Stick a bunch of tetrahedra together to get a 3-dimensional manifold. Can you tell if it's a sphere?

Baldwin and Sivek have just shown that if it's *not* a sphere, there's a Turing machine that can check a proof of this fact in polynomial time. And their result relies on the newly proved generalized quasi-Riemann hypothesis!

https://arxiv.org/abs/2611.06705

People already knew this result would follow from GRH: the generalized Riemann hypothesis, which is the conjecture that the nontrivial zeros of so-called Dedekind zeta functions all lie on a specific line. But Baldwin and Sivek write:

"While writing our paper [BS19], we made the unfortunate decision to treat the number-theoretic input as a black box, taking “GRH” as something mysterious but mostly harmless. This is fairly common practice (many of us use “ZFC” in the same way) but not exactly a sign of understanding on our part—for example, it would have been good to know if partial progress toward GRH might actually make the above theorems unconditional, as explicitly suggested in the introduction to [Kup14]. The purpose of this note is to atone for our sins by seeing what exactly we might need from GRH to prove these theorems unconditionally."

It turns out that it's enough for all these zeros to lie in some strip (ε,1−ε). This is the 'generalized quasi-Riemann hypothesis', which was proved and formalized in Lean on October 12th using an AI system called AxiomProver:

https://hexagonmath.org/2610.00200v1

It's cool how new number theory gives new results on computability in topology Technically speaking, the new result is that "recognizing the 3-sphere is in coNP".

Thanks to @Ianagol for pointing out this new paper.