What was already known
The conjecture extends the lower bound of one for nonsingular integer-matrix determinants to integral group-ring matrices. A related direct-finiteness defect alone does not give this determinant witness.
AI involvement
OpenAI credits an internal model with the determinant manuscript and Lean development, building on its companion direct-finiteness construction.
The release attributes the manuscript and formal proof to an internal model, while the exact level of human steering is unspecified.
- Supporting source (github.com) — How the results were produced
Importance
The witness overturns a broad group-ring determinant conjecture.
Limitations
The reported Lean proof covers the invertible-operator determinant inequality. The paper’s additional finite negative spectral-log integral is outside that formal statement.
Technical verification
The reported Lean theorem constructs the integral matrix, its rational inverse and a bounded regular operator with determinant in (0,1), matching the decisive counterexample. The additional spectral-measure integral identity in the paper is outside the formal statement.
Formal checking
Proof of OAI.GroupRingDeterminant.main against GroupRingDeterminant Comparator MainStatement
- Checker
- Lean 4 kernel, as reported by the OpenAI math release
- Scope
- A finitely generated group and integral group-ring matrix with rational inverse and Fuglede–Kadison determinant strictly between zero and one
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
No named human explanation of the determinant transfer and witness is documented; Few is the evidence default.
Correction and publication history
If correct, the witness overturns a broad group-ring determinant conjecture.
Read this published revision