Idk man, if your having such a strong reaction maybe I don't remember correctly.
But since you're having such a strong reaction maybe it would be interesting to go back to my 2020 notes where I was playing with this idea.
I vaguely remember that it was not so hard to extend \(F_a(n)=a\,{\rm mod}\, n\) from \(n\) integer to \(x\) real. And \(F_a(x)\) had to do with dividing stuff modulo the integers, so in the group of complex roots of units, or something like that. Like the graph of the thing was just a normal hyperbola of the form \(k/x\) but displayed on a quotient of the plane by a 2-lattice, i.e. a torus. In other words it was just division but topologically wrapped up.
Maybe the Fields' medal-part of it was that integer points of the hyperbole were important and maybe there is something hard in finding rational/integer points of algebraic curves... it may be that is critical point.
But I don't remember... need to get back to it.
But since you're having such a strong reaction maybe it would be interesting to go back to my 2020 notes where I was playing with this idea.
I vaguely remember that it was not so hard to extend \(F_a(n)=a\,{\rm mod}\, n\) from \(n\) integer to \(x\) real. And \(F_a(x)\) had to do with dividing stuff modulo the integers, so in the group of complex roots of units, or something like that. Like the graph of the thing was just a normal hyperbola of the form \(k/x\) but displayed on a quotient of the plane by a 2-lattice, i.e. a torus. In other words it was just division but topologically wrapped up.
Maybe the Fields' medal-part of it was that integer points of the hyperbole were important and maybe there is something hard in finding rational/integer points of algebraic curves... it may be that is critical point.
But I don't remember... need to get back to it.
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)