What was already known
Bajpai, Dona and Nitsche listed C-32, C-47 and C-55 as unresolved. An existing arithmeticity criterion reduced the latter two cases to suitable commuting-transvection witnesses.
AI involvement
AlphaEvolve generated the retained witness words. The human authors applied the existing criterion, checked the words with exact rational arithmetic and wrote the mathematical proof.
AlphaEvolve generated the retained witness words needed for the arithmeticity argument; humans selected the criterion, checked the certificates and wrote the proof.
Importance
The two certificates settle C-47 and C-55 and complete their arithmetic classification alongside a separate C-32 result.
Limitations
The C-32 thinness proof uses a different ChatGPT-found cone and is outside this contribution. The exact matrix checks are source-reported.
Technical verification
Theorem 1's explicit words are checked by exact rational matrix arithmetic for rank-one unipotence, commuting and symplectic orthogonality, then the authors apply the cited arithmeticity criterion. The source reports both exact computer checks and independent human mathematical verification; neither was reproduced here.
Formal checking
Deterministic exact matrix arithmetic over Q on the published witness words
- Checker
- Authors' exact-arithmetic computation
- Scope
- C-47 and C-55 witness words satisfy the transvection conditions used in Theorem 1
- Basis
- Reported by the linked checker
Independent mathematical argument and certificate verification written by the authors
- Checker
- Riestenberg, Taha, Trettel and Wienhard
- Scope
- The AlphaEvolve certificates prove arithmeticity of C-47 and C-55 under the stated criterion
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
The authors explain how each witness word yields commuting, independent, symplectically orthogonal transvections and why the prior criterion gives finite index.
Correction and publication history
The two certificates settle C-47 and C-55 and complete their arithmetic classification alongside a separate C-32 result.
Read this published revision