What was already known
Carlen–Lieb–Loss and Bristiel–Caputo established uniform-law permanent inequalities. An earlier OpenAI result established qualitative robustness for a balanced total-variation neighborhood. This project adds the exact radius and sharp linear stability coefficient.
AI involvement
The disclosure credits GPT-6 Astra with the mathematical arguments, exact certificates and Lean proofs; GPT-6 Pro supplied mathematical review and supplementary atom geometry.
The disclosure credits GPT-6 Astra with the sharp mathematical arguments, exact certificates and Lean proofs; GPT-6 Pro supplied review and a supplementary note.
Importance
Replaces qualitative nonuniform robustness by an optimal quantitative four-row bound, with exact equality and related consequences.
Limitations
The theorem concerns four rows and balanced laws; it gives no all-dimension radius. The explicit interior exponent is sufficient and weaker than the known uniform optimum at r=0. A supplementary atom-geometry note is outside the formal contract. The source reports agent review but no independent human peer review.
Technical verification
The source reports sixteen compared lean declarations, exact support coverage and sos certificate soundness, and transfer to all balanced real laws; python generates witnesses. This applies to the selected claim and is not an independent reproduction by AIProofWatch.
Formal checking
The source reports sixteen completed statement comparisons, exact-certificate replay and three-kernel checking in an exact-commit Linux workflow. Lean proves certificate coverage, soundness and real-law transfer; Python supplies witnesses only.
- Checker
- Lean 4 kernel and documented Comparator or independent-kernel replay run by repository authors
- Scope
- For a balanced probability law ν on the 24 permutations of four labels at total-variation distance at most r≤1/24 from uniform, and nonnegative rows fᵢ normalized by the uniform mean of fᵢ² equal to one, 1−Eν∏ᵢfᵢ(σ(i))≥(1−24r)/9 ΣᵢVar(fᵢ). The coefficient is sharp at each radius and 1/24 is the exact admissible radius.
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
The source reports agent review but no independent human proof review or documented human explanation of the sharp certificate mechanism. Few is the evidence default.
Correction and publication history
Claim-level primary-source review of exact theorem, prior baseline, AI contribution and documented checking scope.
Read this published revision