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Phys-53 · physics · mathematics · Physics

Local noninteracting ensemble representation of Coulomb ground densities

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Must every normalizable full Coulomb electronic ground-state density be the density of some ground-state ensemble of a local noninteracting Kohn–Sham Hamiltonian with potential in the admissible source class V? It suffices to test three electrons with two identical finite-charge nuclei and the fixed spin sector $S_z=1/2$.

Short English statement

Must every normalizable full Coulomb electronic ground-state density be the density of some ground-state ensemble of a local noninteracting Kohn–Sham Hamiltonian with potential in the admissible source class V? It suffices to test three electrons with two identical finite-charge nuclei and the fixed spin sector $S_z=1/2$.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Phys-53-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Phys-53

The explicit three-electron, two-equal-nucleus S_z=1/2 Coulomb density admits no local noninteracting ground ensemble in the admissible class V. This is an admissible counterexample to the universal representability question. Within these assumptions, this is a claimed negative answer to the stated question.

Local noninteracting ensemble representation of Coulomb ground densities

OpenAI's claim

OpenAI presents a counterexample in the selected manuscript. A three-electron two-nucleus molecule is claimed to have a density outside all noninteracting ground-state ensembles for a single real spin-independent local Kohn--Sham potential in the stated class.

Mathematical statement of the claimed result

Result 1.
Set $D=10^{200}$, $e_z=(0,0,1)$, and take the source's finitely specified positive integer charge $Z$. For the three-electron Coulomb molecule \[ H_C=\sum_{i=1}^3\left(-\tfrac12\Delta_i- \frac Z{|r_i-(D/Z)e_z|}-\frac Z{|r_i+(D/Z)e_z|}\right) +\sum_{i<j}|r_i-r_j|^{-1}, \] the source claims a normalized absolute ground state $\Psi$ in the $S_z=1/2$ sector. For every real scalar local potential $v\in L^{3/2}(\mathbb R^3)+L^\infty(\mathbb R^3)$, the noninteracting Hamiltonian $H_s(v)=\sum_i(-\Delta_i/2+v(r_i))$ has no trace-one ground-space ensemble $\Gamma$ with \[ n_\Gamma=n_\Psi\quad\text{almost everywhere}. \] The densities are spin-summed, with integral three, and the same scalar potential acts on both spin components.

Outline of the argument

A controlled small-coupling Coulomb limit isolates the two lowest spatial orbitals. Density and occupation constraints forced by a hypothetical noninteracting representation conflict with the interacting ground-state structure; exact finite charge selection makes the obstruction a genuine finite molecule.

Formal verification

Family 278 has no Lean scope page in this release. Its selected result is not presented as formalized. 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.

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