Pith. sign in

REVIEW 3 major objections 4 minor 30 references

The Tammes Problem in $\mathbb{R}^{n}$ and Linear Programming Method

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Polynomial certificates fix exact Tammes distances

desk verdict Sound but mostly a repackaging of the Delsarte bound; the 600-cell certificate is the real content and it is not yet auditable. read the letter →

arxiv 2411.16038 v1 pith:X5Z5P2HK submitted 2024-11-25 math.MG

Yanlu Lian , Qun Mo , Yu Xia This is my paper · ORCID
classification math.MG MSC 52C1790C05
keywords TammesproblemsphericalcodeslinearprogrammingboundGegenbauerpolynomialsoptimalconfigurationscrosspolytopeicosahedron600-cell
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

This paper gives a sufficient condition, in the linear programming framework for spherical codes, for a point configuration on the unit sphere to solve the Tammes problem, which asks how to place $N$ points so the smallest pairwise distance is as large as possible. The condition asks for two polynomials $f$ and $g$ with nonnegative Gegenbauer coefficients: $f$ must be nonpositive on the interval of allowed inner products and must attain $N = f^{\#}=f(1)/c_0$, while $g$ must stay below $N$ on a smaller interval. If such certificates exist, Theorem 3 concludes $d_{n,N}=d_C$. The authors verify these certificates for the cross-polytope in any dimension, the icosahedron with $N=12$ on $\mathbb{S}^2$, and the 600-cell with $N=120$ on $\mathbb{S}^3$, recovering exact Tammes values in each case.

What carries the argument

The central object is the class $\mathcal{P}(k,\tau,n)$: polynomials of degree at most $k$ whose expansion in Gegenbauer polynomials $P_i^{(n)}$ has nonnegative coefficients ($c_0>0$ and $c_i\ge 0$) and which are nonpositive on $[-1,\tau]$. The derived quantity $f^{\#}=f(1)/c_0$ serves as the linear-programming upper bound on the size of a spherical code. Lemma 2 uses the positive definiteness of Gegenbauer kernels to prove $N\le f^{\#}$ for any configuration, with equality forcing $f$ to vanish on every pairwise inner product. Theorem 3 then layers on the second polynomial $g$ and the zero-free interval $(t_2,t_C)$, converting the pair $(f,g)$ into a certificate that the configuration solves the Tammes problem.

What would settle it

Expand the displayed $f$ and $g$ in the Gegenbauer basis and check every coefficient for nonnegativity in exact arithmetic, or evaluate $f(t)$ on $[-1,(\sqrt{5}+1)/4]$ at high precision and test the identity $f(1)=120\,c_0$; a single failed coefficient sign, a single positive value on the interval, or a mismatch in $f^{\#}$ would refute the 600-cell example.

Watch

Extended reading notes

Core claim

The central claim is Theorem 3: a configuration $C\subset\mathbb{S}^{n-1}$ is optimal for the Tammes problem if one can exhibit $f\in\mathcal{P}(k_1,t_C,n)$ with $N=f^{\#}$, a threshold $t_2<t_C$ on which $f$ has no zeros in $(t_2,t_C)$, and a second function $g\in\mathcal{P}(K_2,t_2,n)$ with $g^{\#}<N$. The proof runs a two-case argument against any competing configuration $S$: if $S$'s maximal inner product is at most $t_2$, the Delsarte-type bound forces $N\le g^{\#}$, contradicting condition (iii); if it is above $t_2$, then $f$ is nonzero at the maximizing pair, yet the equality case of the bound forces $f$ to vanish on all pairs. No competitor can beat $d_C$. The paper supplies explicit $f$ and $g$ for the cross-polytope, for the icosahedron, and for the 600-cell, concluding $d_{n,2n}=\sqrt{2}$, $d_{3,12}=4/\sqrt{10+2\sqrt{5}}$, and $d_{4,120}=(\sqrt{5}-1)/2$.

Load-bearing premise

For the 600-cell example, the load-bearing premise is that the displayed high-degree polynomials $f$ and $g$ have all nonnegative Gegenbauer coefficients, are nonpositive on the required intervals, and satisfy the exact identities $f^{\#}=120$ and $120>g^{\#}$; these identities are asserted by direct calculation without a demonstrated proof certificate.

Editorial extensions

If this is right

  • The cross-polytope realizes $d_{n,2n}=\sqrt{2}$ for every $n\ge 2$, certified by $f(t)=t(t+1)$ and $g(t)=t+1$.
  • The icosahedron realizes $d_{3,12}=4/\sqrt{10+2\sqrt{5}}$, certified by a degree-4 polynomial $f$ and a degree-2 polynomial $g$.
  • The 600-cell realizes $d_{4,120}=(\sqrt{5}-1)/2$, certified by a degree-17 polynomial $f$ and a degree-13 polynomial $g$.
  • For any $n$ and $N$, a candidate configuration together with certificates $(f,g)$ satisfying Theorem 3 suffices to prove global optimality, so the Tammes problem reduces to an explicit polynomial-construction task.
  • When $N=f^{\#}$, Lemma 2 forces the certificate $f$ to vanish on every pairwise inner product, so a valid certificate encodes the full distance distribution of the optimal configuration.

Reading between the lines

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

  • A natural next step, not taken by the paper, is to search computationally for certificates: if a numerical optimizer can find $f$ and $g$ of moderate degree satisfying Theorem 3, that would settle the Tammes value for $N$ where only heuristic packings are known.
  • The 600-cell example suggests that highly symmetric regular polytopes may come with low-degree certificates; testing whether the 24-cell or other regular polytopes admit similar pairs would extend the pattern.
  • Because Theorem 3 gives only a sufficient condition, a failed search for certificates proves nothing; computational experiments that cannot produce such a pair should not be read as evidence against optimality.
  • The equality condition in Lemma 2 yields a cheap numerical check: evaluate the candidate $f$ on the configuration's pairwise inner products; any nonzero value would immediately disprove the claimed certificate.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper restates the Delsarte linear programming bound for spherical codes as a sufficient condition (Theorem 3) for a configuration C to solve the Tammes problem. The theorem is then applied to three families: the n-dimensional cross-polytope (Example 5), the icosahedron (Example 6), and the 600-cell (Example 7). The proof of Theorem 3 is a standard and correct application of the Delsarte–Goethals–Seidel method: equality in Lemma 2 forces f to vanish on all pairwise inner products of a competing configuration, and conditions (ii)–(iii) rule out any competitor with larger minimum distance. Examples 5 and 6 are elementary and essentially correct, though Example 6 contains a miscalculated value of g#. Example 7 is the main claimed new application; it rests on high-degree polynomial identities in Q(√5) that are asserted without derivation, code, or a machine-checkable certificate.

Significance. Theorem 3 is a correct but not conceptually novel restatement of the Delsarte LP bound; its contribution is the construction of auxiliary polynomials for particular N. If Example 7 were fully verified, the paper would provide a self-contained certificate for the known optimality of the 600-cell among 120-point configurations in S^3. However, the absence of exact-arithmetic verification for the 600-cell certificate means the headline value d_{4,120} = (√5−1)/2 is not established by the text as written. The paper ships no scripts, worksheets, or proof certificates, which is a meaningful deficiency for a paper whose central claims are computational identities.

major comments (3)
  1. [Section 4, Example 7] The claims that the product form of f equals the displayed degree-17 Gegenbauer expansion, that all Gegenbauer coefficients are nonnegative, that f(1)/c0 = 120 exactly, that g ∈ P(13, 1/2, 4), and that g# < 120 exactly are supported only by 'direct computation' and Remark 8's reference to Maple. No worksheet, script, log, or formal certificate is provided. Because one sign error or one omitted term invalidates the example, these identities are load-bearing. Please supply a complete exact-arithmetic verification over Q(√5) (e.g., a Maple workbook or a Mathematica notebook with exact coefficients and interval checks) and state clearly which of the asserted identities are machine-checked.
  2. [Section 4, Example 7] The printed Gegenbauer expansion of f omits the coefficients of P13 and P12; as printed, the display jumps from P14 to P11. If these coefficients are zero, this should be stated; if the display is incomplete, the identity is not verifiable. Similar care is needed for the list of coefficients of g, which appears complete.
  3. [Section 4, Example 6] The printed value g# = 12/(5√5) is incorrect; direct calculation from the displayed expansion gives g# = 3(√5−1) ≈ 3.708. The inequality 12 > g# remains true with the corrected value, so Example 6 is not invalidated, but the numerical error illustrates that the algebraic details in the paper need independent verification.
minor comments (4)
  1. [Section 4, Example 5] It would help to state explicitly that c0 = 1/n for f(t) = t(t+1), since the computation f# = 2n jumps from the definition.
  2. [General] There are several spacing and OCR-style typos, e.g., 'po ints' in the abstract and 'setz' in Example 7; a careful proofread is needed.
  3. [Section 2] In the Gegenbauer recurrence, the notation P^{(n)}_k(t) is used inconsistently; align the notation across Definition 1, Lemma 2, and the examples.
  4. [References] Reference [5] (Boyvalenkov and Danev) is listed but never cited in the body; since it is directly relevant to the 600-cell example, cite it where appropriate or remove it.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 3 is an independent LP sufficiency certificate, and the examples are algebraic verifications of that certificate, not predictions forced by construction.

full rationale

The paper's central derivation is a standard Delsarte-type linear programming sufficiency argument. Lemma 2 proves N <= f# from the Gegenbauer expansion's positive definiteness and the sign condition f(t) <= 0 on [-1, t_C]; this proof is self-contained and does not assume any conclusion about the Tammes value. Theorem 3 then combines Lemma 2 with two additional conditions: f# = N makes the bound tight on the candidate configuration, condition (ii) rules out competing configurations with t_S in (t2, t_C), and condition (iii) rules out t_S <= t2. The contradiction argument in both cases is internally valid and does not presuppose d_{n,N} = d_C. The auxiliary functions f and g in Examples 5-7 are constructed for each candidate configuration, but that is how an LP certificate works: existence of such functions is a verification, not a circular assumption. In particular, f# = N is a computed equality from the Gegenbauer expansion, not a parameter fitted to data and then reported as a prediction. The only substantive weakness is that in Example 7 the high-degree polynomial identities, coefficient nonnegativity, and the exact inequalities f# = 120 and 120 > g# are asserted rather than accompanied by machine-checkable certificates or derivation scripts; Remark 8 merely says Maple can do it. That is a reproducibility and correctness-risk concern, not circularity. The paper cites no prior work of the present authors, and no load-bearing argument reduces to a self-citation. The theorem and examples do not rename a known empirical pattern, and no normalization or definitional identity smuggles the conclusion into the hypotheses. Therefore no circular step is present.

Assumptions & free parameters 5 free parameters · 2 assumptions · 0 invented entities

The central theorem depends only on standard Delsarte theory and the constructed auxiliary polynomials. The free parameters are the certificate polynomials and threshold t2, which are hand/computer chosen to satisfy the conditions. No new entities are introduced.

free parameters (5)
  • t2 in Theorem 3 = -sqrt(5)/5 (Example 6), 1/2 (Example 7)
    Chosen as a zero of f below tC so that condition (ii) holds and g can rule out configurations with tS <= t2.
  • Coefficients of f in Example 6 = roots -1, -sqrt(5)/5 (double), sqrt(5)/5; c0 = (10+2sqrt(5))/75
    Constructed so that f# = 12 and f <= 0 on [-1, tC].
  • Coefficients of g in Example 6 = c0 = (5+3sqrt(5))/15
    Constructed so that N > g#.
  • Coefficients of f in Example 7 = degree-17 algebraic coefficients, c0 = 36360 + 27840sqrt(5)
    Constructed so that f# = 120 and f <= 0 on [-1, (1+sqrt(5))/4].
  • Coefficients of g in Example 7 = degree-13 algebraic coefficients, c0 = 32064896 + 14094016sqrt(5)
    Constructed so that g# < 120 and g <= 0 on [-1, 1/2].
assumptions (2)
  • standard math Positive definiteness of Gegenbauer polynomials: sum_{i,j} c_l P_l(x_i.x_j) >= 0 for c_l >= 0
    Invoked in the proof of Lemma 2 via [20, Lemma 1] and [29, Page 126].
  • ad hoc to paper The displayed Gegenbauer expansions of f and g in Example 7 satisfy the stated coefficient sign and interval sign conditions
    Load-bearing for the 600-cell example; asserted by direct calculation/Maple without derivation or code.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The Tammes Problem in $\mathbb{R}^{n}$ and Linear Programming Method." pith.science (2026). https://pith.science/paper/X5Z5P2HK

@misc{pith2026241116038,
  author       = {Pith},
  title        = {Pith review of: The Tammes Problem in $\mathbbR^n$ and Linear Programming Method},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X5Z5P2HK}},
  note         = {Machine review of arXiv:2411.16038}
}
abstract

The Tammes problem delves into the optimal arrangement of $N$ points on the surface of the $n$-dimensional unit sphere (denoted as $\mathbb{S}^{n-1}$), aiming to maximize the minimum distance between any two points. In this paper, we articulate the sufficient conditions requisite for attaining the optimal value of the Tammes problem for arbitrary $n, N \in \mathbb{N}^{+}$, employing the linear programming framework pioneered by Delsarte et al. Furthermore, we showcase several illustrative examples across various dimensions $n$ and select values of $N$ that yield optimal configurations. The findings illuminate the intricate structure of optimal point distributions on spheres, thereby enriching the existing body of research in this domain.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

30 extracted references · 29 canonical work pages

  1. [1]

    Ambrus, I

    In: G. Ambrus, I. Bárány, K. Böröczky, G. Fejes Töth, J. Pa ch, New Trends in Intuitive Geometry. Bolyai Society Mathematical Studies, vol 27. Springer, Berlin, He idelberg, (2018)

    work page 2018

  2. [2]

    N. N. Andreev, A spherical code, Russian Math. Surveys, 5 4 (1999), 251-253

    work page 1999

  3. [3]

    Böröczky, The problem of Tammes for n = 11, Studia Sci

    K. Böröczky, The problem of Tammes for n = 11, Studia Sci. Math. Hungar. 18 (1983), 165–171

    work page 1983

  4. [4]

    Boyvalenkov, S

    P. Boyvalenkov, S. Dodunekov and O. R. Musin, A survey on t he kissing numbers, Serdica Math. J., 38 (2012), 507-522

    work page 2012

  5. [5]

    Boyvalenkov and D

    P. Boyvalenkov and D. Danev, Uniqueness of the 120-point spherical 11-design in four dimensions, Arch. Math., 77 (4) (2001), 360-368

    work page 2001

  6. [6]

    Brass, W

    P. Brass, W. Moser and J. Pach, Research Problems in discr ete Geometry, New York: Springer-Verlag, 2005

    work page 2005

  7. [7]

    J. H. Conway and N. J. A. Sloane, Sphere packings, lattice s and groups, New York, Springer-Verlag, (1999)

    work page 1999

  8. [8]

    Cohn, Table of spherical codes, https://dspace.mit

    H. Cohn, Table of spherical codes, https://dspace.mit. edu/handle/1721.1/153543

    work page

Show all 30 references
  1. [9]

    Danzer, Finite point sets on S2 with minimum distance as large as possible, Discrete Math

    L. Danzer, Finite point sets on S2 with minimum distance as large as possible, Discrete Math. 6 0 (1986), 3-66

    1986

  2. [10]

    Delsarte, J

    P. Delsarte, J. M. Goethals and J. J. Seidel, Spherical c odes and designs, Geom. Dedic., 6 (1977), 363-388

    1977

  3. [11]

    Delsarte, Bounds for unrestricted codes by linear pr ogramming, Philips Res

    P. Delsarte, Bounds for unrestricted codes by linear pr ogramming, Philips Res. Rep., 27 (3) (1972), 272-289

    1972

  4. [12]

    G. A. Kabatyanskii and V. I. Levenshtein, On bounds for p ackings on a sphere and in space. Probl. Peredaci. Inform. 14, 1 (1978), 3-25 (in Russian); English translatio n in: Probl. Inform. Transm. 14, 1 (1978), 1-17

    1978

  5. [13]

    X. Lai, D. Yue, J. K. Hao, F. Glover, and Z. Lu, Iterated dy namic neighborhood search for packing equal circles on a sphere, Comput. oper. res, 151, (2023)

    2023

  6. [14]

    Mackay, J

    A. Mackay, J. Finney and K. Gotoh, The closest packing of equal spheres on a spherical surface, Acta Crys- tallogr. Sect. A, 33(1), (1977), 98-100

    1977

  7. [15]

    O. R. Musin, Towards a proof of the 24-cell conjecture, A cta Math Hungar., 155 (1) (2018), 184-199

    2018

  8. [16]

    O. R. Musin, Five essays on the geometry of László Fejes T óth, in New Trends in Intuitive Geometry, 321-333, Bolyai Soc. Math. Stud., 27, János Bolyai Math. Soc., Budape st, (2018)

    2018

  9. [17]

    O. R. Musin and A. S. Tarasov, The Tammes problem for N = 14, Exp. Math. 24 (4) (2015), 460-468

    2015

  10. [18]

    O. R. Musin and A. S. Tarasov, The strong thirteen sphere s problem, Discrete Comput. Geom. 48 (2012), 128-141

    2012

  11. [19]

    O. R. Musin, The kissing number in four dimensions, Ann. Math.,168 (1) (2008), 1-32

    2008

  12. [20]

    O. R. Musin, The kissing problem in three dimensions, Di screte Comput. Geom., 35 (3) (2006), 375-384

    2006

  13. [21]

    A. M. Odlyzko and N. J. A. Sloane, New bounds on the number of unit spheres that can touch a unit sphere in n dimensions, J. Comb. Theory, Ser. A, 26 (2) (1979), 210-214. 8

    1979

  14. [22]

    Pfender, Improved Delsarte bounds for spherical cod es in small dimensions, J

    F. Pfender, Improved Delsarte bounds for spherical cod es in small dimensions, J. Comb. Theory, Ser. A, 114 (6) (2007), 1133-1147

    2007

  15. [23]

    Pfender and G

    F. Pfender and G. M. Ziegler, Kissing numbers, sphere pa ckings, and some unexpected proofs, Notices Amer. Math. Soc., 51 (8) (2004), 873-883

    2004

  16. [24]

    R. M. Robinson, Arrangement of 24 circles on a sphere, Ma th. Ann., 144 (1961), 17-48

    1961

  17. [25]

    Schütte and B

    K. Schütte and B. L. van der Waerden, Auf welcher Kugel ha ben 5, 6, 7, 8 oder 9 Punkte mit Mindestabstand 1Platz?, Math. Ann. 123 (1) (1951), 96-124

    1951

  18. [26]

    R. M. L. Tammes, On the origin number and arrangement of t he places of exits on the surface of pollengrains, Rec. Trv. Bot. Neerl. 27 (1930), 1-84

    1930

  19. [27]

    Hárs, The Tammes problem for n = 10, Studia Sci

    L. Hárs, The Tammes problem for n = 10, Studia Sci. Math. Hungar. 21/3-4 (1986), 439-451

    1986

  20. [28]

    Zong, The mathematical foundation of post-quantum c ryptography, https:// arxiv.org/pdf/2404.19186

    C. Zong, The mathematical foundation of post-quantum c ryptography, https:// arxiv.org/pdf/2404.19186

    arXiv

  21. [29]

    Zong, Sphere packings, Springer-Verlag, New York, ( 1999)

    C. Zong, Sphere packings, Springer-Verlag, New York, ( 1999)

    1999

  22. [30]

    Zong, The kissing numbers of convex bodies-a brief su rvey, Bull

    C. Zong, The kissing numbers of convex bodies-a brief su rvey, Bull. London Math. Soc., 30 (1) (1998), 1-10. 9

    1998

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.