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Bounded Excess Electrons

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
For the $N$-electron Coulomb Hamiltonian with nuclear charge $Z$, let $N_0(Z)$ be the least $N$ after which adding electrons no longer lowers the ground-state energy. Prove that $N_0(Z)-Z$ remains bounded as $Z\to\infty$.

Formulation references: [S1], [S2].

Short English statement

For the $N$-electron Coulomb Hamiltonian with nuclear charge $Z$, let $N_0(Z)$ be the least $N$ after which adding electrons no longer lowers the ground-state energy. Prove that $N_0(Z)-Z$ remains bounded as $Z\to\infty$.

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Phys-29-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Phys-29

For one nucleus the theorem states that adding an electron cannot lower the energy once n>Z+C, with one universal C for every real Z>=1. It therefore bounds N0(Z)-Z uniformly, strengthening the problem statement asymptotic assertion and also treating multiple nuclei. The claimed conclusion supplies the requested case and extends it in the directions just specified.

Bounded Excess Electrons

OpenAI's claim

OpenAI presents a proof in the selected manuscript. Coulomb molecules are claimed to bind at most $Z+CM$ electrons, and neutral atoms to have universally bounded first ionization energy and half-electron exterior radius.

Mathematical statement of the claimed result

Result 1.
For $M$ fixed nuclei at distinct positions $R_j\in\mathbb R^3$, charges $z_j\ge1$, and total $Z=\sum_jz_j$, use the antisymmetric spin-two electron Hamiltonian \[ H_n=\sum_{i=1}^n\left(-\tfrac12\Delta_i-\sum_j\frac{z_j}{|x_i-R_j|}\right) +\sum_{i<j}\frac1{|x_i-x_j|},\qquad E_n=\inf\sigma(H_n),\quad E_0=0. \] The claimed constant $C<\infty$ is universal over charges and positions, and obeys \[ E_n<E_{n-1}\quad\Longrightarrow\quad n\le Z+CM, \qquad n>Z+CM\quad\Longrightarrow\quad E_n=E_{n-1}. \] For a single nucleus the strict-binding bound becomes $n\le Z+C$, including real $Z\ge1$.

Outline of the argument

Ground states are compared across particle sectors using common localization and screening fields. Local count bounds, changes of observation law, and estimates in nuclear voids control excess particles; deterministic comparison fields give both sides of the neutral-atom energy and radius bounds.

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. No Comparator is assigned to the selected manuscript here; the family page selects other papers or supporting statements. 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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