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QMA hardness using only unit-charge nuclei in the continuum

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Is the full antisymmetric continuum electronic Coulomb ground-energy promise problem QMA-hard when every nucleus has charge one, distinct positions are rational, electron count is unary, and rational thresholds are separated by at least inverse input length? Require deterministic polynomial-time reductions producing polynomially many nuclei and electrons, polynomial bit lengths, and gap at least one.

Short English statement

Is the full antisymmetric continuum electronic Coulomb ground-energy promise problem QMA-hard when every nucleus has charge one, distinct positions are rational, electron count is unary, and rational thresholds are separated by at least inverse input length? Require deterministic polynomial-time reductions producing polynomially many nuclei and electrons, polynomial bit lengths, and gap at least one.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Phys-85-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Phys-85

The same continuum hardness is claimed with unit-charge nuclei only, polynomially many particles and rational positions, and no supplied orbital basis or extra potential. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

QMA hardness using only unit-charge nuclei in the continuum

OpenAI's claim

OpenAI presents a proof in the selected manuscript. The same continuum hardness is claimed with unit-charge nuclei only, polynomially many particles and rational positions, and no supplied orbital basis or extra potential.

Mathematical statement of the claimed result

Result 1.
Consider the full nonrelativistic spin-two electronic Coulomb Hamiltonian \[ H=-\tfrac12\sum_i\Delta_i-\sum_{i,\alpha}|x_i-R_\alpha|^{-1} +\sum_{i<j}|x_i-x_j|^{-1} \] with all nuclear charges equal to one. Inputs comprise distinct rational positions, unary electron number, and rational thresholds $a<b$ with $b-a\ge L^{-1}$, where $L$ is total input length. The promise distinguishes $\inf\sigma(H)\le a$ from $\inf\sigma(H)\ge b$. The source claims QMA-hardness via deterministic classical polynomial-time reductions, producing polynomial counts and bit lengths and $b-a\ge1$. Only positions, electron count, and thresholds are output.

Outline of the argument

Singlet-pair gadgets turn the lattice source into a positive spin model. Independently tunable contact lengths encode its couplings in unit-charge molecular geometry, and continuum error estimates control all spin sectors and orbital alternatives while retaining the energy promise.

Formal verification

For family 275, the documented Comparator contains deterministic polynomial reductions for both binary nuclear charges and unit charges in the full antisymmetric spinful continuum with unit threshold separation. The listed Comparator files are ContinuumCoulombHardness.lean. The inventory does not supply a named declaration for these Comparator entries. 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

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