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QMA hardness of continuum electronic energy with binary nuclear charges

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Is the ground-energy promise problem for the full antisymmetric continuum electronic Coulomb Hamiltonian QMA-hard under deterministic classical polynomial-time many-one reductions when nuclear charges are positive binary integers, positions and thresholds are rational, electron count is unary, and the threshold gap is at least inverse input length? Ask for reductions with gap at least one, without auxiliary effective interactions.

Short English statement

Is the ground-energy promise problem for the full antisymmetric continuum electronic Coulomb Hamiltonian QMA-hard under deterministic classical polynomial-time many-one reductions when nuclear charges are positive binary integers, positions and thresholds are rational, electron count is unary, and the threshold gap is at least inverse input length? Ask for reductions with gap at least one, without auxiliary effective interactions.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Phys-46-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Phys-46

Approximating full continuum electronic Coulomb energy is claimed QMA-hard with binary-encoded positive nuclear charges, rational distinct nuclei, and unary electron count. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

QMA hardness of continuum electronic energy with binary nuclear charges

OpenAI's claim

OpenAI presents a proof in the selected manuscript. Approximating full continuum electronic Coulomb energy is claimed QMA-hard with binary-encoded positive nuclear charges, rational distinct nuclei, and unary electron count.

Mathematical statement of the claimed result

Result 1.
An input gives distinct rational nuclear positions, positive integer binary-encoded charges, unary electron number $N$, and rational thresholds $a<b$, with total bit length $L$ and $b-a\ge L^{-1}$. The electronic Hamiltonian on the full spinful antisymmetric continuum space is \[ H=-\tfrac12\sum_i\Delta_i-\sum_{i,\alpha}\frac{Z_\alpha}{|x_i-R_\alpha|} +\sum_{i<j}\frac1{|x_i-x_j|}. \] With $E_0=\inf\sigma(H)$, the promise distinguishes $E_0\le a$ from $E_0\ge b$. The source claims QMA-hardness using classical reductions that are deterministic, many-one, and polynomial in running time, and its reduction achieves $b-a\ge1$. No orbital basis, magnetic field, or preshared electron state is supplied; the separation is between thresholds, not a spectral gap.

Outline of the argument

A finite local-Hamiltonian instance is converted to positive spin interactions through controllable contact geometry. Coulomb orbital and exchange estimates realize that interaction in the antisymmetric continuum, with binary charge scaling preserving polynomial description length and a unit energy separation.

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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