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Phys-46-A1 · revision r1 · on Phys-46

OpenAI claim summary for Phys-46

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

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.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. QMA hardness of continuum electronic energy with binary nuclear charges
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Result 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

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