An improved lower bound for the logarithmic energy on $\mathbb S^2$

Jordi Marzo

REVIEW 3 major objections 4 minor 28 references

An improved lower bound for the logarithmic energy on $\mathbb S^2$

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read The paper proves a new lower bound for the constant $C_{\log}$ in the asymptotic expansion of the minimal logarithmic energy on $\mathbb S^2$, replacing the previous lower bound $\log 2 - \tfrac34$ with $\tilde C = -0.0568456\ldots$ and…

desk verdict A new and honest idea for the smeared-energy lower bound, but the headline constant is not proven as written: the triangle integral ignores the truncation in Φ where the cell radius is below ε/√N. read the letter →

arxiv 2506.01660 v1 pith:B7FHRADX submitted 2025-06-02 math.CA

Jordi Marzo This is my paper

classification math.CA MSC 31C1241A60

keywords logarithmicenergyspherelowerboundasymptoticexpansionWassersteindistancepotentialtheoryoptimaltransportsphericalcaps

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

The paper tightens the known numerical range of the constant $C_{\log}$ in the asymptotic expansion of the minimal $N$-point logarithmic energy on the unit sphere: $E_N = I(\sigma)N^2 - \tfrac12 N\log N + C_{\log} N + o(N)$. The previous best lower bound was $\log 2 - \tfrac34 \approx -0.0568528$; this note proves a smaller one, $\tilde C \approx -0.0568456$, leaving only a narrow window down to the conjectured upper bound $-0.0556053$. The improvement matters because the exact value of $C_{\log}$ is expected to equal the upper bound, and any tightening of the lower side constrains how far optimal point configurations can deviate from the conjectured picture.

What carries the argument

The carrying object is the signed smeared measure $\mu = \frac1N\sum_{i=1}^N \mu_i - d\sigma$, with $\mu_i$ the normalized restriction of surface measure to the cap $B(x_i,\epsilon/\sqrt N)$. Its energy $I(\mu)$ appears in the expansion as a nonnegative remainder; the paper's mechanism is the inequality $W_1(\frac1N\sum_j \frac{\chi_{B_j}}{\sigma(B_j)}\,d\sigma, d\sigma)^2 \le 2I(\mu)$, which makes the Wasserstein-1 distance between the smeared empirical measure and uniform measure a lower bound for that remainder. Duality to Lipschitz functions, the distance-to-union-of-caps test function, and a lower bound on the spherical integral of that distance over an equilateral triangle turn the inequality into an explicit $v(\epsilon)/N$ term, whose coefficient function $u(\epsilon)+v(\epsilon)$ is maximized at $\epsilon=2$.

What would settle it

Pick a small symmetric configuration where both sides can be computed explicitly, for example two antipodal smeared caps, and check whether $W_1(\mu+d\sigma,d\sigma)^2 \le 2I(\mu)$ holds under the paper's normalization of $G(x,y)=\log(1/|x-y|)$ and geodesic distance; if the ratio exceeds 1, the constant 2 is wrong and $\tilde C$ would have to be recomputed with the corrected constant.

Watch

Extended reading notes

Core claim

The central theorem asserts that $C_{\log}$ exists and satisfies $\tilde C \le C_{\log} \le C_{BHS}$, where $$\tilde C = \log 2 - \frac34 + \frac{1}{162}\left(\frac{4\sqrt{3}}{\sqrt{2\pi}}\left(2+3\$tanh^{{-1}}$(1/2)\right)-12\right)^2 = -0.0568456\ldots,$$ and $C_{BHS}=-0.0556053\ldots$ is the previously known upper bound conjectured to be the true value. The proof starts from the standard energy decomposition for point masses smeared over caps of radius $\epsilon/\sqrt N$. Its new step is to replace the crude estimate $I(\mu)\ge 0$ by a quantitative lower bound: a Wasserstein-1 comparison imported from Coulomb-gas theory gives $W_1^2 \le 2I(\mu)$, Kantorovich–Rubinstein duality turns $W_1$ into the integral of a distance-to-caps function, and a spherical covering theorem bounds that integral from below by an explicit integral over an equilateral spherical triangle. Evaluating the resulting contribution and optimizing the cap radius at $\epsilon=2$ yields the improved constant.

Load-bearing premise

The load-bearing premise is that the Wasserstein-energy inequality imported from the literature, $W_1^2 \le 2I(\mu)$, remains correct with the same constant 2 under this paper's normalization of the Green function and the sphere; the paper notes a change of normalization but gives no conversion.

Editorial extensions

If this is right

The interval of possible values of $C_{\log}$ shrinks: $C_{\log}$ is known to lie between $-0.0568456\ldots$ and $-0.0556053\ldots$, roughly a quarter the previous width.
The conjecture that $C_{\log}$ equals the upper bound remains consistent, and any counterexample would have to fit the remaining $0.00024$-wide window.
The new quantitative lower bound $I(\mu)\ge v(\epsilon)/N$ makes precise, for every configuration, that the smeared-energy term is of order $N^{-1}$, exactly the order that affects the linear-term constant.
Because the estimate applies to all configurations, not only minimizers, the same mechanism can control the energy gap of near-optimal configurations, which is the quantity relevant to constructing optimal point sets.

Reading between the lines

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

The numerical value of $\tilde C$ hinges on the constant 2 in the imported Wasserstein-energy inequality, so verifying the normalization conversion is a concrete check before relying on the displayed digits.
A promising extension, flagged by the paper itself, is to replace Wasserstein-1 with Wasserstein-2; computing the asymptotic value of the smeared quantization problem would likely yield a further lower-bound improvement of the same type.
The same three-ingredient scheme (smearing, Wasserstein duality, equilateral-triangle covering bound) should generalize to logarithmic energy on $\mathbb S^n$ or to Riesz-type kernels, where the existing gaps between lower and upper bounds are larger.

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.

Desk Editor's Note

read the letter

Short version: this note has a real idea and an honest small-step result, but the numerical claim as written is not supported. The truncation in Φ(s) = max{s−ε/√N, 0} is dropped in the triangle integral. I checked the stress-test calculation: with C = √(2π/√3) = 1.904..., ε = 2 exceeds C, so for θ < arccos(C/2) the cell boundary hθ is below ε/√N and the true integrand is zero, while the displayed integral ∫_ε^{hθ} r(r−ε) dr is positive. The condition ε < C/cos(π/6) only checks the maximum of hθ. Correcting this changes v(2) downward by roughly a factor of three; the final constant −0.0568456 is therefore not proven. A smaller improvement may survive, but it has to be recomputed.

What is genuinely good: the combination of Wasserstein distance and Fejes Tóth to get a lower bound on the smearing energy I(μ) is new to this problem. The argument is short and the borrowing from [GZ19] and [FT48] is explicit. There is no parameter fitting: ε = 2 is the maximizer of the derived bound, not a number chosen to hit the conjecture. The paper is honest about what it does and does not resolve.

Soft spots, in order. (1) The truncation bug above; it is the main computation, so it is load-bearing. (2) The normalization conversion for [GZ19, Lemma 3.2] is stated but not shown; since the paper explicitly says the normalization differs, the constant 2 in W1² ≤ 2I(μ) needs checking. That is likely fixable but not automatic. (3) The claim that ε = 2 maximizes the final coefficient is justified only by figures; a derivative check would close it. The Fejes Tóth application itself looks fine once the small triangle geometry is accepted.

The citation pattern is solid. The literature is relevant and the provenance is clear. The paper is not a breakthrough and the corrected improvement may be small, but the method is worth a referee's time. I would send it to peer review and ask for the corrected integral, the normalization details, and a proof of the ε = 2 maximization. If those come back clean, it is a publishable short note. If the corrected v(2) drops below the old bound, that should be reported honestly rather than hidden.

Referee Report

3 major / 4 minor

Summary. The paper proposes a new lower bound for the constant C_log in the asymptotic expansion of the minimal logarithmic energy on S^2. Starting from the Beltrán–Lizarte decomposition for smeared point charges, the author uses the transport inequality W1^2 ≤ 2I(μ) and Fejes Tóth's comparison theorem to bound the Wasserstein distance from below by an integral of the distance to the smearing caps over an equilateral spherical triangle. The computed value −0.0568456… is obtained by optimizing the resulting one-variable function at ε=2. The main claim is an improvement over the previous lower bound log 2 − 3/4.

Significance. The problem of determining C_log is a central open question in the theory of discrete logarithmic energy on the sphere, related to Smale's 7th problem. A rigorous improvement of the lower bound, even a small one, is a meaningful contribution. The paper's strategy—coupling the Beltrán–Lizarte decomposition with optimal transport and Fejes Tóth's theorem—is elegant and, if the computational details are corrected, viable. The manuscript is short and relies on known tools; its value depends entirely on the correctness of the final constant.

major comments (3)

[§1, displayed formula for ∫_T Φ dσ] The substitution of the true integrand r max(r−ε/√N,0) by r(r−ε/√N) with lower limit ε/√N is valid only when hθ ≥ ε/√N for every θ in the integration range. From the paper's own formula hθ ≃ α/(2 cosθ) with α/2 = C/√N and C = √(2π/√3) ≈ 1.904, the minimum of hθ√N over θ∈[0,π/6] is C < 2. Hence for ε=2 there is an initial interval θ∈[0, arccos(C/2)] on which the true inner integral is zero while the displayed expression is positive because hθ < ε/√N. This overestimates ∫_T Φ dσ and therefore v(ε); a corrected split at θ0 = arccos(C/2) gives v(2) ≈ 2.4×10^{-6} rather than the stated ≈ 7.3×10^{-6}. The theorem's constant −0.0568456… is not established by the argument as written. The condition for the displayed integral should be ε ≤ C (with a split when C < ε < C/cos(π/6)), not merely ε < C/cos(π/6).
[§1, inequality W1^2 ≤ 2I(μ)] The paper invokes [GZ19, Lemma 3.2] to assert W1(μ+σ,σ)^2 ≤ 2I(μ), but explicitly notes a different normalization of the Green function and the space without providing the conversion. The constant 2 enters directly into the final lower bound through v(ε), so the numerical value of the claimed constant depends on this normalization. The authors should state the precise form of the lemma in their normalization and verify that the constant is indeed 2; if the correct constant differs, all numerical conclusions change.
[§1, final paragraph] The assertion that u(ε)+v(ε) has a maximum at ε=2 is supported only by figures. This is a load-bearing step, since the theorem's constant is the value at the maximizing ε. An explicit derivative or inequality proof for the one-variable function, which is elementary (u is logarithmic plus quadratic and v is a squared cubic), is required. This is especially important after correcting the triangle integral, because the maximizing ε may shift and the value of the maximum changes.

minor comments (4)

[Theorem statement] The statement 'There exists a constant C_log > 0' contradicts the displayed bounds −0.0568528… ≤ C_log ≤ −0.0556053…; the sign is evidently a typo that should be corrected (e.g., 'C_log ∈ R' or the appropriate lower-bound assertion).
[Page 3] There is a missing space in 'configurationx1' in the definition of the smeared measure.
[Fejes Tóth application] The Fejes Tóth theorem is stated for configurations not contained in a hemisphere, but the proof does not explain why it applies to the configurations used to bound E_N; a sentence noting that minimizing configurations are not in a hemisphere (or that configurations in a hemisphere already satisfy the old lower bound) would make the argument complete.
[Notation] In the display after the Fejes Tóth application, the notation 'G(μ)' is introduced for the energy I(μ) without definition; please keep notation consistent with the earlier I(μ) = ∫∫ G dμ dμ.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the improved lower bound is obtained from independent external inequalities and an explicit maximization, not from fitting or self-citation.

full rationale

The paper's central claim is a lower bound for C_log obtained by combining known external ingredients: the energy decomposition and Taylor expansion from Beltran-Lizarte [BL23], the Wasserstein lower bound from Garcia-Zelada [GZ19, Lemma 3.2], the Kantorovich-Rubinstein duality, and Fejes Toth's spherical triangle integral inequality. The parameter epsilon is chosen as the maximum of the explicit univariate function u(epsilon)+v(epsilon), whose defining integrals are evaluated from the stated Fejes Toth geometry; it is not fitted to the conjectured value C_BHS or to the target constant. No equation in the paper is defined in terms of the quantity it is supposed to predict, and no fitted parameter is renamed as a prediction. The only self-citations, [BEMOC21] and [MM21], appear as background or remarks and are not load-bearing for the theorem. The noted normalization caveat in transferring [GZ19, Lemma 3.2] is a possible correctness concern about constants, not a circularity, since the cited lemma is an external, independently stated result and the paper does not derive its target constant from that lemma by construction.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

The central claim rests on three external results: the energy decomposition, the Wasserstein-energy inequality, and the Fejes Tóth lower bound. The first two are cited, not proved, and the normalization match of the second is left vague. No new entities are introduced.

free parameters (1)

epsilon = 2
Chosen by hand to evaluate the bound; the theorem holds for this value, so it is not fitted to the target constant.

assumptions (3)

domain assumption The energy decomposition (1) from [BL23] holds for all configurations and ε>0.
Invoked at the start of the proof to separate the energy into individual, self, and cross terms.
domain assumption W1(1/N Σ χ_{B_j}/σ(B_j) dσ, dσ)^2 ≤ 2 I(μ) from [GZ19, Lemma 3.2] transfers to the paper's normalization with the same constant.
The paper notes a different normalization but gives no conversion, so this is an assumed transfer.
domain assumption Fejes Tóth's theorem applies to the function Φ(s)=max{s−ε/√N,0} extended to [0,π].
The quoted theorem is for increasing functions on [0,π/2]; the paper does not justify the extension, though a truncation argument could fix it.

how reviews work

0 comments

Cite this review

Pith. "Pith review of An improved lower bound for the logarithmic energy on $\mathbb S^2$." pith.science (2026). https://pith.science/paper/B7FHRADX

@misc{pith2026250601660,
  author       = {Pith},
  title        = {Pith review of: An improved lower bound for the logarithmic energy on $\mathbb S^2$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B7FHRADX}},
  note         = {Machine review of arXiv:2506.01660}
}

read the original abstract

In this short note, we employ well-known results to improve the lower bound for the constant associated with the linear term in the asymptotic expansion of the minimal logarithmic energy on the sphere.

Figures

Figures reproduced from arXiv: 2506.01660 by the authors.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages

[1]

Beltr\'an, N

C. Beltr\'an, N. Corral, J. G. Criado del Rey, Discrete and continuous Green energy on compact manifolds. Journal of Approximation Theory, 237, pp. 160–185, 2019

work page 2019
[2]

Beltr\'an, U

C. Beltr\'an, U. Etayo, J. Marzo, J. Ortega Cerd\`a, A sequence of polynomials with optimal condition number. Journal of the American Mathematical Society, 34(1), pp. 219-244, 2021

work page 2021
[3]

Beltr\'an, F

C. Beltr\'an, F. Lizarte, On the minimum value of the condition number of polynomials. IMA Journal of Numerical Analysis, 42(4), pp. 2959–2983, 2022

work page 2022
[4]

Beltr\'an, F

C. Beltr\'an, F. Lizarte, A lower bound for the logarithmic energy on S ^2 and for the Green energy on S ^n . Constructive Approximation, 58, pp. 565–587, 2023

work page 2023
[5]

B\'etermin, E

L. B\'etermin, E. Sandier, Renormalized Energy and Asymptotic Expansion of Optimal Logarithmic Energy on the Sphere. Constructive Approximation, 47(1), pp. 39–74, 2016

work page 2016
[6]

Borodachov, D

S. Borodachov, D. Hardin, E. Saff, Discrete Energy on Rectifiable Sets. Springer Monographs in Mathematics. Springer, New York, 2019

work page 2019
[7]

J. S. Brauchart, D. P. Hardin, E. B. Saff, The next-order term for optimal Riesz and logarithmic energy asymptotics on the sphere. Recent Advances in Orthogonal Polynomials, Special Functions, and Their Applications. Ed. by J. Arvesú and G. Lagomasino. Contemporary Mathematics, 578, American Mathematical Society, pp. 31–61, 2012

work page 2012
[8]

Fejes T\'oth, The isepiphan problem for n -hedra

L. Fejes T\'oth, The isepiphan problem for n -hedra. American Journal of Mathematics, 70, pp. 174–180, 1948

work page 1948

Show all 28 references

[9]

Fejes T\'oth, G

L. Fejes T\'oth, G. Fejes T\'oth, W. Kuperberg, Lagerungen—arrangements in the plane, on the sphere, and in space. Springer, Cham, 2023

2023
[10]

García-Zelada, Concentration for Coulomb gases on compact manifolds

D. García-Zelada, Concentration for Coulomb gases on compact manifolds. Electronic Communications in Probability, 24, Paper No. 12, 18 pp., 2019

2019
[11]

S. Graf, H. Luschgy, Foundations of Quantization for Probability Distributions. Lecture Notes in Mathematics, 1730. Springer-Verlag, Berlin, 2000

2000
[12]

Kloeckner, Approximation by finitely supported measures

B. Kloeckner, Approximation by finitely supported measures. ESAIM: Control, Optimisation and Calculus of Variations, 18, pp. 343–359, 2012

2012
[13]

A. B. J. Kuijlaars, E. B. Saff, Asymptotics for Minimal Discrete Energy on the Sphere. Transactions of the American Mathematical Society, 350(2), pp. 523-538, 1998

1998
[14]

A. B. Lauritsen, Floating Wigner crystal and periodic Jellium configurations. Journal of Mathematical Physics, 62, 083305, 2021

2021
[15]

E. H. Lieb, H. Narnhofer, The thermodynamic limit for jellium. Journal of Statistical Physics, 12, pp. 291–310, 1975

1975
[16]

Marzo, A

J. Marzo, A. Mas, Discrepancy of minimal Riesz energy points. Constructive Approximation, 54(3), pp. 473–506, 2021

2021
[17]

Peyre, Comparison between W_2 distance and H _ -1 norm, and localization of Wasserstein distance

R. Peyre, Comparison between W_2 distance and H _ -1 norm, and localization of Wasserstein distance. ESAIM: Control, Optimisation and Calculus of Variations, 24(4), pp. 1489–1501, 2018

2018
[18]

E. A. Rakhmanov, E. B. Saff, Y. M. Zhou, Minimal Discrete Energy on the Sphere. Mathematical Research Letters, 1(6), pp. 647–662, 1994

1994
[19]

Sandier, S

E. Sandier, S. Serfaty, From the Ginzburg–Landau Model to Vortex Lattice Problems. Communications in Mathematical Physics, 313(3), pp. 635–743, 2012

2012
[20]

R. R. Sari, D. Merlini, On the -dimensional one-component classical plasma: The thermodynamic limit problem revisited. Journal of Statistical Physics, 14, pp. 91–100, 1976

1976
[21]

M. Shub, S. Smale, Complexity of Bezout’s theorem. III. Condition number and packing. Journal of Complexity, 9(1), pp. 4–14, 1993

1993
[22]

Skinner, Logarithmic potential theory on Riemann surfaces (PhD thesis), 2015

B. Skinner, Logarithmic potential theory on Riemann surfaces (PhD thesis), 2015

2015
[23]

Smale, Mathematical problems for the next century

S. Smale, Mathematical problems for the next century. Mathematics: Frontiers and Perspectives, American Mathematical Society, Providence, RI, pp. 271–294, 2000

2000
[24]

Steinerberger, On the Logarithmic Energy of Points on S ^2

S. Steinerberger, On the Logarithmic Energy of Points on S ^2 . Journal d’Analyse Mathématique, 148, pp. 187–211, 2022

2022
[25]

Tsuji, Potential Theory in Modern Function Theory

M. Tsuji, Potential Theory in Modern Function Theory. Maruzen, Tokyo, 1959

1959
[26]

Villani, Topics in Optimal Transportation

C. Villani, Topics in Optimal Transportation. Graduate Studies in Mathematics, 58. American Mathematical Society, Providence, RI, 2003

2003
[27]

Wagner, On the product of distances to a point set on a sphere

G. Wagner, On the product of distances to a point set on a sphere. Journal of the Australian Mathematical Society, 47(3), pp. 466–482, 1989

1989
[28]

Wolff, Fekete points on spheres, Unpublished manuscript

T. Wolff, Fekete points on spheres, Unpublished manuscript

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Beltr\'an, N

C. Beltr\'an, N. Corral, J. G. Criado del Rey, Discrete and continuous Green energy on compact manifolds. Journal of Approximation Theory, 237, pp. 160–185, 2019

work page 2019

[2] [2]

Beltr\'an, U

C. Beltr\'an, U. Etayo, J. Marzo, J. Ortega Cerd\`a, A sequence of polynomials with optimal condition number. Journal of the American Mathematical Society, 34(1), pp. 219-244, 2021

work page 2021

[3] [3]

Beltr\'an, F

C. Beltr\'an, F. Lizarte, On the minimum value of the condition number of polynomials. IMA Journal of Numerical Analysis, 42(4), pp. 2959–2983, 2022

work page 2022

[4] [4]

Beltr\'an, F

C. Beltr\'an, F. Lizarte, A lower bound for the logarithmic energy on S ^2 and for the Green energy on S ^n . Constructive Approximation, 58, pp. 565–587, 2023

work page 2023

[5] [5]

B\'etermin, E

L. B\'etermin, E. Sandier, Renormalized Energy and Asymptotic Expansion of Optimal Logarithmic Energy on the Sphere. Constructive Approximation, 47(1), pp. 39–74, 2016

work page 2016

[6] [6]

Borodachov, D

S. Borodachov, D. Hardin, E. Saff, Discrete Energy on Rectifiable Sets. Springer Monographs in Mathematics. Springer, New York, 2019

work page 2019

[7] [7]

J. S. Brauchart, D. P. Hardin, E. B. Saff, The next-order term for optimal Riesz and logarithmic energy asymptotics on the sphere. Recent Advances in Orthogonal Polynomials, Special Functions, and Their Applications. Ed. by J. Arvesú and G. Lagomasino. Contemporary Mathematics, 578, American Mathematical Society, pp. 31–61, 2012

work page 2012

[8] [8]

Fejes T\'oth, The isepiphan problem for n -hedra

L. Fejes T\'oth, The isepiphan problem for n -hedra. American Journal of Mathematics, 70, pp. 174–180, 1948

work page 1948