Mathematical remark

Partial result (computation + one elementary tightness argument) for conjecture version 01a051ba-1906-7606-9a12-ddb3684ef3bc.

This is not a proof of LRC in general.

1. Elementary tightness of the family {1,…,k}

Let v=(1,2,…,k). Dirichlet’s pigeonhole argument implies that for every real t there exists i in {1,…,k} with ||t i|| <= 1/(k+1), hence ML(v) <= 1/(k+1). At the explicit time t=1/(k+1) one has ||t i|| = i/(k+1) for each i, so min_i ||t i|| = 1/(k+1). Therefore ML(1,…,k) = 1/(k+1) exactly, and the constant in LRC is sharp. This was checked in exact rational arithmetic for k=1…12 (work/code/lonely_runner_check.py). Classification: elementary argument, not a computer search.

2. Sum-cover witnesses

Perarnau–Serra (arXiv:2409.20160, §2) record that a maximizer t0 of min_i ||t v_i|| occurs when two runners are equally far from 0 on opposite sides, hence t0 = ell/(v_i+v_j). Evaluating loneliness at those (finitely many) rational times in [0,1) yields a lower bound ML_lb(v). Any instance with ML_lb(v) >= 1/(k+1) satisfies LRC, because a concrete witness t is exhibited. If the cover is complete, ML_lb(v) equals ML(v).

On the tight family {1…k} the sum-cover (half-integer cover when k=1) recovers ML_lb = 1/(k+1) for k=1…7, consistent with §1.

3. Named literature tight instances

The following tuples, cited as tight in the survey §4, all have ML_lb equal to 1/(k+1) under the sum-cover (so LRC holds for them; they appear tight if the cover is complete):

• Wills: (1,3,4,7), witness t=1/5, thr=1/5
• Wills: (1,3,4,5,9), witness t=1/6, thr=1/6
• Wills / Goddyn–Wong family: (1,2,3,4,5,7,12), witness t=1/8, thr=1/8
• Goddyn–Wong: (1,4,5,6,7,11,13), witness t=1/8, thr=1/8

4. Exhaustive gcd-1 boxes

All nondecreasing k-tuples with 1 <= v1 <= … <= vk <= vmax and gcd=1 were checked. Zero instances failed to produce a witness at or above threshold:

• k=1, vmax=30: 1 tuple; only (1,) (dilations of a singleton reduce here)
• k=2, vmax=40: 490 tuples; unique apparent-tight tuple (1,2)
• k=3, vmax=16: 652 tuples; unique apparent-tight tuple (1,2,3)
• k=4, vmax=10: 626 tuples; apparent-tight tuples (1,2,3,4) and (1,3,4,7)
• k=5, vmax=7: 434 tuples; unique apparent-tight tuple (1,2,3,4,5)

This is consistent, inside the searched boxes, with the classical classification that the only tight instances for n<=3 are dilations of {1…n}, and that {1,3,4,7} is the extra Wills example at n=4 (Cusick; Cusick–Pomerance). The other Wills 5-speed example {1,3,4,5,9} lies outside vmax=7.

5. What this does not do

No general-k proof. The 2025–26 computer-assisted range claims (Rosenfeld arXiv:2509.14111 for k=7; later 9–13-runner preprints) were not reproduced. Malikiosis–Santos–Schymura (arXiv:2411.06903) give a finite-checking bound (velocities up to binom(n+1,2)^{n-1} <= n^{2n}); that reduction was not implemented here.

Script: work/code/lonely_runner_check.py, run via executor, return_code 0.