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Math-2601-A1 · revision r1 · on Math-2601

OpenAI claim summary for Math-2601

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.

The main theorem claims that a real circulant sign matrix satisfying $HH^{\mathsf T}=nI$ exists exactly for orders one and four. This gives a positive answer to the complete stated order-restriction question.

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

Manuscript

Contents
  1. Circulant Hadamard conjecture
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Circulant Hadamard conjecture

OpenAI's claim

Circulant Hadamard orders and Barker lengths: The manuscript claims that real circulant Hadamard matrices occur only at orders one and four. Its further consequence restricts Barker sequences longer than one to the seven lengths $2,3,4,5,7,11,13$.

Mathematical statement of the claimed result

Mathematical claim 1.
A real circulant sign matrix $H\in\{-1,1\}^{n\times n}$ with $HH^{\mathsf T}=nI$ is claimed to exist exactly for $n\in\{1,4\}$. The Barker corollary gives an aperiodic sign sequence $(a_0,\ldots,a_{n-1})$ with $|\sum_{j=0}^{n-k-1}a_ja_{j+k}|\le1$ for every $1\le k<n$, at an integer length $n>1$, exactly when \[n\in\{2,3,4,5,7,11,13\}.\] Thus the longer-length impossibility assertions are proved according to the manuscript; they are not counterexamples to those assertions.

Outline of the argument

Circulant Hadamard orders and Barker lengths: The first row is written in an integral cyclic group ring, where orthogonality becomes a scalar norm equation. Cyclotomic coefficient divisibility first reduces the order to $4u^2$ with odd $u$. An alternating product of character evaluations over the prime components is then followed through descent, ruling out $u>1$; standard correlation reductions give the Barker consequence.

Formal verification

Family 179 describes Lean coverage for circulant order classification and even Barker lengths; the odd Barker classification is not selected.

Circulant Hadamard orders and Barker lengths: The manuscript links CirculantHadamard.lean (declaration OAI.CirculantHadamard.exists_iff_order_one_or_four), EvenBarker.lean.

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources