What was already known
The classical drawing gave this upper bound, but its original optimality proof had a gap. The formula was proved when one part has at most six vertices; earlier published general claims have not been adjudicated here.
AI involvement
OpenAI credits its internal model with the manuscript and Lean work. Item-specific autonomy is not documented.
The internal-model release and institutional OpenAI authorship support model leadership, while item-specific autonomy is unproved.
Importance
Claims the exact all-size bipartite crossing number and a drawing that attains it.
Limitations
The Lean statement counts spatial crossing points and repeated crossings of an edge pair separately. Its match to every classical drawing convention has not been independently established; earlier priority claims remain unresolved.
Technical verification
The released scope note states the full K_{m,n} crossing formula for continuous simple edge paths and links a bipartite Comparator statement; the Lean main theorem exposes the equality and drawing attainment. The formal development is source-reported; the correspondence with every informal drawing convention is not documented.
Formal checking
For all positive m,n, ordinary crossing number of K_{m,n} equals the Zarankiewicz product, with a matching drawing. Formal equality and attainment theorem with scope note and Comparator target
- Checker
- Lean 4 formal development in the OpenAI math release
- Scope
- For all positive m,n, ordinary crossing number of K_{m,n} equals the Zarankiewicz product, with a matching drawing
- 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 central universal lower-bound mechanism was found in the released release; Few is the evidence default.
Correction and publication history
Reports the all-size Zarankiewicz formula within the stated drawing model.
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