Rumors have it that the best methods to recognize a fake from an authentic Pollock is multifractal analysis. He seems to have a special signature in terms of 1/f-noise.

perhaps it is best that i link to an aborted write-up of a slightly expanded version of my Reims-talk. in the trivial case of the multiplicative group it shows how one gets the Habiro ring from the coalgebra picture. Anyway, here it is.

First point would be choosing a category for “finite dimensional F1-algebras”, which in my opinion should be the one of finite monoids (I know you prefer groups, but evaluating functors of points only in groups sounds to me like taking points only over fields in classical schemes). I guess this is something similar to what you did in your talk in Reims to describe the F1 nc-variety of “dessins des enfants”, but I don’t recall you mentioning anything about the correspondence between such schemes and coalgebras. In this case a coalgebra would be a coalgebra in the category of pointed sets with wedge-cartesian product as a tensor product, never heard of something like that but surely somebody already called it “comonoid”.

For this to make any sense and being nontrivial, I ‘d say one has to admit that the counit should be able to take values 0 or 1, which would leave us with a “field with one element” containing two elements.. Maybe going further along Connes-Consani last proposal, take monoids with 0, and force the counit to take values in the trivial monoid with 0, i.e. {0,1} with the multiplicative structure.

Is it something like that what you had in mind?