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Thomas–Fermi asymptotics for growing atomic ionization number

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For integer $Z>m\geq 1$, define $I_m(Z)=E(Z-m,Z)-E(Z,Z)$ for full nonrelativistic Coulomb atoms with two spin states. Does $I_m(Z)/m^{7/3}$ approach the positive Thomas–Fermi constant $a_TF$ whenever m$\to$infinity and Z/m$\to$infinity? Require the same constant for the iterated m$\to$infinity limits of fixed-m Z-limsup and Z-liminf, without asserting any fixed-m convergence.

Short English statement

For integer $Z>m\geq 1$, define $I_m(Z)=E(Z-m,Z)-E(Z,Z)$ for full nonrelativistic Coulomb atoms with two spin states. Does $I_m(Z)/m^{7/3}$ approach the positive Thomas–Fermi constant $a_TF$ whenever m$\to$infinity and Z/m$\to$infinity? Require the same constant for the iterated m$\to$infinity limits of fixed-m Z-limsup and Z-liminf, without asserting any fixed-m convergence.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Phys-60-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Phys-60

Removing $m$ electrons from a full Coulomb atom is claimed to cost asymptotically $a_{\mathrm{TF}}m^{7/3}$ when $m\to\infty$ and $Z/m\to\infty$. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

Thomas–Fermi asymptotics for growing atomic ionization number

OpenAI's claim

OpenAI presents a proof in the selected manuscript. Removing $m$ electrons from a full Coulomb atom is claimed to cost asymptotically $a_{\mathrm{TF}}m^{7/3}$ when $m\to\infty$ and $Z/m\to\infty$.

Mathematical statement of the claimed result

Result 1.
On the full two-spin-state antisymmetric continuum space, set \[ H_{Z,N}=\sum_i(-\Delta_i/2-Z/|x_i|)+\sum_{i<j}|x_i-x_j|^{-1}, \quad E_Z(N)=\inf\sigma(H_{Z,N}),\quad I_m(Z)=E_Z(Z-m)-E_Z(Z). \] For integer $Z>m\ge1$, the claimed joint asymptotic is \[ I_m(Z)/m^{7/3}\to a_{\mathrm{TF}}>0 \qquad\text{whenever }m\to\infty,\ Z/m\to\infty. \] The same coefficient appears in the iterated upper and lower limits with $Z\to\infty$ first at fixed $m$, then $m\to\infty$. It is the Thomas--Fermi ionization coefficient for kinetic normalization $\tfrac3{10}(3\pi^2)^{2/3}\int\rho^{5/3}$, characterized by $\lim_ZI_m^{\mathrm{TF}}(Z)=a_{\mathrm{TF}}m^{7/3}$. No fixed-$m$ quantum limit is asserted.

Outline of the argument

Positive-price particle sectors and an exterior cutoff isolate the outer electrons. Conditional screening comparison continues a lower barrier, identifies the limiting field and charge, and then returns to fixed particle sectors to recover the Thomas--Fermi coefficient.

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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