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REVIEW 3 major objections 4 minor 30 references

On the Excess Charge Problem of Atoms

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Pith's one-line read This paper proves that every fermionic atom of nuclear charge Z≥4 binds fewer than 1.1185Z electrons, with explicit lower-order terms, improving the previous best bound and giving the first finite-Z separation between fermionic and…

desk verdict Genuine improvement of the Lieb/Nam excess-charge bound with a clean, mostly self-contained argument; two loose ends—an unquoted Hardy endpoint at p=3 and a non-reproducible computer bound in the appendix—should be fixed, but neither looks fatal. read the letter →

arxiv 2504.18487 v1 pith:UA4KANF3 submitted 2025-04-25 math-ph math.FAmath.MPmath.SPphysics.atom-ph

Dirk Hundertmark , Nikolaos Pattakos , Marvin Raimund Schulz This is my paper · ORCID
classification math-phmath.FAmath.MPmath.SPphysics.atom-ph MSC 81Q1081V45
keywords excesschargeionizationconjecturefermionicatomsbosonicBenguria–Lieb–Namargumentmean-fieldfunctionalradialsymmetrizationHardyinequality
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 establishes a new rigorous upper bound on the maximum number of electrons $N_c(Z)$ a single atom of nuclear charge $Z$ can bind: for $Z \ge 4$, $N_c(Z) < 1.1185Z + 3.90Z^{1/3} + 0.0134 + 0.184Z^{-1/3} + 0.0196Z^{-2/3}$. This improves on the long-standing bounds of Lieb ($2Z+1$) and Nam ($1.22Z+3Z^{1/3}$), and is the first finite-$Z$ quantitative result showing that fermionic atoms behave differently from bosonic atoms, since bosonic atoms can bind about 1.21 particles per nuclear charge in the large-$Z$ limit. The proof extends the Benguria–Lieb–Nam weighting argument to a general power weight $|x|^p$, proves a radial-symmetrization result for the resulting mean-field functional, and controls the weighted kinetic energy explicitly. If correct, the result shows that the excess charge of real atoms is quantitatively distinct from the bosonic case already at moderate nuclear charge.

What carries the argument

The central object is the mean-field functional $\beta_p(\mu) = \frac{\iint \frac{|x|^p+|y|^p}{2|x-y|}\,d\mu(x)d\mu(y)}{\int |x|^{p-1}\,d\mu(x)}$, together with its $N$-particle analogue $\alpha_{N,p}$. The key identity is the reduction $\beta_p = \beta_p^{\mathrm{rad}}$ for $2\le p\le 3$: the paper proves that the infimum over probability measures is already attained among radially symmetric measures, by showing that the non-radial part contributes non-negative energy. That reduction relies on an improved Hardy inequality with constant $d^2/4$ for functions orthogonal to radial functions, which fixes the allowed range $p\le 3$. Once $\beta_p$ is reduced to a one-dimensional optimization problem, explicit constants $F(p)^{-1}$ are obtained, and refined comparison lemmas connect $\alpha_{N,p}$ to $\beta_p$, while a separate lemma bounds the weighted kinetic-energy term by an explicit power of $Z N^{-2/3}$.

What would settle it

Evaluate the mean-field constant β₃ by numerical optimization over non-radial atomic measures; if a measure is found with the quotient below 1/F(3) ≈ 0.8943, the radial reduction theorem would be false. Alternatively, test the improved Hardy inequality directly: find a smooth compactly supported function orthogonal to all radial functions whose kinetic energy is less than (9/4 − ε) ∫ |u|²/|x|² for ε > 0.

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Extended reading notes

Core claim

The paper's central claim is that for $p \in (1,3]$, the critical electron number satisfies $N_c(Z) < F(p)Z + C(p)Z^{1/3}$, where $F(p) = \max_{0\le t\le 1}\frac{1+t^p}{1+t^{p-1}}$. For the two explicitly worked exponents, the authors prove: for $p=2$, $N_c(Z) < \frac12(\sqrt{2}+1)Z + 2.96Z^{1/3}$ for all $Z\ge 2$, with $\frac12(\sqrt{2}+1) \approx 1.2071$; and for $p=3$, $N_c(Z) < F(3)Z + 3.90Z^{1/3} + 0.0134 + 0.184Z^{-1/3} + 0.0196Z^{-2/3}$ for all $Z\ge 4$, with $1.1184 < F(3) < 1.1185$. The argument resolves Nam's conjecture that the mean-field constant $\beta_2$ is attained by radially symmetric probability measures, and extends this radial reduction to all $2\le p\le 3$. The proof then transfers the mean-field lower bound to the $N$-particle quantity $\alpha_{N,p}$ through sharp comparison inequalities, and bounds the remaining weighted kinetic-energy term using Lieb's moment inequality, the fermionic kinetic-energy estimate, and the virial theorem.

Load-bearing premise

The proof relies on an improved Hardy inequality that holds only for functions orthogonal to radial functions; if that inequality fails at or near p=3, the leading coefficient 1.1185 would not be established by this argument.

Editorial extensions

If this is right

  • Every atom with nuclear charge at least 4 has at most about 1.1185 electrons per unit nuclear charge, a rigorous improvement over the previous 1.22 coefficient.
  • For large but finite $Z$, fermionic atoms are quantitatively distinct from bosonic atoms, which can bind about 1.21 particles per nuclear charge.
  • The bound for $p=2$ improves Nam's result for all $Z\ge 2$ and Lieb's $2Z+1$ bound for $Z>5.3$.
  • Nam's conjecture that the mean-field constant $\beta_2$ is attained by radial measures is confirmed, and the radial reduction is extended to the full range $2\le p\le 3$.

Reading between the lines

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

  • The same weighting scheme could be applied to higher spatial dimensions $d\ge 4$, where the improved Hardy constant is $d^2/4$ and the method might allow exponents $p>3$, yielding still smaller leading coefficients.
  • The explicit constants $C(p)$ are not claimed optimal; numerical experiments in the paper suggest $F(3)$ is within a few percent of the true value, so a more refined optimization could lower the rigorous leading coefficient.
  • Because Lieb's original bound holds for molecules, the generalized weighting argument may transfer to multi-nuclear systems, giving excess-charge bounds for molecules as well.
  • The radial-symmetrization lemma might extend to $3<p<4$ with a different Hardy-type estimate, which would further separate the fermionic bound from the bosonic value 1.21.
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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 studies the maximal number N_c(Z) of electrons that a nucleus of charge Z can bind, and proves new explicit upper bounds. Theorem 2.2 states that for every p in (1,3] there is C(p)>0 such that N_c(Z) < F(p)Z + C(p)Z^{1/3}, with F(p) = max_{0<=t<=1}(1+t^{p-1})/(1+t^p). The two concrete cases are p=2, giving N_c(Z) < 1.2072Z + 2.96Z^{1/3} for Z>=2, and p=3, giving the stronger bound N_c(Z) < 1.1185Z + 3.90Z^{1/3} + 0.0134 + 0.184Z^{-1/3} + 0.0196Z^{-2/3} for Z>=4. The proof adapts the Benguria-Lieb-Nam weighted-multiplier method to a general weight |x|^p; the main ingredients are a radial-symmetrization theorem for the mean-field functional beta_p (Theorem 4.2 and Lemma 4.3), comparison lemmas between the many-particle constant alpha_{N,p} and beta_p (Lemmas 5.3, 5.5, 5.9), and an upper bound for the weighted kinetic term (Lemma 6.4). The paper claims the first finite-Z separation between fermionic and bosonic atoms, since the fermionic bound has leading coefficient below the bosonic value t_c approximately 1.21.

Significance. If the proof can be completed at the points flagged below, this is a significant result: it improves the best rigorous upper bound on excess charge for fermionic atoms from 1.22Z + 3Z^{1/3} to 1.1185Z + O(Z^{1/3}), proves Nam's radial conjecture for the range 2 <= p <= 3, and gives a quantitative separation from bosonic atoms at finite nuclear charge. The main chain is modular, uses standard inequalities and classical theorems, and contains no fitted target data; the final constants are derived rather than adjusted. The paper also contains useful methodological contributions, including the multipole-expansion estimates in Section 5 and the refined comparison lemmas. However, the significance is conditional on two verifiability issues: the unstated endpoint Hardy lemma used at p=3, and a computer-assisted tail estimate in Appendix A that is not reproducible from the manuscript.

major comments (3)
  1. [Section 4, Lemma 4.3 and Eq. (4.15)] At p=3 the proof of Lemma 4.3 (and hence Theorem 4.2 and Proposition 4.5) relies on the endpoint of the improved Hardy inequality with constant d^2/4 for functions orthogonal to radial functions, cited as [EF06, Lemma 2.4]. Since d_H - p^2/4 = 9/4 - 9/4 = 0 at p=3, the nonnegativity of the regularized quartic form C_3(nu), and therefore the lower bound beta_3 >= F(3)^{-1}, depends on the exact value of that constant. The lemma is not stated in the manuscript, and its hypotheses are not checked for the potentials Phi_{epsilon,lambda} psi_{nu,lambda} used in the IMS localization step. Please state the lemma explicitly and verify the orthogonality and H^1 membership conditions in the present regularization; without this, the leading coefficient 1.1185 in Proposition 2.5 is not established by the written proof.
  2. [Appendix A, Lemma A.5] The tail bound in Lemma A.5 is computer-assisted but not reproducible: the proof selects N=1000, states 'computing those terms on a computer explicitly', and reports the aggregate values 0.0242, 0.0203, and 0.0003 without code, pseudocode, or a table of intermediate sums. This estimate is used in the proof of Lemma 5.9 and contributes to the constants E_3 and E_4 in Proposition 2.5, so it is part of the claimed explicit bounds. To make the proof verifiable, include the computer code or a rigorous interval-arithmetic version with a table of the tail computation.
  3. [Section 7.3, after Eq. (7.33)] The claim that the supremum in Eq. (7.33) is attained at r = beta_3^{-1} is not correct as stated. For g(r) = 3(3/10)^{1/3} beta_3^{-2/3} r^{1/3} + lambda beta_3^{-1} r^{-2/3} on the interval [beta_3^{-1}, 5/2], the derivative vanishes at approximately r = 1.27, which lies inside the interval when the numerical values in the paper are used; the maximum is then at an endpoint, and comparison of the endpoints shows that r = 5/2 gives the larger value. The final bound E_1 < 3.893 still appears to hold, but the stated reason does not; please replace this by a correct monotonicity or endpoint argument, or by an explicit numerical bound on the full interval.
minor comments (4)
  1. [Section 2, discussion after Proposition 2.4] The sentence 'In particular Proposition 2.4 shows N < 1.12Z + 4Z^{1/3}, Z >= 4' appears to refer to Proposition 2.5, since Proposition 2.4 has leading coefficient 1.2072; please correct the cross-reference.
  2. [Figure 1 caption] The caption contains the typo 'fat' for 'fact', and it would be helpful to state explicitly that the apparent non-monotonicity of alpha_{N,1} in the plot is a numerical artifact and not in contradiction with Lemma 3.2.
  3. [Section 6.3] The phrase 'quantum mechanic virial theorem' should read 'quantum mechanical virial theorem'.
  4. [Abstract and Introduction] The statement that the bounds show the fundamental difference between fermionic and bosonic atoms 'for finite Z' is imprecise: the separation holds for sufficiently large finite nuclear charge under the stated explicit bounds, not for every finite Z. Please qualify this wording.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main bound is derived from stated comparison inequalities and independent external estimates; the delicate endpoint input [EF06] is external, not self-referential.

full rationale

Proposition 2.5 is obtained by a genuine derivation chain: the binding assumption is transformed by the Benguria–Lieb–Nam identity (3.10); α_{N,p} is bounded below by the mean-field constant β_p through comparison lemmas (5.6), (5.17) and (5.50); β_p is bounded below by the explicit minimization in Proposition 4.5 after the radial-reduction Theorem 4.2; and the kinetic term is bounded via the virial theorem plus Lieb's inequality in Section 6. None of these steps uses the desired conclusion N_c(Z) < F(3)Z + O(Z^{1/3}) as an input. The constants F(3), 3.90, 0.0134, 0.184 and 0.0196 are produced by optimizing auxiliary parameters and by elementary estimates, not fitted to N_c data. The most delicate external input is [EF06, Lemma 2.4], the improved Hardy inequality with constant d^2/4 for functions orthogonal to radial functions; at p = 3 the proof uses the exact endpoint, so the final constant is only as secure as that external lemma. That is a correctness risk, not circularity: the cited inequality is a prior external result, not a restatement of the excess-charge bound. There are no load-bearing self-citations; [ALHK23] is cited only for the standard virial theorem alongside [Wei67]. The derivation is self-contained relative to its stated inputs, so the circularity score is 0.

Assumptions & free parameters 1 free parameters · 8 assumptions · 0 invented entities

No free parameters are fitted to empirical ionization data; the only hand-chosen quantity is the test-function power p, which is universal over all Z. The proof relies on standard mathematical facts and on the fixed nonrelativistic Hamiltonian model. The load-bearing input with the narrowest validity is the improved Hardy inequality, which is exactly why the weight exponent is capped at p=3.

free parameters (1)
  • Weight power p = 3 (any p in (1,3] is allowed by the theorem)
    The proof works for every p in (1,3]; p=3 is selected to minimize the leading-order coefficient F(p). This is a methodological choice, not a fit to empirical ionization data.
assumptions (8)
  • domain assumption Standard nonrelativistic many-body Schrodinger model with point nucleus and Coulomb interactions
    The analysis starts from equation (2.1); the result is a theorem about this model, and model error is not quantified.
  • standard math HVZ theorem: the essential spectrum of H_{N,Z} starts at E_{N-1,Z}
    Used to define binding via the strict inequality in equation (2.5).
  • standard math Improved Hardy inequality with constant d^2/4 for functions orthogonal to radial functions
    Used in Lemma 4.3 to prove C_p(mu) at least C_p(mu_rad), and it forces the restriction p at most d = 3.
  • standard math Newton's theorem for spherically symmetric potentials
    Used in the radial reduction and in evaluating off-diagonal Coulomb integrals.
  • standard math Lieb-Thirring inequality for fermions with spin q=2 and constant from Frank-Nam-Van Den Bosch
    Bounds the L^{5/3} norm of the one-particle density in Lemma 6.3.
  • standard math Virial theorem for ground states of Coulomb Hamiltonians
    Converts the kinetic energy to -E_{N,Z} in Lemma 6.3.
  • standard math Lieb's prior bound N_c(Z) < 2Z+1
    Used to restrict the range of N/Z when optimizing constants in Propositions 2.4 and 2.5.
  • standard math Ground-state energy bound -E_{N,Z} at most A Z^2 N^{1/3}
    Proved in Lemma A.7 by filling hydrogen levels and dropping the repulsive interaction.

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Pith. "Pith review of On the Excess Charge Problem of Atoms." pith.science (2026). https://pith.science/paper/UA4KANF3

@misc{pith2026250418487,
  author       = {Pith},
  title        = {Pith review of: On the Excess Charge Problem of Atoms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UA4KANF3}},
  note         = {Machine review of arXiv:2504.18487}
}
abstract

This paper establishes new bounds on the maximum number of electrons $ N_c(Z) $ that an atom with nuclear charge $Z$ can bind. Specifically, we show that \begin{equation*} N_c(Z) < 1.1185Z + O(Z^{1/3}) \end{equation*} with an explicit bound on the lower order term $O(Z^{1/3})$. This result improves long--standing bounds by Lieb and Nam obtained in 1984, respectively 2012. Our bounds show the fundamental difference between fermionic and bosonic atoms for finite $Z$ since for bosonic atoms it is known that $\lim N_c(Z)/Z = t_c \approx 1.21$ in the limit of large nuclear charges $Z$.

Figures

Figures reproduced from arXiv: 2504.18487 by the authors.

Figure 1
Figure 1. Numerical approximation of the values of , for various and . Starting from an initial sample set of vectors the values of , have been obtained using a Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm im￾plemented in Python. For = 2.5 and = 3.0 the values of , seem to be almost identical for ≤ 20. In the plot 16,1 > 18,1 which is contrary to Lemma 3.2 and due to the fat that numerical approximation is difficult for … view at source ↗
Figure 2
Figure 2. Values of () according to Proposition 4.5, where 0 was computed numerically. The lower bounds () on −1 have been found by choosing explicit measures in (4.3) and numerical optimization. The exact value of −1 [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗

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