A proof of a Dodecahedron conjecture for distance sets

Hiroshi Nozaki; Masashi Shinohara

REVIEW 3 major objections 5 minor 20 references

A proof of a Dodecahedron conjecture for distance sets

T0 review · 3 major / 5 minor · reviewed 2026-08-27 · deepseek-v4-flash

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Pith's one-line read This paper proves the dodecahedron conjecture: every 20-point 5-distance set in R^3 is a regular dodecahedron, so g3(5)=20.

desk verdict Very likely the correct resolution of the dodecahedron conjecture, but the manuscript as written has a load-bearing unproved assertion in Lemma 2.3 and non-reproducible computer searches; it needs revision, not desk rejection. read the letter →

arxiv 2009.13111 v1 pith:FS2AF36R submitted 2020-09-28 math.MG math.CO

Hiroshi Nozaki , Masashi Shinohara This is my paper · ORCID

classification math.MGmath.CO MSC 05D0505B05

keywords s-distancesetsregulardodecahedronmaximumcardinalityfiniteEuclideandistancediametergraphquasi-representablecoloringsoptimalconfigurationsg3(5)=20

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

An $s$-distance set is a finite point set in Euclidean space in which exactly $s$ different distances occur. This paper proves that in three-dimensional space a 5-distance set can have at most 20 points, and that every 20-point example is an exact scaled or rotated copy of the vertices of a regular dodecahedron. The result settles a long-standing open problem and gives both the maximum size and the uniqueness of the extremal configuration. A reader should care because it turns a suspected extremal shape into a proven statement, and because the reduction technique—showing that a large enough set must contain a smaller subset with fewer distances—could apply to related questions.

What carries the argument

Two mechanisms carry the proof. The first is the diameter graph of a point set, whose edges join pairs at the largest distance in the set. A known theorem for diameter graphs in $\mathbb{R}^3$—any two odd cycles must share a vertex—implies that every 20-point set has an independent set of size at least 8; deleting the distances inside that independent set yields an 8-point subset with at most four distances. The second mechanism is a graph-coloring representation of few-distance sets: color the pairs of an 8-point set by their distance, treat the squared distance values as variables, and ask whether the associated matrix can have rank 3 in $\mathbb{R}^3$. The paper calls such colorings quasi-representable, and enriches Szöllősi and Östergård's classification by adding one vertex at a time through a clique-and-loop search. Exhaustive computer enumeration isolates the cube, 116 four-distance subsets of the dodecahedron, and two exceptional non-dodecahedral four-distance sets; the exceptional ones cannot be extended to 20 points, while the dodecahedral subsets force the full dodecahedron.

What would settle it

Directly: exhibit a 20-point 5-distance set in $\mathbb{R}^3$ not similar to the regular dodecahedron, or a 21-point 5-distance set; either would contradict Theorem 1.2. Procedurally: independently re-run the exhaustive enumeration of all quasi-representable 3- and 4-colorings of 8 points and check whether the 63 colorings with a 12-vertex clique are reproduced; any discrepancy would locate the failure of the proof's computational core.

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Extended reading notes

Core claim

The paper's central claim, Theorem 1.2, is that every 20-point 5-distance set in $\mathbb{R}^3$ is isomorphic to the vertex set of a regular dodecahedron, and consequently $g_3(5)=20$, where $g_3(5)$ denotes the maximum possible size of a 5-distance set in $\mathbb{R}^3$. In particular, no 21-point 5-distance set exists in three-dimensional space, and any configuration attaining the maximum is forced to be the dodecahedron. To reach this, the paper proves Theorem 1.3: every 5-distance set in $\mathbb{R}^3$ with at least 20 points contains an 8-point subset that is an $s$-distance set for some $s\le 4$. The classification then reduces to understanding which 8-point 3- or 4-distance sets can sit inside a 20-point 5-distance set; the only such extendable sets are subsets of the dodecahedron itself.

Load-bearing premise

The proof's conclusion depends on the correctness and completeness of the paper's exhaustive computer searches in Section 5; if those searches missed an 8-point configuration that can extend to 20 points, the claimed uniqueness would collapse.

Editorial extensions

If this is right

The maximum cardinality of a 5-distance set in $\mathbb{R}^3$ is exactly 20, so no configuration with 21 or more points using only five distances exists.
Every 20-point 5-distance set in $\mathbb{R}^3$ is similar to the vertex set of a regular dodecahedron, giving a complete uniqueness classification of the extremal case.
Any 5-distance set in $\mathbb{R}^3$ with at least 20 points contains an 8-point subset with at most 4 distances; this reduction is a reusable structural tool for other sizes and dimensions.
The only 8-point 3- or 4-distance sets that can be extended to a 20-point 5-distance set are subsets of the dodecahedron, so the dodecahedron's own distance structure is a forced part of the classification.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same diameter-graph reduction could be applied to $s=6$ in $\mathbb{R}^3$, since the independence-number bound does not depend on the number of distances; the main new difficulty would be classifying 8- or larger subsets with fewer distances.
If future readers want to verify the computational core without trusting the original searches, an independent open enumeration of the 1074 quasi-representable 4-colorings and the 63 colorings with a 12-vertex clique would be a natural check; a discrepancy there would pinpoint exactly where the classification could fail.
The dodecahedron's appearance as the unique extremal shape suggests that other regular polytopes may serve as forced extremal configurations for larger $s$ in higher dimensions, though the paper does not address that case.
The count of 116 dodecahedral 8-point 4-distance subsets is obtained by a separate antipodal-pair argument and Burnside's lemma, so a reader who distrusts the computer search can still verify the dodecahedron-side half of the classification independently.

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Formalized claims in Lean

Claim #1: The paper's central claim, Theorem 1.2, is that every 20-point 5-distance set in $\mathbb{R}^3$ is isomorphic to the vertex set of a regular dodecahedron, and consequently $g_3(5)=20$, where $g_3(5)$ denotes the maximum possible size of a 5-distance set in $\mathbb{R}^3$. In particular, no 21-point 5-distance set exists in three-dimensional space, and any configuration attaining the maximum is for

/-- @claim 1 The paper's central claim, Theorem 1.2, is that every 20-point 5-distance set in $\mathbb{R}^3$ is isomorphic to the vertex set of a regular dodecahedron, and consequently $g_3(5)=20$, where $g_3(5)$ denotes the maximum possible size of a 5-distance set in $\mathbb{R}^3$. In particular, no 21-point 5-distance set exists in three-dimensional space, and any configuration attaining the maximum is for -/
def central_claim : Prop :=

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The paper convinces me that the dodecahedron conjecture is true, but not that this write-up proves it as it stands. Theorem 1.2 — g3(5)=20 with the dodecahedron as the unique extremal configuration — is the real thing: it resolves a long-standing open problem listed in Croft-Falconer-Guy, and it is not a routine parameter sweep. The proof genuinely combines Dol'nikov's diameter-graph theorem with the Szollosi-Ostergard coloring classification, and adds original independence-number arguments. Section 3 is particularly clean; the f(n) lemmas are self-contained and a pleasure to read.

The soft spots are real and load-bearing. Lemma 2.3 asserts without proof that an 8-point 4-distance subset of a regular dodecahedron contains no antipodal pair. The cube, which is an 8-point 3-distance subset of the dodecahedron and does contain antipodal pairs, shows why this needs an argument: containing an antipodal pair does not automatically destroy the 4-distance property, it merely rules out some cases. If there are 4-distance subsets with an antipodal pair, the Burnside count in Lemma 2.3 counts the wrong family, and the key identification in the last paragraph of Section 5 — that the remaining 116 realizations are dodecahedron subsets — loses its ground. This is not a cosmetic gap; it is the hinge between the geometric enumeration and the algebraic classification.

The second weakness is reproducibility. Lemmas 5.3 and 5.4 and the candidate enumeration are described as Magma/Maple calculations, but no code, logs, or certificates are provided. For a computer-assisted proof of this importance, a serious referee needs to be able to check the programs. The external Szollosi-Ostergard classification is published, so that part is fine; the problem is the paper's own searches.

None of this makes me doubt the theorem. The proof strategy is sound, the graph theory is rigorous, and the computational results look plausible. But as submitted, the paper is not fully verifiable. A serious referee should be sent this; it deserves a round of revision asking for a proof of Lemma 2.3's antipodal assertion and for the computational artifacts. For a specialist in distance sets, this is an important paper to engage with. I would cite it once the details are pinned down.

Thanks.

Referee Report

3 major / 5 minor

Summary. The manuscript proves Theorem 1.2: every 20-point 5-distance set in R^3 is similar to the vertex set of a regular dodecahedron, so g_3(5)=20. The proof first establishes Theorem 1.3, which states that any such set contains an 8-point subset with at most 4 distances, using Dol'nikov's theorem on diameter graphs and a new independence-number bound f(n). It then classifies all 8-point 3-/4-distance sets in R^3 that can be extended to a 5-distance set with at least 20 points, using quasi-representable colorings, a graph G(C) encoding possible extensions, and extensive computer searches in Magma/Maple. The classification yields the dodecahedron subsets as the only extendable configurations, which gives the theorem.

Significance. If the proof is valid, the result resolves a long-standing open problem and gives both the exact maximum cardinality and the uniqueness of the extremal configuration. The graph-theoretic part in Section 3 is elegant and nearly self-contained, and the use of coloring matrices and quasi-representability is a powerful method. However, the paper's central classification depends on two external pillars: the unproved antipodal-pair assertion in Lemma 2.3 and the exhaustive computer searches in Section 5, which are not accompanied by code or certificates. These gaps need to be addressed before the result can be fully verified.

major comments (3)

[Section 2, Lemma 2.3] The lemma begins with the assertion that an 8-point 4-distance subset of a regular dodecahedron contains no antipodal pair, 'otherwise X is not 4-distance.' No proof is given, and the claim is not evident: the inscribed cube is a 3-distance 8-point subset that does contain antipodal pairs, so containing an antipodal pair does not by itself force five distances. The rest of the lemma (choosing one vertex from each of 8 antipodal pairs, and the Burnside count) counts the intended configuration only if this assertion is true. Since the final paragraph of Section 5 explicitly invokes Lemma 2.3 to conclude that 116 of the 118 realizable 4-distance sets are dodecahedron subsets, this gap is load-bearing for Theorem 1.2. The same lemma also asserts without proof that removing any two antipodal pairs cannot break all 12 faces of the dodecahedron; please provide a rigorous proof or a verifiable computer verification for both claims.
[Section 5, Lemmas 5.3 and 5.4 and the enumeration after (5.1)] The exhaustive computer searches in this section are described only as 'By exhaustive computer search' with Magma/Maple, with no code, input data, logs, certificates, or a detailed description of the search space. These searches are the sole basis for: (i) the classification of loops in G(C) (Lemma 5.3), (ii) the claim that exactly 63 colorings in CG(8,4) have omega*(C) >= 12 with a unique clique of order 12 (Lemma 5.4), and (iii) the enumeration of candidate vectors (a_1,...,a_8) satisfying (5.1) that underlies the definition of G(C). Because Theorem 5.1 and hence Theorem 1.2 rest directly on these computations, the manuscript should include the complete code and enough documentation to reproduce the searches, or replace them with independently verifiable certificates or mathematical proofs.
[Section 4, Proposition 4.3] Proposition 4.3 is introduced as 'essentially proved' in [19], but the paper modifies the statement to use only principal submatrices and the determinant condition in (4.2) without proving the modification. Since this proposition is the criterion used to generate all candidate colorings in Section 5, a complete proof (or a precise statement of the exact result in [19] that covers this version) is needed. As written, the reader cannot tell whether the weak quasi-representability condition has been correctly translated into the system (4.2).

minor comments (5)

[Section 2, Lemma 2.3] In the Burnside count, the identity transformation is said to fix (10 choose 2) * 2^8 = 11520 sets; the binomial should be (10 choose 8), although the values are numerically equal.
[Lemma 3.7] The proof says the inequalities f(8)=4, f(10)=5, f(13)=6, and f(16)=7 'can be proved by a similar way' but does not give the details; since these values are used in Lemmas 3.8 and 3.9, please include the finite case checks or explicit certificates.
[Lemma 3.8] The chain 'f(16)+f(4)>=f(15)+f(3)=6+2=8' uses f(15)=6 without stating it; the inequality is justified by f(15)>=f(13)=6, but the equality as written is misleading.
[Section 5, Figure 3] Figure 3, which contains the realizations of C1 and the candidate extension points, is referenced in Section 5 but does not appear in the arXiv version of the manuscript; please ensure the figure is included.
[Section 2, notation] The notation d(x,y)=d_i for the dodecahedron graph uses d_i both for geometric distances and for graph distance; this minor clash could confuse readers, and the sentence introducing it would benefit from explicit indexing.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the dodecahedron conclusion is derived, not assumed; the load-bearing external inputs do not contain the target theorem.

full rationale

The proof chain is: (1) Dol'nikov's diameter-graph theorem gives alpha(DG(X)) >= ceil((n-m)/2) when an odd cycle of length m exists; elementary f(n) bounds then show alpha(DG(X)) >= 8 for any 20-point X in R^3, so every such X has an 8-point subset with at most 4 distances (Theorem 1.3). (2) The paper then uses Szollosi-Ostergard's independent classification of quasi-representable colorings [19] to enumerate all 8-point 3- and 4-distance colorings C that can be extended by 12 vertices to a weakly quasi-representable 5-coloring (Lemma 5.4). No condition in this enumeration asks for a dodecahedron or for the target theorem; the dodecahedral sets emerge only after solving the realization systems and matching the independent count of dodecahedron 4-subsets obtained by Burnside in Lemma 2.3. (3) The final identification that the remaining 116 4-distance sets should be subsets of the dodecahedron is a counting argument from two independently computed lists, not a definitional identification or a fitted parameter. The paper's self-citations [17,18] appear only in background Theorem 1.1 and are not invoked in the proofs of Theorems 1.2, 1.3, or 5.1. The legitimate concerns are correctness and reproducibility issues, not circularity: Lemma 2.3's first assertion that an 8-point 4-distance subset cannot contain an antipodal pair is stated without proof, and the Section 5 computer searches are described as Magma/Maple runs without scripts or certificates. These would be gaps if the enumeration is wrong, but they do not make the conclusion equivalent to the input.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central theorem rests on independent geometric theorems (Dol'nikov, Gram-matrix characterization), an external computational classification from [19], and the paper's own unverified computer searches. There are no fitted numerical parameters and no invented geometric entities. The main burden is the completeness of the Magma/Maple searches, which the paper treats as axioms by assertion.

assumptions (5)

standard math Dol'nikov's theorem: diameter graphs of sets in R^3 cannot contain two vertex-disjoint odd cycles.
Invoked as Theorem 3.3 and used to derive Corollary 3.4 and the independence-number lower bounds in Section 3.
standard math Positive semidefinite and Gram matrix characterization of Euclidean representability (Theorem 4.1).
Used to define representability, quasi-representability, and the rank conditions in Proposition 4.3.
domain assumption Szollosi-Ostergard classification: exactly 19 quasi-representable 3-colorings and 1074 quasi-representable 4-colorings of 8 points in R^3 (Lemma 5.2).
Borrowed from [19]; no code or certificate is provided here, and the subsequent clique search is filtered through this classification.
ad hoc to paper Completeness of the paper's exhaustive computer searches (Lemmas 5.3, 5.4, and the candidate enumeration after (5.1)).
The lemmas are asserted from Magma/Maple runs without scripts, logs, or certificates; the final classification depends on these searches being exhaustive and correct.
standard math Distance-regular graph theory and the eigenmatrix of the dodecahedron graph (Section 2).
Used for Lemma 2.1 and the analysis of the dodecahedron's distance matrices and spherical representations.

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Pith. "Pith review of A proof of a Dodecahedron conjecture for distance sets." pith.science (2026). https://pith.science/paper/FS2AF36R

@misc{pith2026200913111,
  author       = {Pith},
  title        = {Pith review of: A proof of a Dodecahedron conjecture for distance sets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FS2AF36R}},
  note         = {Machine review of arXiv:2009.13111}
}

abstract

A finite subset of a Euclidean space is called an $s$-distance set if there exist exactly $s$ values of the Euclidean distances between two distinct points in the set. In this paper, we prove that the maximum cardinality among all 5-distance sets in $\mathbb{R}^3$ is 20, and every $5$-distance set in $\mathbb{R}^3$ with $20$ points is similar to the vertex set of a regular dodecahedron.

Figures

Figures reproduced from arXiv: 2009.13111 by the authors.

**Figure 1.** Maximal planar 4-distance sets For a 2-distance set X, we consider the graph on X where two vertices are adjacent if they have the smallest distance in X. We can construct the 2-distance set that has the structure of a given graph [8]. Lisonˇek [13] gave an algorithm for a stepwise augmentation of representable graphs (adding one vertex per iteration), and classified the optimal 2-distance sets in R d for d ≤ 7 by a… view at source ↗

**Figure 2.** Dodecahedron graph X and Y are in the same sphere, there exists an isometry σ in the orthogonal group O(3) such that Xσ = Y . This implies that (±X) σ = ±Y , namely the set of 8 antipodal pairs of ±X are isomorphic to that of ±Y . Since a regular dodecahedron in a given sphere is uniquely determined after one face is fixed, if each set of 8 antipodal pairs makes a face of the dodecahedron, then σ becomes an isometry… view at source ↗

**Figure 3.** the 8-point subsets which are realizations of [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

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Reference graph

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Pith tools

Reviewed August 27, 2026 · model on record in the stance chip above.

[1] [19]

Sz¨ oll˝ osi and P.R.J.¨Osterg˚ ard, Constructions of maximum few-distance sets in Eu- clidean spaces, Electron

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work page 2020

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Bannai, E

E. Bannai, E. Bannai, and D. Stanton, An upper bound for the cardinality of an s- distance subset in real Euclidean space, II, Combinatorica 3 (1983), 147–152

work page 1983

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Blokhuis, Few-distance sets, CWI Tract, 7 (1984), 1–70

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Bosma, J

W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user lan- guage, J. Symbolic Comput. 24 (1997), 235–265

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Brouwer, A.M

A.E. Brouwer, A.M. Cohen, and A. Neumaier, Distance-regular Graphs, Springer-Verlag, Berlin, (1989)

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H. T. Croft, K. J. Falconer, and R. K. Guy, Unsolved Problems in Geometry, Problem Books in Mathematics , Springer, New York (1994)

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E.R. van Dam, J.H. Koolen, and H. Tanaka, Distance-regular graphs, Electron. J. Comb. (2016), #DS22

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V. L. Dol’nikov, Some properties of graphs of diameters. The Branko Grnbaum birthday issue. Discrete Comput. Geom. 24 (2000), 293299

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