Failure of Kohn–Sham ensemble representation

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Constructs a three-electron Coulomb molecule with two equal positive-integer-charge nuclei whose absolute ground-state density has no noninteracting ground-state ensemble representation by a single real spin-independent local potential in \(L^{3/2}(\mathbb R^3)+L^\infty(\mathbb R^3)\). This disproves Kohn–Sham ensemble representability for that potential class; the required nuclear charge is specified nonnumerically.

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The paper states

Theorem 1.1

In A Coulomb ground-state density without Kohn-Sham ensemble representation, 25 September 2026

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

Set D=10^{200} and e_z=(0,0,1). There is a finite positive integer Z, specified by (7.10), for which the Coulomb Hamiltonian

H_C=\sum_{i=1}^3 \left(-\frac12\Delta_i -\frac{Z}{|r_i-(D/Z)e_z|} -\frac{Z}{|r_i+(D/Z)e_z|}\right) +\sum_{1\le i<j\le3}\frac1{|r_i-r_j|}

has a normalizable absolute ground state \Psi in the sector S_z=1/2, normalized by \|\Psi\|_2=1, but there is no v\in\mathcal V and no ground-state ensemble \Gamma of H_s(v) such that

n_\Gamma=n_\Psi\qquad\text{almost everywhere on }\R^3.

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Papers

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A Coulomb ground-state density without Kohn-Sham ensemble representation

25 September 2026, current edition.

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We construct a finite three-electron Coulomb molecule whose spin-summed ground-state density cannot be reproduced by any ground-state ensemble of noninteracting electrons in a single real, spin-independent local potential in \(L^{3/2}(\mathbf R^3)+L^\infty(\mathbf R^3)\). The molecule has two equal positive integer nuclear charges, specified by an exact finite nonnumerical formula.

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Yixin Ma, OpenAI mathematical physics, 25 families: a Chinese guided reading, 7 October 2026. Source
Chinese guided reading of this family that compares the manuscripts’ main theorems with their abstracts and flags boundary conditions and limit orders that change the physical reading; drafted with Codex and not a re-proof.

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22 references in A Coulomb ground-state density without Kohn-Sham ensemble representation

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