@PedroLauridsenRibeiro Thanks!! Just to be clear, I also don't have a proposal for a nontrivial topology, but rather for a generalization of "topology" that does apply to $\mathcal B(\mathbb R)$. It does go in the direction of considering stacks, but the resulting object is not actually stacky (no nontrivial automorphisms).

(And all other $\mathrm{Ext}$ groups vanish.)

@PedroLauridsenRibeiro Of course there are no compactly supported real-analytic functions. But one can still take the compactly supported cohomology of the sheaf of real-analytic functions, and it will have cohomology only in degree $1$, where one will find a nonzero $H^1_c(\mathbb R^n,\mathcal A)$ (with bad topological properties, but it's possible to consider it as condensed $\mathbb R$-vector space). Then $\mathcal B(\mathbb R^n)$ is given by $\mathrm{Ext}^1(H^1_c(\mathbb R^n,\mathcal A),\mathbb R)$.

By the way, in the condensed formalism, $\mathcal B(\mathbb R)$ is, as expected, the dual space of the space of compactly supported real-analytic functions $\mathcal A_c(\mathbb R)$, if the latter is understood to be a complex (in this case, supported in degree $1$).

@C.F.G This is a good reference! But it effectively discusses the topology of point 1 in my answer, and then in Note 4.1 remarks that it does not seem possible to put any topology on $\mathcal B(\mathbb R)$: "Not only do we not have, at present, a reasonable way of defining a topology for the space of hyperfunctions on an open set, but it is the common understanding that there exists no topology in the ordinary sense." My question is whether someone found a non-ordinary sense.

@FedorPetrov Use the cover of $\mathbb C$ by $\mathbb C\setminus \mathbb R$ and $U$, and that the corresponding Cech complex is exact as $\mathbb C$ is Stein.

There is a fully faithful embedding of etale sheaves into pro-etale sheaves, and this preserves cohomology, see Proposition 5.2.6 (2) of The pro-etale topology for schemes. This applies in particular to $\mathbb Z$ or $\mathbb G_m$, so their pro-etale cohomology agrees with the usual etale cohomology. The pro-etale $\pi_1$ is a red herring for these questions.

To summarize: There is a clear problem with Joshi's proof, as there is a contradiction between Proposition 6.10.7 and the local inequality proved in the proof of Theorem 9.11.1. The mistake could be in Proposition 6.10.7 (and, given that the proof isn't written down, is the first suspicious place) but it might as well be a mistake in the proof of Theorem 9.11.1. In any case, this whole discussion is only about Joshi's proof, not Mochizuki's; I do not think that there is a real error internally in IUT IV.

I'm more afraid that this is an instance where the cited reference does not match the statement that is claimed. The critical difference between Joshi and Mochizuki is that "Joshi's version of Mochizuki's Corollary 3.12" (=Joshi's Theorem 9.11.1) has a purely local proof and hence cannot have the same content as Mochizuki's Corollary 3.12. However, it may be correct on its own; then the mistake is a mismatch between what Joshi has to compute in Proposition 6.10.7, and what Mochizuki actually computed in IUT IV. But I agree with Sam Hopkins that this discussion is not fruitful.

I think 2. is not true as stated, actually. Taking $X=\mathbb N$, this would be saying that for any null-sequence $(\xi_n)_n$ and any $\ell^2$-sequence $(\xi'_n)_n$, the sum $\sum_n \xi_n \xi'_n$ exists, which is not true. To make it true, you should assume that $\xi: X\to F$ is itself $\ell^2$.

As the argument shows, there are no "one prime at a time" versions of the $abc$ conjecture: It's a purely global phenomenon!