arXiv:2608.13599v1

Single-speed modifications of the tight Lonely Runner instance: an effective bound and the complete classification for r = 2

Yuhan Zhang

math.CO11J7111B7505C1552C07

Paper metadata

Original submission: August 1, 2026
Claimed MSC: 11J71, 11B75, 05C15, 52C07
arXiv categories: math.CO
Paper license: CC BY 4.0 ↗
Comments: 10 pages. Code and data: https://doi.org/10.5281/zenodo.21695561

Abstract

For a set V of n-1 distinct positive integers write LR(V) = max_t min_{v in V} ||v t||, where ||x|| is the distance from x to the nearest integer; V is tight if LR(V) = 1/n, the value predicted by the Lonely Runner Conjecture. Goddyn and Wong (Integers 6 (2006), #A38) classified the tight sets obtained from the baseline [n-1] by replacing one speed r with a multiple mr, and proved that for a fixed r only finitely many non-multiple replacements can be tight, remarking that this "partially explains" why the two sporadic tight sets {1,3,4,7} and {1,3,4,5,9}, in which the speed 2 is replaced by an odd number, have no analogues. We make their finiteness effective and settle the case they singled out. Let U(n,r) be the region left uncovered when speed r is deleted from the baseline. We compute the length of every connected component of U(n,r) exactly, in both regimes 2r > n-1 and 2r <= n-1, in terms of an arithmetic quantity I(n,r). Since a connected set on which the inserted speed w must stay 1/n-close to the integers cannot be longer than 2/(wn), this yields the explicit necessary bound w <= 4rI/(2s-I) with s = n-r, and hence: if ([n-1] minus {r}) union {w} is tight with 2r <= n-1, then n <= 6r. This improves the constant implicit in Goddyn and Wong's finiteness theorem from 12 to 6 and makes the classification of single-speed modifications a finite computation for each n. Carrying the computation out for r = 2 we obtain the complete classification: ([n-1] minus {2}) union {w} with w > n-1 is tight if and only if (n,w) = (5,7) or (6,9); the same method disposes of r = 3 entirely. We also report exhaustive censuses in exact rational arithmetic, and note that the natural guess that tight sets have all speeds below 2n is false, a counterexample being the Goddyn-Wong set {1,...,29,31,90} with n = 32.

AI-generated audit

Audit summary

Audited against arXiv v1

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated August 18, 2026

01Statements4 reported findingsContains unsupported statements

The component-length formula, effective bound, complete classifications for deleted speeds $2$ and $3$ , the one-speed census, the bounded-window census, and the explicit tight counterexample are correct. The two-speed census in Theorem 1.8(ii) is not able to be verified because its released exhaustive-search implementation omits all cases in which speed $1$ is deleted.

Theorems 1.4 and 1.6Correct

Exact component lengths and the effective replacement bound

Pages 3 and 5–6 · Theorems 1.4 and 1.6 · arXiv:2608.13599v1

Dirichlet's lemma confines every component of $U(n,r)$ to a $1/(rn)$ -neighborhood of a reduced fraction $p/r$ . The runner $2r$ supplies the inner puncture exactly when $2r\leq n-1$ , while the largest legal representatives in the residue classes $\pm p^{-1}\pmod r$ supply the two outer walls. All other runner constraints are weaker. This yields the two displayed component-length formulas. Covering a longest component with the inserted speed gives $w\leq4rI/(2s-I)$ , and $I\leq n-1$ together with $w\geq n$ gives $n\leq6r$ .

Theorem 1.7Correct

The deleted-speed $2$ and $3$ classifications

Pages 3 and 6–7 · Theorem 1.7 · arXiv:2608.13599v1

For $r=2$ , the exact value of $I(n,2)$ reduces the effective bound to five possible values of $n$ and finitely many $w\leq13$ ; exact breakpoint evaluation leaves only $(n,w)=(5,7)$ and $(6,9)$ . For $r=3$ , the same formula reduces to $7\leq n\leq16$ with an explicit finite range for $w$ , and exact evaluation finds no tight set. The range $n\leq6$ follows from the verified divisibility classification of Goddyn and Wong.

Theorem 1.8(ii)Not able to verify

The released search does not cover replacements that delete speed $1$

Page 4 · Theorem 1.8(ii); computational proof in Section 4 and the archived file twoswap_hunt.py · arXiv:2608.13599v1

The theorem quantifies over all distinct $r_1,r_2\in[n-1]$ and claims that, for $6\leq n\leq34$ and $w_i\leq4n$ , the only tight two-speed modification is $\{1,4,5,6,7,11,13\}$ . The released driver enumerates deleted pairs with combinations(range(2,n),2), so it never tests any pair $\{1,r\}$ . Neither the manuscript nor another released search supplies a general argument excluding those omitted cases throughout the theorem's range. No counterexample was found, but the full quantified census is not established by the supplied computation. A repair requires enumerating the missing $r_1=1$ branch with the same exact certification, or proving an independent exclusion theorem for it.

Archived verification repository, twoswap_hunt.py ↗

Theorem 1.8(i), (iii), and (iv); Proposition 1.9Correct

The remaining exact censuses and the speed- $90$ example

Pages 4 and 7–8 · Theorem 1.8(i), (iii), (iv), Proposition 1.9, and Section 4 · arXiv:2608.13599v1

The one-speed driver covers every deleted speed and every inserted $w$ in the stated box. The window driver enumerates every primitive $(n-1)$ -subset of $\{1,\ldots,2n-1\}$ in the stated range. Survivors of the proved-safe floating-point filter are evaluated exactly at the complete breakpoint set. The independently implemented wall-subdivision and branch-and-bound checks agree with the component formulas and certify every listed tight set, including $\{1,2,\ldots,29,31,90\}$ at $n=32$ .

02Proofs4 reported findingsContains incorrect or incomplete proofs

The analytic proofs and the exact-certification method are correct. The computational proof of Theorem 1.8(ii) is incomplete because a quantified family of deleted-speed pairs is absent from the enumeration.

Section 3Correct and complete

Wall analysis for $U(n,r)$

Pages 5–6 · Equations (4)–(7) and proof of Theorem 1.4 · arXiv:2608.13599v1

Fractions with non-coprime numerator are excluded by a surviving divisor of $r$ . For a reduced fraction, multiples of $r$ yield exactly the inner wall $|\delta|>1/(2rn)$ when present. Among nonmultiples, the constraints with residue distance one are binding, and inequality (7) shows every residue-distance-at-least-two constraint is weaker. The proof also blocks the remaining portion of each Dirichlet interval, so no unlisted components occur.

Breakpoint and numerical certificationCorrect and complete

Every candidate maximum is checked exactly and the pre-filter is safe

Pages 4 and 7–8 · Lemmas 2.2 and 4.1; Section 4.3 · arXiv:2608.13599v1

The minimum of finitely many sawtooth functions can attain a positive maximum only at a peak of an active function or at a crossing of two active functions, giving the complete rational breakpoint set. Exact rational evaluation at that set determines $LR(V)$ . Lemma 4.1 bounds the floating-point error far below the discard margin, and every survivor is recomputed exactly. An independent wall-subdivision implementation matched all tested component sets and formulas, while adaptive branch-and-bound independently certified the listed tight sets.

Theorem 1.8(ii) enumerationIncomplete as written

One deletion branch is absent from the purported exhaustive search

Page 8 · Section 4.2 and Table 2; archived twoswap_hunt.py and reproduce_all.py · arXiv:2608.13599v1

Both released entry points loop over combinations(range(2,n),2). The proof obligation includes pairs with $r_1=1$ , because Theorem 1.8(ii) imposes no lower bound $r_i\geq2$ . The window census covers only a smaller parameter window and does not fill the omitted range $n\leq34$ , $w_i\leq4n$ . Repair classification: no repair supplied for the full statement; extending the exact enumeration to combinations(range(1,n),2) is a plausible repair only until its output is produced and independently checked.

Archived verification repository ↗

Goddyn–Wong inputCorrect and complete

The divisibility classification is applied in its stated range

Pages 2–3 and 6 · Theorem 1.1, Proposition 1.3, and proof of Theorem 1.7 · arXiv:2608.13599v1

The cited theorem gives exactly the criterion for tight replacements $r\mapsto mr$ . Proposition 1.3 correctly shows that a deleted speed $r$ with $2r>n-1$ must divide an inserted speed: at $t=1/r$ , all surviving baseline speeds are farther than $1/n$ , while a nonzero value of $\|w/r\|$ is at least $1/r>1/n$ .

Goddyn–Wong, Tight instances of the lonely runner, Theorem 2.3 ↗

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

Detailed audit reportFull reasoning, manuscript locations, and references.

Open report PDF ↗

Author response

Challenge an audit finding

Local workflow preview

A listed author may submit formal evidence that an audit is inaccurate. The response would be considered in a fresh AI re-evaluation; it would not edit the audit automatically.

Paper: arXiv:2608.13599v1
Authors listed: Yuhan Zhang
Audit date: August 18, 2026

01Establish identityMatch an authenticated scholarly identity to this paper.
02Submit evidenceIdentify the finding and give a formal mathematical response.
03Re-evaluateA separate agent checks the response and records a disposition.

Email fallback for papers without a linked ORCID

A production fallback could send a one-time link only when the submitted address matches an independently maintained author-contact allowlist for this paper. MathAudit must return the same message for every address so the form cannot reveal which contacts are on that list.

Structured response preview

This form remains unavailable until production identity verification succeeds. Nothing entered here is submitted.

Audit finding challengedWhy the evaluation is inaccurateFormal evidence or repairVerifiable sources, if anyI am submitting evidence, not instructions for the reviewing agent.

This panel never establishes authorship in the local prototype. A production result should be described narrowly as an authenticated ORCID match or control of a separately allowlisted author-contact mailbox.

Audit controls

Status: Report ready
Source: arXiv v1
Prompt: Standard prompt · locked

The report is available below and in the public summary.

Review prompt

You are a mathematical auditor. Evaluate only the correctness of the paper's central statements, the correctness and completeness of its proofs, and evidence-backed non-novelty. Do not make editorial comments or evaluate importance, exposition, style, or suitability for publication. Keep the report neutral, precise, and professional. PART 1 — CORRECTNESS OF THE MAIN STATEMENTS Identify the paper's central statements and assess each one separately using exactly one of these outcomes: - Correct: the statement has a correct, verifiable proof; or, despite a defect in the printed proof, you can give a fully verified repair or independent proof. - Incorrect: you can give strong formal evidence that the statement as written is false, preferably a counterexample or a complete contradiction. - Not able to verify: the supplied proof does not establish the statement, and you can neither supply a fully verified proof nor disprove it. For every adverse or unverified outcome, restate the exact claim and hypotheses, identify its location in the reviewed version, and give the decisive reasoning. Do not infer that a theorem is false merely because its printed proof is defective. Assign the overall statements status by the following priority: 1. Contains wrong statements, if at least one central statement is Incorrect. 2. Contains unsupported statements, if no central statement is Incorrect but at least one is Not able to verify. 3. Correct, if every central statement is Correct, apart from any typographical or minor formal findings allowed by Part 4. PART 2 — CORRECTNESS AND COMPLETENESS OF THE PROOFS Audit the proofs of the central statements and every material internal or cited input on which they depend. For each proof or proof segment, use the narrowest supported outcome: - Correct and complete: every substantive step is verifiable and no case or crucial nontrivial step is missing. - Incorrect as written: a displayed calculation, inference, lemma, or cited application is formally false or inapplicable. - Incomplete as written: a necessary substantive step or case is missing, even though no explicit contradiction has been proved. - Not able to verify: the available paper and sources are insufficient to decide whether the step is valid. Do not flag omitted routine details, compressed arguments, or “analogously” steps when the omitted reasoning is recoverable and correct. For every defect, identify its smallest exact location, explain its downstream dependency, and classify any proposed repair as one of: - Verified repair: supply enough reasoning to verify that the repair works throughout every affected use. - Plausible repair only: describe the possible repair but do not treat it as established. - No repair supplied. Assign the overall proofs status by the following priority: 1. Contains incorrect or incomplete proofs, if at least one proof is Incorrect as written or Incomplete as written. 2. Contains unverified proofs, if there is no proved incorrectness or incompleteness but at least one material proof is Not able to verify. 3. Correct, if all audited proofs are Correct and complete, apart from any typographical or minor formal findings allowed by Part 4. PART 3 — NOVELTY OF THE MAIN STATEMENTS This is a one-sided, evidence-only check. Evaluate only central results. Report a result only when a verifiable primary source, publicly available before the earliest public version of the paper that contains the result, proves the same statement up to a change of notation, or when the result follows directly and immediately from such an earlier theorem. For each non-novelty finding, provide the earlier source, its public date, the exact theorem or proposition, and an explicit hypothesis-and-conclusion mapping. A later source cannot establish non-novelty. Do not flag a result merely because an easier proof exists or because it follows through a nontrivial argument. Do not flag a result that the paper itself explicitly presents as known, a survey item, or an alternative proof. Mention only statements that meet this non-novelty threshold. Say nothing about statements for which no such evidence was found: do not certify novelty, list unsuccessful searches, or add statement-level “no prior source found” entries. If the list is empty, the section may say only “No non-novelty findings.” PART 4 — TECHNICAL AND EVIDENTIARY RULES 1. Criticism must never be overstated. Call a statement incorrect, a proof incorrect or incomplete, or a result non-new only when strong formal evidence supports that exact judgment. Otherwise use Not able to verify where that outcome exists, or remain silent in the novelty section. 2. Review a specified paper version. Give exact page, theorem, equation, or proof locations; restate relevant quantifiers and hypotheses; and check domains, endpoints, parameter ranges, constants, and dependencies explicitly. Apply a defect only to the regimes and downstream results that actually depend on it. 3. Immediate-consequence closure: when deciding whether a conclusion is supported, treat the argument as including every immediate formal consequence of facts established earlier in the paper and available at that point. An immediate formal consequence is a local, mechanically checkable inference using only definitions already in force, elementary logic, direct substitution or renaming, weakening of an inequality or set inclusion, or a monotonicity or symmetry property explicit in a definition or already proved. 4. Such an inference must introduce no new lemma, estimate, construction, case analysis, limiting or compactness argument, external result, or new issue of uniformity or parameter dependence. Check the direction of monotonicity and the dependence of every threshold or constant explicitly. Do not exchange pointwise and uniform conclusions, change exceptional sets, interchange quantifiers or limits, or extend an endpoint under this rule unless the required fact was already established. 5. If the claimed conclusion follows after immediate-consequence closure, it is fully supported. Do not report a gap, unsupported statement, minor formal correction, or recommendation to add an explanatory sentence, and do not mention the suppressed item in the report. This suppression does not apply to a genuine typographical error as defined below, because a typo is a literal notation defect that still requires a textual correction. For example, if

P(delta)

is proved for every sufficiently small

delta>0

and

P

is immediately monotone in the required direction, then the claim that

P(delta)

holds for every

delta>0

requires no correction; the omitted monotonicity step is exposition rather than an audit finding. 6. Immediate-consequence closure permits reconstruction of an omitted inference; it does not permit silent alteration of the printed domain, hypotheses, or quantifiers of an intermediate statement later invoked outside its stated scope. If formal validity requires changing the text of that statement or its stated range, report the appropriate finding even when the repair is easy. Thus a result stated only for

t>0

and subsequently applied at

t=0

remains a formal scope mismatch unless the endpoint case was separately established before the application. 7. Apply this mandatory gate before recording any adverse, unverified, typographical, or minor formal finding: (a) identify the exact assertion or inference alleged to be defective; (b) reconstruct the strongest conclusion following from the preceding argument after all admissible immediate formal consequences are added; (c) identify the specific defect that remains — a formally demonstrated falsehood, a concrete nontrivial unresolved proof obligation, or a genuine domain, hypothesis, quantifier, or notation mismatch requiring a textual correction; and (d) state the evidence and minimal repair. If no specific defect remains, or if the only possible addition would be explanatory, suppress the finding entirely. Use Not able to verify only when a precise, nontrivial unresolved obligation can be identified. 8. A typographical error is an incorrect symbol, index, exponent, sign, variable name, or notation mismatch whose unique intended correction is mechanically determined from the surrounding text and does not alter the mathematical argument. Report it in yellow as Typo, with the printed text, correction, exact location, and reason it is harmless. It does not lower an overall status. 9. A minor formal correction is a boundary, endpoint, or displayed parameter-range defect that can be repaired consistently in every affected statement by a local change, without a new argument, without invalidating any related result or application, and without creating another substantive defect. Report it in yellow as Minor formal correction, with the same details. It does not lower an overall status. 10. Do not use either minor category if more than one materially different repair is possible; if the repair changes the substance or scope of a central result; if an excluded case is used later; if the correction fails to propagate consistently; or if a new mathematical idea is required. In such cases use the strongest substantive classification actually supported by evidence. 11. Write every mathematical expression as valid, compilable LaTeX. Use

...

for inline mathematics and

...

for displayed mathematics, with balanced delimiters and braces. Do not leave raw LaTeX commands outside math delimiters. 12. Separate theorem truth from the correctness of the printed proof. A theorem can be Correct in Part 1 because of a fully verified independent proof while its printed proof is adverse in Part 2. Conversely, a defective proof alone never licenses an Incorrect theorem classification. Organize the report under Parts 1–3 and apply Part 4 throughout. Do not add editorial commentary, scores, rankings, recommendations, or a list of suppressed observations.

This panel is visibly marked as owner-only for the concept prototype. Authentication is not enabled locally.

Single-speed modifications of the tight Lonely Runner instance: an effective bound and the complete classification for r = 2

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Exact component lengths and the effective replacement bound

The deleted-speed 222 and 333 classifications

The released search does not cover replacements that delete speed 111

The remaining exact censuses and the speed-909090 example

Wall analysis for U(n,r)U(n,r)U(n,r)

Every candidate maximum is checked exactly and the pre-filter is safe

One deletion branch is absent from the purported exhaustive search

The divisibility classification is applied in its stated range

Challenge an audit finding

Authenticate with ORCID, then require an exact arXiv match

The deleted-speed $2$ and $3$ classifications

The released search does not cover replacements that delete speed $1$

The remaining exact censuses and the speed- $90$ example

Wall analysis for $U(n,r)$