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Phys-76 · physics · mathematics · Physics

The universal radius containing all but m electrons of neutral atoms

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every choice of normalized full Coulomb neutral-atom ground states $Psi_Z$, let $R_m(Psi_Z)$ be the radius at which the expected number of electrons outside the ball is m. Do $m^{1/3}\limsup _{Z\to infinity,Z>m}$ $R_m$ and $m^{1/3}\liminf _{Z\to infinity,Z>m}$ $R_m$ both tend to $(81\pi ^2/2)^{1/3}$ as integer m$\to$infinity? The inner limits keep m fixed.

Short English statement

For every choice of normalized full Coulomb neutral-atom ground states $Psi_Z$, let $R_m(Psi_Z)$ be the radius at which the expected number of electrons outside the ball is m. Do $m^{1/3}\limsup _{Z\to infinity,Z>m}$ $R_m$ and $m^{1/3}\liminf _{Z\to infinity,Z>m}$ $R_m$ both tend to $(81\pi ^2/2)^{1/3}$ as integer m$\to$infinity? The inner limits keep m fixed.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Phys-76-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Phys-76

Neutral-atom exterior-mass radii are claimed to have the Thomas--Fermi asymptotic $(81\pi^2/2)^{1/3}m^{-1/3}$ in the stated large-charge limits. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

The universal radius containing all but m electrons of neutral atoms

OpenAI's claim

OpenAI presents a proof in the selected manuscript. Neutral-atom exterior-mass radii are claimed to have the Thomas--Fermi asymptotic $(81\pi^2/2)^{1/3}m^{-1/3}$ in the stated large-charge limits.

Mathematical statement of the claimed result

Result 1.
For each integer nuclear charge $Z$, let $\Psi_Z$ be any normalized neutral ground state of \[ H_Z=\sum_{i=1}^Z(-\Delta_i/2-Z/|x_i|)+\sum_{i<j}|x_i-x_j|^{-1} \] on the spin-two antisymmetric space. With spin-summed density $\rho_{\Psi_Z}$, define $R_m(\Psi_Z)=\inf\{r:\int_{|x|>r}\rho_{\Psi_Z}\le m\}$. The source claims \[ \lim_{m\to\infty}m^{1/3}\limsup_{Z\to\infty,\ Z>m}R_m(\Psi_Z) =\lim_{m\to\infty}m^{1/3}\liminf_{Z\to\infty,\ Z>m}R_m(\Psi_Z) =\left(\frac{81\pi^2}{2}\right)^{1/3}. \] The inner limit holds $m$ fixed; the state remains neutral and has $Z$, rather than $Z-m$, electrons. Both $Z$ and $m$ tend through positive integers in the iterated limits.

Outline of the argument

Conditional screening data are retained on expanding annuli. Compactness and a uniqueness argument identify the neutral limiting profile, whose integrated electron tails then determine both the upper and lower radius limits uniformly over ground states.

Formal verification

For family 263, the documented entries give the large-$m$ ionization-energy coefficient and the generalized neutral outer-radius asymptotics. Fixed-$m$ limits and the general uniform excess-charge manuscript are not selected. The listed Comparator files are CoulombIonization.lean, CoulombRadii.lean. Named declarations include OAI.CoulombAtom.generalized_ionization. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

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