What was already known
Matolcsi, Ruzsa and Weiner conjectured this vanishing; Maxwell and Brierley proved it for the Karlsson family. The arbitrary-phase case outside Tao’s class remained.
AI involvement
OpenAI is the named author of the model-produced manuscript. The available release does not name a human checker or human explanation of its certificate argument.
The September 24 paper credits OpenAI as author, and the release identifies the manuscript collection as model-produced.
Importance
Extends a family result to the full non-Tao order-six class.
Limitations
The selected Lean clause and exact integer/rational certificates address this Fourier statement. They do not establish the stronger N(6)=3 mutually unbiased bases claim.
Technical verification
The paper specifies exact integer and rational moment certificates and provides a complete standard-library verifier. A proof-bearing Lean endpoint states the same Hadamard vanishing result outside Tao equivalence. The selected Lean statement also has a weaker at-most-five MUB clause; it must not be read as a proof of N(6)=3. The source reports these machine checks, but this review did not run them.
Formal checking
Proof-bearing Lean declaration with a separate comparator statement
- Checker
- Lean 4 in the OpenAI formalization release
- Scope
- The Fourier-vanishing clause of OAI.MUB6.fourier_and_family_bound for every non-Tao-equivalent order-six Hadamard matrix
- Basis
- Reported by the linked checker
Formal checking
Exact integer/rational certificate verification with a checksum wrapper
- Checker
- The manuscript's standard-library Python verifier
- Scope
- Finite invariant-moment identities used for the single-matrix Fourier dichotomy
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
The paper explains the moment-certificate and cubic-class dichotomy, but no human-authored explanation of that mechanism is identified. Few is the evidence default.
Correction and publication history
If correct, extends a family result to the full non-Tao order-six class.
Read this published revision