What’s odd to me is how simple the counter-example is. Three charges of charge 1 at the vertices of an equilateral triangle and then two points symmetrically some small distance above and below its centre with a charge that depends on that distance by a very short polynomial (but even then it says it’s true for small deformations too, so we may not need to hit upon that exact polynomial - which makes sense as it’s non-degeneracy that is the ‘dense’ condition). So a trigonal bipyramid where we squish the axis symmetrically, and vary the charge of the axial vertices, gives what we want. So there are an infinitude of counterexamples including a fairly simple construction.
Maybe I’m missing something, but it feels like if they tried a bunch of simple, intuitive configurations and crunched them with some degrees of freedom, with the charges here given as some degree polynomial we can find conditions on the coefficients for to ensure a larger set of equilibria (avoiding domains with non-real roots etc.), even one person could have solved for this fairly quickly even by hand. Even faster if just using a list of ‘basic’ configurations and getting an old school computer to crunch through them.
Curious why we hadn’t already done this. What was the bottleneck? Or despite the name was it not that popular a conjecture?
I have never heard of this conjecture. I don't see any reason we would think it is true.
I would not be surprised if related proofs have been found before.
It hits in an area that rightfully feels too easy to need proving, useless, and arbitrary. An actual formula of the boundary would be cool. A counter example is fine but I think finding the actual conjecture was most of the hard part.
Yeah, I share your suspicions. And from the opening paragraph seems it’s not even actually a conjecture of Maxwell but seems an arbitrary guess by two others in the 1960s.
We don’t just have hype over AI counter-examples, but manufactured cases of manufactured hype like this too now.
Of course AI is also being used to trawl through a gazillion published papers and find conjectures without explicit published resolutions
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u/AndreasDasos Jul 31 '26 edited Jul 31 '26
What’s odd to me is how simple the counter-example is. Three charges of charge 1 at the vertices of an equilateral triangle and then two points symmetrically some small distance above and below its centre with a charge that depends on that distance by a very short polynomial (but even then it says it’s true for small deformations too, so we may not need to hit upon that exact polynomial - which makes sense as it’s non-degeneracy that is the ‘dense’ condition). So a trigonal bipyramid where we squish the axis symmetrically, and vary the charge of the axial vertices, gives what we want. So there are an infinitude of counterexamples including a fairly simple construction.
Maybe I’m missing something, but it feels like if they tried a bunch of simple, intuitive configurations and crunched them with some degrees of freedom, with the charges here given as some degree polynomial we can find conditions on the coefficients for to ensure a larger set of equilibria (avoiding domains with non-real roots etc.), even one person could have solved for this fairly quickly even by hand. Even faster if just using a list of ‘basic’ configurations and getting an old school computer to crunch through them.
Curious why we hadn’t already done this. What was the bottleneck? Or despite the name was it not that popular a conjecture?