What was already known
The problem asks whether odd collections of unit disks have odd-covered area at least π, or at least any positive universal constant. Steinerberger’s separate seven-disk theorem gives a rigorous example below π.
AI involvement
Steinerberger designed the objective and constraints; AlphaEvolve searched for the retained 51- and 151-disk configurations. He reports independent Mathematica checks of their numerical areas.
AlphaEvolve found the retained numerical disk configurations under the author’s problem description and constraints; the rigorous seven-disk theorem is human work.
Importance
The searched configurations suggest lower areas than the rigorous seven-disk example, but remain numerical candidates.
Limitations
The numerical estimates are not rigorous area certificates. The 151-disk value is sensitive to coordinate precision; rounding its coordinates to three digits changes the reported estimate from about 2.51 to 2.72. The seven-disk theorem is Steinerberger’s human result.
Technical verification
The author reports independent Mathematica checks and Monte Carlo estimates for the 51- and 151-disk candidates. These are numerical checks with sensitive coordinates, not a complete rigorous area certificate; no qualifying machine proof or explicit human proof check of the numerical upper bounds is documented.
Independent checking
Steinerberger reports Mathematica checks and Monte Carlo area estimates for the 51- and 151-disk configurations; these are numerical estimates.
- Checker
- Stefan Steinerberger
- Scope
- Numerical odd-covered areas of the 51- and 151-disk candidates
- Basis
- Reported by the linked checker
Human understanding
The author describes broad shapes but says it is unclear whether the 151-disk pattern reflects a family or an optimization artifact. Little understanding of the mechanism behind the stronger numerical values is documented.
Correction and publication history
The searched configurations suggest lower areas than the rigorous seven-disk example, but remain numerical candidates.
Read this published revision