QMA-hardness of continuum Coulomb energy with unit nuclear charges

OpenAI

The electronic Coulomb ground-energy promise problem is QMA-hard for distinct rational positions of unit-charge nuclei, unary electron count and rational thresholds separated by at least an inverse polynomial, with minimization over the full three-dimensional spinful fermionic continuum.

Sources for this record

What was already known

A published OpenAI result established continuum hardness with binary-encoded positive nuclear charges. Earlier human results used designed continuum potentials or supplied finite orbital bases.

AI involvement AI-led

OpenAI is credited as author and reports internal-model production. No named human check or explanation of this specific unit-charge reduction was documented.

Institutional AI authorship and the collection's production account support AI-led work; autonomy from all human task selection is not established.

Importance Major

The unit-charge theorem strengthens continuum hardness to a more constrained nuclear model.

Limitations

The reduction is deterministic polynomial time and proves hardness only; QMA membership is not claimed. The reported Lean check relies on a published source-hardness theorem.

Technical verification Formal (Lean 4)

The repository identifies a Lean theorem and comparator for the unit-charge continuum promise problem. The reduction uses a published source-hardness theorem as an input.

Formal checking

Lean proof of the unit-charge reduction and final hardness theorem

Checker
Lean kernel via OAI.ContinuumCoulomb.unit_coulomb_qmaHard
Scope
QMA-hardness of the unit-charge full-continuum electronic energy promise problem, using the published CMP source-hardness theorem
Basis
Reported by the linked checker
Evidence source

Checking a formal statement does not establish that it matches the original problem.

Human understanding Few

Default: no named human account explaining the new continuum reduction is documented.

Correction and publication history

2026-10-18 · publication

The unit-charge theorem strengthens continuum hardness to a more constrained nuclear model.

Read this published revision