; not used.

Companion code

• vzsky/13-lonely-runners (ST26) and t-tanupat/nine-and-ten-lonely-runners (Tra25). Recorded 2026-08-17: LrcVerifier<13>/<14> templates exist but results/ stops at k=12, and their sat/README.md estimates ~1.5 machine years for a k=14 find-cover. Not re-fetched on 2026-08-22; nothing in this session depends on their numbers.

Lookups that came back empty

• No published statement of ST26 Proposition 1.4 or 4.4 with the primality hypothesis on k+1 removed. Their §7 gestures at extending "the algebraic method in Proposition 4.1 … to other classes of speed tuples", but the composite-k+1 case is not treated.
• No OEIS lookup was useful: the object here is a covering condition on (Z/14Z)^13, not an integer sequence.