Indeed, this fundamental difference between inter-universal Teichm¨uller theory and
conventional research on geometry over F1 might give rise to various interesting
questions and hence stimulate further research. Finally, in this context, it is perhaps
of interest to note that although there is no specific mathematical object in inter-universal Teichm¨uller theory that may be said to correspond to “F1”, in some sense

the notion of a “field with one element” may, at a more conceptual level,
be thought of as corresponding to the notion of coricity/cores/coric
objects — that is to say, objects that are invariant with respect to
[i.e., “lie under, in a unique way”] various operations such as links [cf.
the discussion of §2.7, (iv)] — a notion which is indeed central to inter-universal Teichm¨uller theory.