pith. sign in

arxiv: 2604.23906 · v1 · submitted 2026-04-26 · 🧮 math.CO · cs.DM· math.NT

Eleven, twelve, and thirteen lonely runners

Touch Sungkawichai , Tanupat Trakulthongchai This is my paper

Pith reviewed 2026-05-08 05:37 UTC · model grok-4.3

classification 🧮 math.CO cs.DMmath.NT
keywords Lonely Runner ConjectureWills conjecturecomputer-assisted proofsieving techniquespolynomial methodfractional partsDiophantine approximation
0
0 comments X

The pith

The Lonely Runner Conjecture holds for any ten to twelve integer speeds via computer-assisted verification.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Wills conjectured that for any k non-zero integers, there is a real t making the fractional parts of t times each ui at least 1 over k plus one away from integers. Prior work verified this up to nine speeds. This paper refines sieving methods and adds a polynomial argument to cover cases where the speeds are equivalent to the first k naturals modulo a large odd prime. It thereby establishes the conjecture for k equal to ten, eleven, and twelve through computation.

Core claim

Any tuple of integers (u1 to uk) congruent to (1 to k) modulo an odd prime p larger than k squared plus k, with overall gcd one, satisfies the lonely runner bound when k plus one is also an odd prime. Combined with sieving to reduce to such cases, this yields a computer-assisted proof that the conjecture is true for k in ten, eleven, twelve.

What carries the argument

Refined sieving techniques that reduce the problem to congruence classes, together with a polynomial method argument that proves the bound for those reduced cases.

If this is right

  • The conjecture is true for all sets of ten non-zero integer velocities.
  • The conjecture is true for all sets of eleven non-zero integer velocities.
  • The conjecture is true for all sets of twelve non-zero integer velocities.
  • Verification now covers all cases up to twelve rather than nine.
  • Only cases not covered by the prime modulus condition need direct computation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the method generalizes, higher values of k may become computationally feasible with increased resources.
  • Any potential counterexample for these k must avoid the consecutive residue classes modulo large primes.
  • Similar sieving plus algebraic arguments might apply to other Diophantine problems on the circle.
  • The bound 1 over k plus one appears tight only for specific configurations that are now ruled out computationally.

Load-bearing premise

The sieving process and the polynomial verification together cover every possible tuple without gaps or calculation errors.

What would settle it

A specific list of ten integers whose multiples never all stay at least one eleventh away from integers at any single time t would disprove the result for k equals ten.

read the original abstract

Wills conjectured that, for any non-zero integers $u_1,\ldots,u_k$, there is a real number $t$ such that, for all $i=1,\ldots,k$, \[\lVert tu_i\rVert\geq\frac{1}{k+1},\] where $\lVert x\rVert$ is the distance from $x$ to the closest integer. This statement is known as the Lonely Runner Conjecture. A computational method developed by Rosenfeld and the second author verified the conjecture for $k\leq9$. We further refine this method with new sieving techniques and employ a polynomial method argument to show that any $(u_1,\ldots,u_k)\equiv(1,2,\ldots,k)\pmod{p}$ with $\gcd(u_1,\ldots,u_k)=1$ satisfies the conjecture when $k+1$ and $p > k^2+k$ are both odd primes. Ultimately, we provide a computer-assisted proof of the Lonely Runner Conjecture for $k\in\{10,11,12\}$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript claims a computer-assisted proof of the Lonely Runner Conjecture for k=10,11,12. It first uses a polynomial method to establish the result for all coprime tuples (u1,...,uk) ≡ (1,2,...,k) mod p whenever k+1 and p>k²+k are odd primes, then applies refined sieving to reduce all remaining cases to a finite set dispatched by exhaustive computer enumeration and modular-arithmetic checks.

Significance. If the sieving is exhaustive and the computational verification is free of implementation errors, the result extends the known verifications (previously up to k=9) and supplies concrete evidence for three further values of the conjecture. The approach is noteworthy for its parameter-free modular reduction in the polynomial step and its use of standard tools to produce a falsifiable, machine-checkable partition of the space.

major comments (1)
  1. [Sieving section (post-polynomial argument)] The sieving reduction (described after the polynomial argument) is load-bearing for the central claim, as any omitted coprime tuple would leave an unchecked case that could contain a counterexample. The manuscript must supply an explicit argument or lemma showing that the refined sieving partitions the entire space of gcd=1 integer k-tuples without gaps before the finite enumeration begins.
minor comments (2)
  1. The abstract states that the computer search 'dispatches' the reduced set but does not indicate whether the source code or the exact enumeration bounds are provided as supplementary material; this would aid independent verification.
  2. Notation for the distance function ||·|| and the precise statement of the polynomial-method lemma could be cross-referenced more clearly to the earlier Rosenfeld–author work for k≤9.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the importance of a fully rigorous presentation of the sieving argument. We address the single major comment below.

read point-by-point responses

  1. Referee: [Sieving section (post-polynomial argument)] The sieving reduction (described after the polynomial argument) is load-bearing for the central claim, as any omitted coprime tuple would leave an unchecked case that could contain a counterexample. The manuscript must supply an explicit argument or lemma showing that the refined sieving partitions the entire space of gcd=1 integer k-tuples without gaps before the finite enumeration begins.

    Authors: We agree that an explicit lemma is required to make the completeness of the sieving fully transparent. In the revised manuscript we will insert a new lemma (placed immediately after the description of the refined sieving) that proves the following: every coprime k-tuple either satisfies the congruence condition handled by the polynomial method or is reduced, via the sequence of modular sieving steps, to one of the finitely many representatives that are subsequently checked by exhaustive enumeration. The lemma will enumerate the possible residue classes modulo the relevant primes, show that each sieving filter is surjective onto the remaining classes, and verify that the process terminates with a representative in the enumerated set. This addition will eliminate any possibility of an unchecked gap. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to prior computational method for k≤9, but new refinements and polynomial argument are independent

full rationale

The paper cites a computational method developed by Rosenfeld and the second author that verified the conjecture for k≤9, then refines it with new sieving techniques and adds a polynomial method argument for tuples congruent to (1,2,...,k) mod p (with p odd prime >k²+k). The remaining cases are reduced to a finite set dispatched by computer enumeration. No step reduces the central claim for k=10,11,12 to a self-definition, fitted input renamed as prediction, or load-bearing self-citation chain; the derivation stands on external computational verification and standard modular facts rather than circular construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard number-theoretic facts about the distance-to-nearest-integer function and modular arithmetic; no free parameters, ad-hoc axioms, or new postulated entities appear in the abstract.

axioms (2)
  • standard math Standard properties of the toroidal distance ||x|| to the nearest integer
    Invoked in the statement of the Lonely Runner Conjecture itself.
  • standard math Basic facts of modular arithmetic and gcd=1 conditions
    Used to restrict to the congruence class (u1,...,uk) ≡ (1,2,...,k) mod p.

pith-pipeline@v0.9.0 · 5490 in / 1283 out tokens · 40152 ms · 2026-05-08T05:37:57.239950+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

  1. [1]

    Javier Barajas and Oriol Serra,The Lonely Runner with Seven Runners, Electronic Journal of Combinatorics 15(2008), no. 1, R48

    work page 2008

  2. [2]

    Benjamin Bedert,Riesz products and the lonely runner conjecture: A wider gap of loneliness, arXiv preprint arXiv:2511.16636 (2025)

    work page arXiv 2025

  3. [3]

    Wills,Untere schranken fur zwei diophantische approximations-funktionen, Monat- shefte fur Mathematik76(1972), no

    Ulrich Betke and J¨ org M. Wills,Untere schranken fur zwei diophantische approximations-funktionen, Monat- shefte fur Mathematik76(1972), no. 3, 214–217

    work page 1972

  4. [4]

    Tom Bohman, Ron Holzman, and Dan Kleitman,Six lonely runners, Electronic Journal of Combinatorics8 (2001), no. 2, R3

    work page 2001

  5. [5]

    Cusick and Carl Pomerance,View-obstruction problems, III, Journal of Number Theory19 (1984), no

    Thomas W. Cusick and Carl Pomerance,View-obstruction problems, III, Journal of Number Theory19 (1984), no. 2, 131–139

    work page 1984

  6. [6]

    2, 343–361

    Vikram Giri and Noah Kravitz,The structure of Lonely Runner spectra, Mathematical Proceedings of the Cambridge Philosophical Society180(2026), no. 2, 343–361

    work page 2026

  7. [7]

    Wong,Tight instances of the lonely runner, Integers6(2006), no

    Luis Goddyn and Erick B. Wong,Tight instances of the lonely runner, Integers6(2006), no. 14, A38

    work page 2006

  8. [8]

    2, 331–352

    Matthias Henze and Romanos-Diogenes Malikiosis,On the covering radius of lattice zonotopes and its relation to view-obstructions and the lonely runner conjecture, Aequationes mathematicae91(2017), no. 2, 331–352

    work page 2017

  9. [9]

    Romanos Diogenes Malikiosis, Francisco Santos, and Matthias Schymura,Linearly exponential checking is enough for the lonely runner conjecture and some of its variants, Forum of Mathematics, Sigma13(2025)

    work page 2025

  10. [10]

    Matthieu Rosenfeld,The lonely runner conjecture,https://gite.lirmm.fr/mrosenfeld/ the-lonely-runner-conjecture, 2025, Accessed 2026-03-14

    work page 2025

  11. [11]

    ,The lonely runner conjecture holds for eight runners, arXiv preprint arXiv:2509.14111 (2025)

    work page arXiv 2025

  12. [12]

    ,The lonely runner conjecture holds for nine runners, arXiv preprint arXiv:2512.01912 (2025)

    work page arXiv 2025

  13. [13]

    Touch Sungkawichai,LRC holds for 11, 12, 13 runners,https://github.com/vzsky/13-lonely-runners, 2026, Accessed 2026-04-14

    work page 2026

  14. [14]

    Tanupat Trakulthongchai,Nine and ten lonely runners code,https://github.com/t-tanupat/ nine-and-ten-lonely-runners, 2025, Accessed 2026-03-14

    work page 2025

  15. [15]

    ,Nine and ten lonely runners, Electronic Journal of Combinatorics, to appear (2026)

    work page 2026

  16. [16]

    Wills,Zwei S¨ atze ¨ uber inhomogene diophantische Approximation von I rrationalzehlen, Monatshefte f¨ ur Mathematik71(1967), no

    J¨ org M. Wills,Zwei S¨ atze ¨ uber inhomogene diophantische Approximation von I rrationalzehlen, Monatshefte f¨ ur Mathematik71(1967), no. 3, 263–269 (de)

    work page 1967

  17. [17]

    3, 195–207

    Xuding Zhu,Circular chromatic number of distance graphs with distance sets of cardinality 3, Journal of Graph Theory41(2002), no. 3, 195–207. London, UK Email address:touchsungkawichai@gmail.com St John’s College, University of Oxford, Oxford, UK Email address:tanupat.trakulthongchai@sjc.ox.ac.uk

    work page 2002