What was already known
Colombo proved the boundary exponent D=N−1 in 1928. Dyn, Goodman and Micchelli settled even exponents in 1986. Li, Tie, Wang and Liu gave a concurrent independent proof of odd-exponent nonsingularity by a different route.
AI involvement
WuJie AI and several language models substantially identified the odd-exponent proof strategy and drafted an initial argument. Qianli Ma chose the route, checked and revised the mathematics, and completed the Lean proof and manuscript.
The original paper explicitly credits AI systems with substantial work on the retained odd-exponent proof strategy and initial proof, while Ma directed and completed the project.
Importance
Closes the remaining odd-exponent case of Colombo’s determinant question and proves a strict Pfaffian sign.
Limitations
Ma’s reported Lean check covers the new odd-exponent theorem, not the classical even-exponent branch or the whole classification as one formal theorem. The independent proof establishes nonsingularity by another argument, not Ma’s stronger sign.
Technical verification
Ma's public Lean 4 proof chain checks the odd-exponent determinant theorem at the cited revision; the source reports a complete build and axiom audit. This does not mechanically prove the historically known even branch, and the original-problem-to-formal-statement match lacks named independent specialist audit. The separate human proof checks the result by another route, not Ma's proof.
Formal checking
Complete Lean proof-chain build, 2817/2817 jobs, source scan and axiom audit; theorem odd_colombo_determinant_positive. Strict positivity of the difference-power determinant for even size, strictly increasing real nodes and odd exponents 2r+1 with r≥m−1
- Checker
- Lean 4 kernel in Qianli Ma's audited build
- Scope
- For even N=2m≥2, pairwise distinct real numbers x₁,…,x₂ₘ and odd D=2r+1≥N−1, Ma proves that the matrix [(xⱼ−xᵢ)^D] is nonsingular. After sorting the nodes, the proof gives the stronger sign (−1)^(m choose 2) Pf[((xⱼ−xᵢ)^D)]>0.
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Ma reports checking and revising the arguments, and his manuscript explains the paired-split, total-nonnegativity, Volterra and Pfaffian steps through a complete written proof. This is evidence of human grasp of this contribution, while Fei Yu's earlier guidance has narrower scope.
Correction and publication history
The VibeMathed and Palomar listings describe the same odd-exponent result, counted here once.
Read this published revision