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Phys-30-A1 · revision r1 · on Phys-30

OpenAI claim summary for Phys-30

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status at unsolvedmath.com: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

The source gives positive universal upper and lower bounds for the first ionization energy. It does not assert convergence or an exact Z→infinity asymptotic for removal of one electron. This entry records partial support; complete coverage of the question is not established by this catalogue.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Asymptotics of Atomic Ionization Energy
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Result 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Asymptotics of Atomic Ionization Energy

OpenAI's claim

OpenAI presents a mathematical result relevant to this question 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