Exactly three mutually unbiased bases in dimension six
OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Proves \(N(6)=3\), resolving Zauner’s dimension-six mutually unbiased bases conjecture: three such bases exist in ℂ6, but four cannot. The exclusion is a complete certified computation under the stated binary64 arithmetic and compiler conditions. An independent companion proves the Matolcsi–Ruzsa–Weiner Fourier-vanishing conjecture for order-six complex Hadamard matrices outside Tao’s cubic equivalence class. (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.
The independent assessments find that the Lean statements express only part of the result described above, or a narrower statement.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.
Three other possible matches are listed below the card, among them Theorem 1.1(Fourier vanishing) in Exact Fourier certificates for complex Hadamard matrices of order six and Corollary 1.3(Adjoint-closed Cartan subalgebras) in the same paper.
Under the arithmetic and compiler conditions in Appendix A, the fresh complete execution documented in Section 7.5 certifies that there are no four pairwise mutually unbiased orthonormal bases in \C^6. Consequently, N(6)=3.
No Lean statement in the release is paired with this theorem.
Three more possible matches
Theorem 1.1(Fourier vanishing)in Exact Fourier certificates for complex Hadamard matrices of order six, 24 September 2026.
Corollary 1.3(Adjoint-closed Cartan subalgebras)in The maximum number of mutually unbiased bases in dimension six, 24 September 2026.
Theorem 1.2(No complete family)in Exact Fourier certificates for complex Hadamard matrices of order six, 24 September 2026.
Reported proof verification
Two Lean statements in the release; independent groups report that Comparator accepted runs for all these files at the version shown here.
We prove that the maximum number of mutually unbiased orthonormal bases in ℂ6 is three, resolving Zauner’s dimension-six MUB conjecture. The upper bound is computer-assisted: under the stated binary64 arithmetic and compiler conditions, a complete execution of the documented verification pipeline excludes four arbitrary complex bases.
We prove the Fourier-vanishing conjecture of Matolcsi, Ruzsa, and Weiner: every complex Hadamard matrix of order six outside Tao’s cubic equivalence class has a vanishing character sum at every permutation of \((1,1,1,-1,-1,-1)\). We also give an exact certificate excluding seven mutually unbiased bases in ℂ6; by Weiner’s completion theorem, this yields an upper bound of five. Both certificates use only integer and rational arithmetic, and the complete verifier is included.
Two Lean statements in this family
Choose one to see it beside the paper theorem it is paired with.
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.
HadamardCubeFiber
The paper states
Proposition 4.1(Classification of the cubic alternative)
Suppose that H and its entrywise square H^{\circ2} are Hadamard matrices of order six. By multiplying rows and columns by unit complex numbers and permuting rows and columns, one can transform H into the matrix T in (2). In particular, every entry of the transformed matrix is a cube root of unity.
The words they share: cube, rows, entrywise, square, complex, matrix, hadamard, order, six.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
HadamardCubeFiber.lean, the release lists it as a supporting statement.
lemmarowRatio_cubeFiber_sum_zero {H : Matrix6} (hH : IsComplexHadamard H) (hs : IsComplexHadamard (entrywiseSquare H)) {i j : Index} (hij : i ≠ j) (a b : ℂ) (hne : a ≠ b) (hab : ∀ k, rowRatio H i j k ^ 3 = a ∨ rowRatio H i j k ^ 3 = b) : ∑ k ∈ cubeFiber (rowRatio H i j) a, H i k * star (H j k) = 0 := bysorry
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.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program
cert-machine records it as “CERTIFIED”.
MathVet records it as “comparator-sandboxed-pass”.
Neal McBurnett, openai-math-lean-tooling records it as “accepted”.
MUBSix.lean, the release lists it as a supporting statement.
theoremfourier_and_family_bound : (∀ H : CMatrix, IsHadamard H → ¬Equivalent H tao → ∀ π : Equiv.Perm Coord, g H (permuteCharge π alpha) = 0) ∧ (∀ n : ℕ, Attainable n → n ≤ 5) := bysorry
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.
One independent group reports that Comparator accepted a run for this Lean statement file: Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program
Neal McBurnett, openai-math-lean-tooling records it as “accepted”.
Other mathematical statements in OpenAI’s Lean library
Atlas found these propositions by scanning the text of the project’s Model.lean file. Each name links to its definition, with any associated statement file beside it. This list does not record running Lean to interpret or verify those definitions.
Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.
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 a weaker statement than the advertised result (their term: “weaker-statement”).
Their note:MUB6.fourier_and_family_bound states (∀ H, IsHadamard H → ¬ Equivalent H tao → ∀ π, g H (permuteCharge π alpha) = 0) ∧ (∀ n, Attainable n → n ≤ 5): the Fourier-vanishing statement (the Matolcsi-Ruzsa-Weiner companion) and only the weak family bound n ≤ 5 for mutually unbiased orthonormal bases of ℂ⁶ (IsMUBFamily B := ∀ r ≠ s, ∀ i j, |⟨B r i, B s j⟩|² = 1/6). Not stated: N(6) = 3, i.e. neither Attainable 3 nor the exclusion of four bases (docs: 'does not establish the paper’s upper bound of three or its computer-assisted exclusion of four'). n ≤ 5 is the non-existence of 6 MUBs (with Weiner’s gap theorem, not in Lean, this would give N(6) <= 4); the binary64 certificate for excluding four is outside the Lean statement. Cone is 2.83M lines of OAI, the largest in this range.
“weaker-statement”, in their words: The challenge statements do not imply the headline claim: they state a nontrivially weaker statement.
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 paper claims that at most three mutually unbiased orthonormal bases exist in $\mathbb C^6$. The linked formalization proves a weaker family bound: every family in its mutually unbiased bases model has at most five members. It also proves a Fourier character-sum vanishing statement for order-six complex Hadamard matrices not equivalent to the Tao matrix, uniformly over coordinate permutations.
The selected statement does not establish the paper’s upper bound of three or its computer-assisted exclusion of four arbitrary bases.
The linked formalization proves a cancellation lemma used in the order-six complex Hadamard analysis. Let $H$ and its entrywise square both be complex Hadamard matrices, and fix two distinct rows. If the cubes of their six entrywise ratios take only two distinct values, then the sum of the row-ratio terms over either specified cube fiber is zero.
This is a supporting Fourier cancellation statement. The paper’s full character-sum vanishing theorem and its mutually unbiased bases bound are outside this selected statement.
AI-drafted comparisons, with no recorded human review
Cube-fiber cancellation for order-six Hadamard row ratiosHadamardCubeFiber
Unreviewed AI draft: states a supporting result. States a cancellation lemma for row ratios of order-six complex Hadamard matrices, a supporting step towards the Fourier vanishing theorem.
Order-six Hadamard Fourier vanishing and a five-basis upper boundMUBSix
Unreviewed AI draft: states a weaker form of a paper’s theorem. States a bound of five mutually unbiased bases in C⁶ where the main paper claims three, together with the Fourier vanishing theorem of the companion paper (whose no-complete-family theorem the bound of five also implies).
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.
Cube-fiber cancellation for order-six Hadamard row ratiosHadamardCubeFiber
Three reports, all at the release’s recorded version.
Reported by
Their word
Their account
Date
Original report
cert-machine
“CERTIFIED”
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
These reports examine claims in the papers using exact calculations for finite claims, independently of the release’s Lean proofs. Read each report for the exact claim it covers.
Propositions 3.4, 5.2 and 6.3 of Exact Fourier certificates for complex Hadamard matrices of order six, 24 September 2026
Reported by
cert-machine
Method
Exact computation
Their verdict
“CERTIFIED” Exact code written independently from the paper confirms that the published object has the claimed property.
What it covers
“a finite component: the exact finite core of both theorems — the single-matrix certificate (Prop. pair-certificate, both q), the finite arithmetic of the cubic-pair obstruction (Prop. cubic-exclusion), and the mixed-moment certificate (Prop. mixed-certificate: Z_lambda positive definite and a strictly negative exact bound). Theory (the lemmas behind the identities, the cubic classification, the 2-design identity, Weiner completion) is not decided.”
Reports of errors and corrections from outside readers, as they state them. The atlas has not judged them.
hiraditya, [Tracking] Re-running the computer checks of openai/math results in Vx (issue 1489), 10 October 2026. Source
The mutually unbiased bases in dimension six result cannot be re-run because the paper cites a verification/code directory that is not in the repository.
Explanations by others
Explanations of this family written elsewhere, by the people named. Each is their own reading of the mathematics.
mo271, OpenQuantumProblems/13: mutuallyUnbiasedBases_dim6 has answer 3 (issue 6918), 7 October 2026. Source
The release claims at most 3 mutually unbiased bases in dimension six by a computer-assisted argument, but its Lean proof gives only at most 5.
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 this family’s papers cite
This family’s papers cite no paper elsewhere in the release.
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
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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.
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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 “Major math journal”. The page says that nothing on it is verified.
Moritz Firsching, anxkhn, Paul Lezeau, Formal Conjectures: mark problems solved in the OpenAI math release and link formal proofs, 11 October 2026. Source
A Formal Conjectures maintainer, after source inspection, links the dimension six result to the open quantum problem on mutually unbiased bases.
Hao-Yu Sun, Mutually unbiased bases: common states, completion, and extension obstructions, 8 October 2026. Source
Sun cites OpenAI’s N(6)=3 result without certifying its computer-execution conditions, says the comparison of its local computation with an independent theorem is unresolved, and does not rely on either OpenAI preprint.
Questions awaiting review
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Read the Lean statement against the paper
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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 266The 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
Compare the proposed paper and Lean matches
Family 266Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing.Open the form
Investigate a calculation that did not decide the claim
Family 266An independent calculation did not decide the claim, or needs data absent from its source.Open the form
Why this passage is shown
Theorem 1.1 (Computer-assisted)
In The maximum number of mutually unbiased bases in dimension six, 24 September 2026
How it was found
Its words were compared with the words of OpenAI’s catalogue description.
The challenge statements do not imply the headline claim: they state a nontrivially weaker statement.
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 less than the headline, and OpenAI’s Lean note says so.
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 in a sandboxed virtual machine: accepted.
Date
8 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.
Re-run through Comparator with Lean’s kernel: accepted (the check took 54 s; peak memory 6.1 GiB).
Date
10 October 2026
Version
at the release’s recorded version
Their method
Each challenge’s solution is built and checked by Comparator at the pinned release, on Lean leanprover/lean4:v4.34.1, recording the verdict, the check time and the peak memory.
The solution was built and its dependency closure checked; Comparator was not run.
Date
not recorded
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.
Re-run through Comparator with Lean’s kernel: accepted (the check took 16981 s; peak memory 26.1 GiB).
Date
10 October 2026
Version
at the release’s recorded version
Their method
Each challenge’s solution is built and checked by Comparator at the pinned release, on Lean leanprover/lean4:v4.34.1, recording the verdict, the check time and the peak memory.
Exact calculations for finite claims, cert-machine
Their verdict
“NEEDS DATA”
Claim tested or proved
Cannot be decided from the release: the specific object the argument relies on is not published.
Detail
Algorithm and error analysis in build/sections/{seed,completion,phase-cover,refinement,exclusion,rounding,execution}.tex (5,886 tex lines). The code and data are NOT PUBLISHED in the clone: build/sections/execution.tex:16-20 names verification/code/ (seed.cpp, covout.cpp, ub.cpp, patchx.cpp, atlas.cpp, stageB.cpp, stageC.cpp, extraction.json) and execution.tex:380-387 an execution archive (verification/computation/README.md, full-run-01/...); neither exists anywhere in the clone at fd4aeeb2.
Propositions 3.4, 5.2 and 6.3 of Exact Fourier certificates for complex Hadamard matrices of order six, 24 September 2026
Exact calculations for finite claims, cert-machine
Their verdict
“CERTIFIED”
Claim tested or proved
Exact code written independently from the paper confirms that the published object has the claimed property.
What it covers
“a finite component: the exact finite core of both theorems — the single-matrix certificate (Prop. pair-certificate, both q), the finite arithmetic of the cubic-pair obstruction (Prop. cubic-exclusion), and the mixed-moment certificate (Prop. mixed-certificate: Z_lambda positive definite and a strictly negative exact bound). Theory (the lemmas behind the identities, the cubic classification, the 2-design identity, Weiner completion) is not decided.”
Cube-fiber cancellation for order-six Hadamard row ratios
Role
the release lists it as a supporting statement
Paired with
Proposition 4.1 (candidate pairing)
Who ran the proof-verification program
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling.