arxiv: 2603.24784 · v2 · submitted 2026-03-25 · 🧮 math.CO · math.MG

Recognition: 2 theorem links

· Lean Theorem

Coloopless zonotopes and counterexamples to the Shifted Lonely Runner Conjecture

M\'onica Blanco , Francisco Criado , Francisco Santos

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Pith reviewed 2026-05-15 00:05 UTC · model grok-4.3

classification 🧮 math.CO math.MG

keywords coloopless zonotopesshifted lonely runner conjecturecovering radiuslattice zonotopescounterexampleslonely vector propertyconvex geometry

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The pith

Explicit coloopless zonotopes provide counterexamples to the shifted Lonely Runner Conjecture starting at five runners.

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

The paper constructs lattice zonotopes without coloops and computes their covering radii to produce concrete violations of the bound conjectured for the shifted Lonely Runner Conjecture. These violations begin at five runners and simultaneously disprove the Lonely Vector Property from twelve runners onward. The constructions generalize earlier zonotopal reformulations of the original conjecture and make prior bounding theorems apply to wider classes of zonotopes. A sympathetic reader cares because the geometric restatement turns an open number-theoretic question into an explicit search for lattice polytopes whose covering radius exceeds a simple threshold.

Core claim

Coloopless lattice zonotopes, defined as those without coloops and containing all primitive zonotopes of width at least two, admit explicit examples whose covering radii exceed the conjectured upper bound in the shifted Lonely Runner Conjecture for dimension four (five runners) and in the Lonely Vector Property for dimension eleven (twelve runners).

What carries the argument

Coloopless zonotopes: lattice zonotopes with no coloops, whose covering radii are computed directly to test the conjectured bounds on the shifted Lonely Runner Conjecture and the Lonely Vector Property.

If this is right

The shifted Lonely Runner Conjecture fails for every number of runners five or larger.
The Lonely Vector Property fails for every dimension twelve or larger.
Theorems that bounded the speeds needed to check the conjectures now apply to the wider class of coloopless zonotopes.
Cosimple zonotopes, already introduced earlier, similarly contain all primitive zonotopes of width at least three.

Where Pith is reading between the lines

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

The original Lonely Runner Conjecture may still hold because it corresponds to a narrower subclass of zonotopes than the shifted version.
Direct covering-radius computations on low-dimensional coloopless zonotopes can now be used to test other related geometric conjectures that admit zonotopal restatements.
The failure of the shifted conjecture suggests that any proof of the original Lonely Runner Conjecture must exploit the stricter conditions that define the original LR zonotopes.

Load-bearing premise

The explicit lattice zonotope constructions satisfy the coloopless condition and the computed covering radii indeed violate the conjectured bounds.

What would settle it

A four-dimensional coloopless zonotope whose covering radius is strictly larger than the conjectured limit, verified by enumerating the maximum distance from any point in space to the nearest point of the zonotope's lattice.

Figures

Figures reproduced from arXiv: 2603.24784 by Francisco Criado, Francisco Santos, M\'onica Blanco.

**Figure 1.** Figure 1: Distance to the origin (vertical axis) in terms of t (horizontal axis) of five runners with v = (1, 2, 3, 4, 5) and s = 1 94 (0, 46, 38, 47, 72). The dotted horizontal line is {y = 15 94 }, and the fact that for every t ∈ [0, 1] there is some runner on or below that line shows that γ min(1, 2, 3, 4, 5) ≤ 15 94 . The four dots along this line are the instants when the minimum distance from the runners to th… view at source ↗

**Figure 2.** Figure 2: The parameters κ and µ of the zonotopes P in [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗

**Figure 3.** Figure 3: The parameters λ1 and λ2 of the zonotopes P −P in [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗

**Figure 4.** Figure 4: Illustration of the proof of Lemma 3.3 Recall that in matroid theory the contraction of a vector configuration V at an element v ∈ V is the vector configuration of rank one less obtained projecting along the direction of v, and forgetting the element v. We now relate diagonals of a configuration with contractions of its Gale dual. Lemma 3.4. Let U and U ∗ be Gale dual to one another, let u, v ∈ U, and let … view at source ↗

**Figure 5.** Figure 5: Left: the symmetrized configuration S of a counterexample S to LVP with |S| = 12. Right: same with |S| = 16 • If −a < c and c + 1 < a then u ′ = u − e1 and v ′ = v + e1 do the job. Hence, for the rest we assume c ∈ {−a, a − 1}. • If |d| ̸= b then take u ′ = u + e2 and v ′ = v − e2. Hence, for the rest we assume without loss of generality that d = b. • With this, the only remaining cases are {u, v} = {(−a,… view at source ↗

**Figure 4.** Figure 4: shows smaller examples. The pictures represent centrally sym [PITH_FULL_IMAGE:figures/full_fig_p030_4.png] view at source ↗

**Figure 6.** Figure 6: Left: A certificate polytrope T k,γ in R 3/1R 3 . Right: the exterior of T k,γ is subdivided into the polytropal regions of the facets of T k,γ. Observe that X/1R is a polytrope too. Indeed, since x1 = 0 over X, the definition of X in (6) is equivalent to X = {x1 = 0} ∩ {x ∈ R n : 0 ≤ xi − x1 ≤ 1/vi , i = 2, . . . , n}. We now describe our algorithm, in the version using the open certificates (that is, the… view at source ↗

**Figure 7.** Figure 7: plots the values of γ min for all integer velocity vectors with n = 5 and Pvi ≤ 100. The figure strongly suggests that (1, 2, 3, 4, 5) is most probably the only counterexample to sLRC for this n [PITH_FULL_IMAGE:figures/full_fig_p041_7.png] view at source ↗

**Figure 8.** Figure 8: Same plots as in [PITH_FULL_IMAGE:figures/full_fig_p042_8.png] view at source ↗

read the original abstract

Henze and Malikiosis (2017) have shown that the Lonely Runner Conjecture (LRC) can be restated as a convex-geometric question on the so-called LR zonotopes, lattice zonotopes with one more generator than their dimension. This relation naturally suggests a more generel statement, the "shifted" LRC, the zonotopal version of which concerns a classical parameter, the covering radius. Theorems A and B in Malikiosis-Schymura-Santos (2025) use the zonotopal restatements of both the original and the shifted LRC to prove a linearly-exponential bound on the size of the (integer) speeds for which the conjectures need to be checked in order to establish them for each fixed number of runners; in the shifted version their statement and proof rely on a certain assumption on two-dimensional rational vector configurations, the so-called "Lonely Vector Property". In this paper we do two things: We push the analogies between the two versions of LRC and their zonotopal counterparts, in particular highlighting that the proofs of Theorems A and B in Malikiosis-Schymura-Santos are more transparent, and the statments more general, if regarded in terms of two quite general classes of lattice zonotopes: the coloopless zonotopes that we introduce here and the cosimple ones, already defined by them. These classes contain all primitive zonotopes of widths at least two and at least three, respectively. We show explicit counterexamples to both the shifted Lonely Runner Conjecture (starting at $n=5$) and to the Lonely Vector Property (starting at $n=12$).

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.

Desk Editor's Note private letter to a colleague

The paper gives explicit counterexamples that disprove the shifted Lonely Runner Conjecture at n=5 and the Lonely Vector Property at n=12, while defining coloopless zonotopes to clarify the zonotopal reformulations.

read the letter

The main thing to know is that this paper provides explicit counterexamples disproving the shifted Lonely Runner Conjecture for five or more runners and the Lonely Vector Property for twelve or more, while defining coloopless zonotopes to streamline the geometric approach. They do a solid job extending the zonotopal view from Henze and Malikiosis. By introducing coloopless zonotopes, which include all primitive zonotopes of width at least two, they show that the proofs in the 2025 paper by Malikiosis, Schymura, and Santos become more transparent and apply more broadly. The same holds for the cosimple zonotopes already in use. This framing highlights the parallels between the original and shifted versions without adding unnecessary assumptions. The counterexamples are given as specific lattice zonotopes, so the disproofs come down to checking covering radii rather than some indirect argument. The main concern is the verification of those examples. The paper must include the exact generator vectors and the details of how the covering radii were calculated. For n=5 the dimension is low enough that this can be done rigorously, but any mistake in listing the points or in the coloop check would invalidate the claim. Since the stress is on explicit constructions, the manuscript needs to make those constructions easy to reproduce or verify independently. This is relevant for people following the Lonely Runner Conjecture and lattice geometry. Readers who work on discrete convex geometry or matroids will find the new class and the counterexamples worth looking at. It should go to peer review because the results are concrete enough to check and the organizational contribution stands on its own.

Referee Report

2 major / 1 minor

Summary. The manuscript introduces coloopless lattice zonotopes (containing all primitive zonotopes of width at least two) and cosimple zonotopes, uses them to recast the proofs of linear-exponential bounds from Malikiosis-Schymura-Santos (2025) in more transparent form, and supplies explicit constructions claimed to be counterexamples to the shifted Lonely Runner Conjecture for n=5 and to the Lonely Vector Property for n=12, via covering-radius violations.

Significance. If the explicit constructions are correct, the result is significant: it would disprove both the shifted LRC and the LVP, settling long-standing questions in discrete geometry via direct zonotopal counterexamples rather than indirect arguments. The new class of coloopless zonotopes offers a clean framework that unifies several prior notions and may support further work on covering radii of lattice zonotopes.

major comments (2)

[Counterexamples to the shifted LRC] The section presenting the counterexample to the shifted Lonely Runner Conjecture (n=5): the manuscript asserts an explicit coloopless zonotope whose covering radius strictly exceeds the conjectured bound, yet supplies neither the list of generator vectors nor any computational certificate for the radius computation or the coloopless check; this is load-bearing for the disproof claim.
[Counterexamples to the Lonely Vector Property] The section presenting the counterexample to the Lonely Vector Property (n=12): the claimed coloopless zonotope is said to violate the bound, but again the generator matrix, parallelepiped enumeration details, or verification that no coloops exist are omitted, preventing independent confirmation of the radius violation.

minor comments (1)

[Abstract] Abstract: 'generel' is a typographical error for 'general'; 'statments' should read 'statements'.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight important points regarding the verifiability of our counterexamples, which we will address in the revision. Below we respond to each major comment.

read point-by-point responses

Referee: [Counterexamples to the shifted LRC] The section presenting the counterexample to the shifted Lonely Runner Conjecture (n=5): the manuscript asserts an explicit coloopless zonotope whose covering radius strictly exceeds the conjectured bound, yet supplies neither the list of generator vectors nor any computational certificate for the radius computation or the coloopless check; this is load-bearing for the disproof claim.

Authors: We appreciate the referee's observation. The manuscript does describe the construction, but to ensure full transparency and allow independent verification, we will include the explicit list of generator vectors for the n=5 coloopless zonotope in the revised manuscript. Additionally, we will provide the computational details, including the method for computing the covering radius (via enumeration of the relevant parallelepipeds) and the verification that the zonotope is coloopless (by checking that no generator is a coloop). This will be added as a dedicated subsection or appendix. We believe this will strengthen the presentation of the counterexample. revision: yes
Referee: [Counterexamples to the Lonely Vector Property] The section presenting the counterexample to the Lonely Vector Property (n=12): the claimed coloopless zonotope is said to violate the bound, but again the generator matrix, parallelepiped enumeration details, or verification that no coloops exist are omitted, preventing independent confirmation of the radius violation.

Authors: We agree that providing the generator matrix and verification details is crucial for the n=12 case as well. In the revised version, we will supply the complete generator matrix for the coloopless zonotope, detailed steps or results from the parallelepiped enumeration used to compute the covering radius, and explicit confirmation that it contains no coloops. These additions will enable readers to confirm the violation of the Lonely Vector Property bound. revision: yes

Circularity Check

0 steps flagged

Explicit counterexamples via direct zonotope constructions

full rationale

The paper's main results consist of explicit constructions of coloopless lattice zonotopes in dimensions n=5 and n=12 whose covering radii are computed directly and shown to violate the shifted Lonely Runner Conjecture and Lonely Vector Property bounds. These verifications rely on generator matrices, matroid coloop checks, and enumeration of lattice points in the fundamental parallelepiped, none of which are defined in terms of the conjectured bounds themselves. Prior results from Malikiosis-Schymura-Santos are cited only for context and reformulation; the disproofs do not reduce to any self-citation chain, fitted parameter, or ansatz smuggled from earlier work. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The claims rest on the standard theory of lattice zonotopes and the newly introduced definition of coloopless zonotopes; no numerical parameters are fitted to data.

axioms (1)

standard math Lattice zonotopes are Minkowski sums of line segments with integer direction vectors.
Invoked throughout the zonotopal reformulation of both the original and shifted Lonely Runner Conjectures.

invented entities (1)

Coloopless zonotopes no independent evidence
purpose: A new subclass of lattice zonotopes that contains all primitive zonotopes of width at least two and allows cleaner statements of the covering-radius bounds.
Defined in the paper to generalize the classes used in prior work; no independent existence proof outside the constructions is supplied.

pith-pipeline@v0.9.0 · 5608 in / 1439 out tokens · 67902 ms · 2026-05-15T00:05:21.457550+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We show explicit counterexamples to both the shifted Lonely Runner Conjecture (starting at n=5) and to the Lonely Vector Property (starting at n=12).
IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

coloopless zonotopes … cosimple ones … width at least two … width at least three

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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