OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Proves QMA-hardness of approximating the electronic Coulomb energy infimum in three dimensions, minimizing over the full spinful fermionic continuum space. Deterministic polynomial-time reductions work even with only unit-charge nuclei at distinct rational positions, polynomially many electrons and an energy-threshold separation of at least one. (OpenAI’s Lean note)
Lean is a language for mathematical statements and proofs that a computer can verify. The assessments below ask whether those statements express the result OpenAI describes here.
Both independent assessments find that the Lean statements express or imply the whole result described above.Read the assessments
Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.
The electronic promise problem for (H) on \mathcal H_N, with distinct rational nuclear positions, all nuclear charges equal to one, unary electron number, and rational thresholds separated by at least L^{-1}, is QMA-hard under deterministic classical polynomial-time many-one reductions. The reduction produces polynomially many nuclei and electrons, polynomial rational bit lengths, and threshold separation at least one. The only output data are the nuclear positions, electron number, and thresholds.
The paper’s main theorem is shown as a possible match to OpenAI’s description. No review confirming the selection is recorded.
The promise problem above is QMA-hard under deterministic classical polynomial-time many-one reductions for c=1. The reduction can produce instances with b-a\ge1.
No Lean statement in the release is paired with this theorem.
The release does not say which paper theorem its catalogue describes. Both statements above are shown whole, the closer match to the description’s wording first. The atlas has not reviewed either choice.
Reported proof verification
One Lean statement in the release; no independent acceptance report is recorded for the version shown here.
We prove that approximating the electronic Coulomb spectral infimum in three-dimensional space is QMA-hard when positive integer nuclear charges are encoded in binary. The nuclei have distinct rational positions, the electron number is unary, and the energy is minimized over all antisymmetric continuum states and spin sectors. A deterministic classical polynomial-time reduction produces instances with threshold separation at least one. The nuclear charges may be exponentially large, but every output has polynomial bit length.
We prove that approximating the electronic ground-energy infimum for clamped unit-charge nuclei is QMA-hard on the full spinful fermionic continuum space. A deterministic classical polynomial-time reduction produces polynomially many nuclei at distinct rational positions and polynomially many electrons, with polynomial rational bit lengths and threshold separation at least one. No orbital basis, magnetic field, or additional external potential is supplied, and no binding assumption is imposed.
One Lean statement in this family
Lean is a language for mathematical statements and proofs that a computer can verify. The Lean statement below is shown beside its possible paper counterpart, with the reason they were linked.
For a configured theorem, the Comparator program tests whether OpenAI’s separate proof proves the exact Lean statement using only the allowed axioms. Some statement files instead configure definitions for comparison by name and type. Acceptance does not establish that a formal statement means the same thing as the paper’s theorem.
The electronic promise problem for (H) on \mathcal H_N, with distinct rational nuclear positions, all nuclear charges equal to one, unary electron number, and rational thresholds separated by at least L^{-1}, is QMA-hard under deterministic classical polynomial-time many-one reductions. The reduction produces polynomially many nuclei and electrons, polynomial rational bit lengths, and threshold separation at least one. The only output data are the nuclear positions, electron number, and thresholds.
The words they share: distinct, unary, positions, data, electrons, lengths, coulomb, many, hard, eq, nuclei, nuclear, promise, qma, time, bit, charge, unit, rational, one, polynomial, energy, electronic, separation, produces, thresholds, output, hardness, least, reductions, deterministic.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
Atlas has no report of an independent group using Comparator to verify OpenAI’s proof of this Lean statement at the version shown here. Who ran the proof-verification program
Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
MathVet and cert-machine agree on how much of the advertised result the Lean statements express.
MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.
MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”).
Their note:unit_coulomb_qmaHard : QMAHard unitCoulombCodec.encode unitCoulombPromise and binary_coulomb_qmaHard: every promise problem in QMA (verifier circuits over {H, T, CNOT} generated by a Mathlib Turing.TM2ComputableInPolyTime, completeness 2/3, soundness 1/3) has a deterministic polynomial-time many-one reduction (PolynomialManyOne, also Mathlib TM2 poly-time) to deciding groundEnergy ≤ lower versus upper ≤ groundEnergy with 1 ≤ upper - lower, distinct rational nuclear positions, unary electron count and (for the unit version) unit charges; groundEnergy is the sInf over normalized antisymmetric H¹ states of the two-spin Coulomb form kinetic - nuclearEnergy + pairEnergy. External package RellichKondrachov is in the cone.
Definitions the group asks readers to compare with the paper:Custom complexity setup: hand-written QMAGate/QMACircuit/QuantumVerifier/InQMA (QMA over gate set {H, T, CNOT}, acceptance = Born probability of the last qubit being 1), PromiseProblem, QMAHard, and a Codec library for binary/unary encodings; polynomial time itself is Mathlib’s Turing.TM2ComputableInPolyTime (not custom). groundEnergy is an EReal-valued sInf of Bochner-integral forms on H¹ vectors, which is fine because Hardy’s inequality makes the Coulomb terms integrable.
“full”, in their words: The challenge statements imply the headline claim after unpacking definitions or one routine step.
Their method: An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.
The formalization proves QMA-hardness of approximating the electronic Coulomb spectral infimum in three-dimensional continuum space when positive integer nuclear charges are encoded in binary. The instances have distinct rational nuclear positions and a unary electron count, and the energy ranges over antisymmetric continuum states and all spin sectors. A deterministic polynomial-time many-one reduction produces the promise gap with threshold separation at least one, while keeping the complete output length polynomial. The same Comparator file also contains the companion unit-charge result.
The formalization proves QMA-hardness of approximating the electronic ground-energy infimum for clamped unit-charge nuclei in the full spinful fermionic continuum. The reduction is deterministic and polynomial time, with rational nuclear positions and separated rational energy thresholds. No orbital basis, magnetic field, additional external potential, or binding premise is part of the input.
The same Comparator file also includes the companion hardness result when positive integer nuclear charges are encoded in binary.
AI-drafted comparisons, with no recorded human review
QMA-hardness of continuum Coulomb energy with binary chargesContinuumCoulombHardness
Unreviewed AI draft: states a paper’s theorem as the paper does. The file states the main theorems of both papers: QMA-hardness of the continuum Coulomb ground energy with binary-encoded nuclear charges and with unit charges.
Who ran the proof-verification program?
These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.
These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.
QMA-hardness of continuum Coulomb energy with binary chargesContinuumCoulombHardness
Two reports, all at the release’s recorded version.
Reported by
Their word
Their account
Date
Original report
cert-machine
“REFUSED”
Re-run through Comparator did not reach a verdict (the job ended (failure) in the step "comparator, Lean’s kernel and nanoda" without uploading a result: the runner terminated it (GitHub: Process completed with exit code 143.)).
Explanations of this family written elsewhere, by the people named. Each is their own reading of the mathematics.
seaworld008, seaworld008/openai-math-top30: thirty selected families explained in Chinese, 7 October 2026. Source
Chinese-language explainer for general readers, with a beginner’s section, a specialist section, background on why the problem was hard and a diagram; it ends by saying it paraphrases OpenAI’s summary and does not represent consensus.
Yixin Ma, OpenAI mathematical physics, 25 families: a Chinese guided reading, 7 October 2026. Source
Chinese guided reading of this family that compares the manuscripts’ main theorems with their abstracts and flags boundary conditions and limit orders that change the physical reading; drafted with Codex and not a re-proof.
What these papers cite in the release, and what cites them
These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.
Papers elsewhere in the release that cite this family’s papers
No paper elsewhere in the release cites this family’s papers.
References outside the release
Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release
Explain a result, compare statements or propose a correction
Atlas has no contributed explanation for this family yet. You can offer one with a GitHub account. You receive named credit and disclose any AI assistance.
Each button opens a GitHub form with this page and its source version filled in. Opening the form does not reserve the work.
Commentary on this family by others, dated and credited. It is commentary, not a finding.
An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Solid publication, could lead to tenure”. The page says that nothing on it is verified.
Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.
Run Comparator on OpenAI’s proof
Atlas records no independent report that Comparator accepted the proof at the release version shown here.Open the form
Read the Lean statement against the paper
An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded.Open the form
Identify the theorem the catalogue describes
Family 275The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means.Open the form
Address a question raised in an independent assessment
The challenge statements imply the headline claim after unpacking definitions or one routine step.
Their method
An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.
The challenge statements state the headline as OpenAI wrote it.
Their method
Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners…
Re-run through Comparator did not reach a verdict (the job ended (failure) in the step "comparator, Lean’s kernel and nanoda" without uploading a result: the runner terminated it (GitHub: Process completed with exit code 143.)).
Date
10 October 2026
Version
at the release’s recorded version
Their method
Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.
The solution was built and its dependency closure checked; Comparator was not run.
Date
10 October 2026
Version
at the release’s recorded version
Their method
An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.