What was already known
Green and King had computed the candidate group’s cohomology and depth two. Okuyama’s associated-prime–centralizer theorem did not establish the required depth bounds for this group’s centralizers.
AI involvement
The paper credits the TARS agent system with finding the counterexample through autonomous search. Xinan Dai then checked, supplemented and reorganized the argument and wrote the paper.
The source explicitly attributes initial counterexample discovery to TARS's autonomous mathematical search, while crediting Dai's later verification and writing separately.
Importance
Refutes Carlson’s general associated-prime depth equality with an explicit finite group.
Limitations
The exact finite computations and software environment are source-reported. Xinan Dai independently checked the argument and some computations, not every finite table.
Technical verification
The source reports exact exhaustive group enumeration and exact ideal-quotient/Gröbner checks for every finite premise, plus Dai's independent examination of the argument and some calculations. The machine checks are source-reported.
Formal checking
exhaustive finite-group enumeration and exact ideal-quotient and Gröbner normal-form checks over F_2
- Checker
- Dai et al.'s exact-computation procedure (software not named)
- Scope
- All 75 rank-two subgroups, six centralizer types, ambient depth-two certificate, and depth-at-least-three certificates for the exceptional centralizers
- Basis
- Reported by the linked checker
independent examination and verification with supplemented logical chains
- Checker
- Xinan Dai (human coauthor)
- Scope
- The counterexample argument, part of its finite computations, and presentation certificates
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Dai reorganized and completed the logical chains, and the paper explains why Okuyama's bridge, exhaustive enumeration, and centralizer depth certificates refute the equality.
Correction and publication history
Refutes Carlson’s general associated-prime depth equality with an explicit finite group.
Read this published revision