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QMA-hardness of the electronic Coulomb ground energy in the full continuum, with unit nuclear charges

Given clamped nuclei at rational positions in R3\mathbb R^3 and NN electrons, the electronic Hamiltonian is H=−12∑iΔxi−∑i,αZα/∣xi−Rα∣+∑i<j1/∣xi−xj∣H=-\frac12\sum_i\Delta_{x_i}-\sum_{i,\alpha}Z_\alpha/|x_i-R_\alpha|+\sum_{i<j}1/|x_i-x_j| on antisymmetric spinful wave functions. O'Gorman, Irani, Whitfield and Fefferman showed that estimating the ground energy is QMA-complete when the many-electron states are restricted to a supplied finite orbital basis, and raised the question for the complete infinite-dimensional space, where an unrestricted continuum state can lie below every basis state. Is approximating the continuum ground-energy infimum inf⁡spec H\inf\mathrm{spec}\,H to inverse-polynomial precision QMA-hard?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Construction
Field
Quantum complexity; electronic structure
Posed by
Bryan O'Gorman, Sandy Irani, James Whitfield and Bill Fefferman (PRX Quantum 3, 2022, Section V)
Year posed
2022
Years open
4y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Lean-checked, statement unaudited
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
15 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Unit-charge paper, Theorem 1.1: the promise problem for the electronic Coulomb infimum with distinct rational nuclear positions, all charges 1, unary electron number and thresholds separated by at least L−1L^{-1} is QMA-hard under deterministic polynomial-time many-one reductions; the reductions output polynomially many nuclei and electrons and threshold gap at least 1, with no basis, magnetic field or extra potential. Companion: the same with binary-encoded, possibly exponentially large charges. No membership in QMA, no spectral-gap or binding assumption, and no claim about chemically natural instances.

What the AI did

The release README says all results were produced by an unreleased internal OpenAI model using one fixed procedure, about three hours of ChatGPT Pro thinking compute per result on average. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). Both manuscripts are authored 'OpenAI' and name no human author. The unit-charge paper does not invoke the binary-charge companion's hardness theorem; both start from the Cubitt-Montanaro-Piddock Heisenberg source.

Verification

No independent mathematician has checked this yet. Checked here: the introductions and main theorems of both manuscripts were read against the posed question. lean/docs/275.md points to ComparatorChallenges/ContinuumCoulombHardness.json (OAI.ContinuumCoulomb.unit_coulomb_qmaHard and binary_coulomb_qmaHard, module OAI.MathematicalPhysics.ContinuumCoulomb.Reduction.Hardness). The solution file exists at the pinned commit; the challenge is not in the formalization catalogue formalization.yaml. The statement was read here: every promise problem in QMA (defined in the file via uniform polynomial-size gate circuits) reduces by polynomial-time many-one maps to the promise problem whose instances are distinct rational positions of unit-charge nuclei, a unary electron count and thresholds at least 1 apart, with energy the infimum of the Coulomb quadratic form over normalized antisymmetric spinful H1 states. That is the headline claim. Not rebuilt here. Membership in QMA is not claimed.

Sources

Changelog1 change

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