The paper: John A. Baldwin and Steven Sivek, 3-manifold decision problems and Riemann hypotheses, arXiv:2611.06705, posted 2026-11-04, math.GT. It's a short note. This page says what it claims, what that claim stands on, and where the abstract and the body read differently.
You're handed a knot drawn as a diagram, or a 3-dimensional space cut into tetrahedra. Is the knot really unknotted? Is the space really the 3-sphere? Computers can decide this, but the question here is about certificates: if the answer is yes, is there a short proof you can check quickly (that puts the problem in NP)? And if the answer is no, is there a short proof of that (coNP)?
For "yes, it's the unknot" and "yes, it's the 3-sphere", short proofs have been known for years. For "no", the known short proofs needed the generalized Riemann hypothesis (GRH), starting with Kuperberg on the unknot.
The certificate for "no" is a piece of algebra: a representation of the space's fundamental group into a group of 2×2 matrices over a finite field of size p. For these problems, a nonabelian one exists when the answer is "no" (that part is the topology: Kronheimer–Mrowka for knots, Zentner for the 3-sphere). The matrices are easy to check. The catch is p: if it has too many digits, the certificate isn't short. In the authors' words, "GRH is used to ensure that the certificate has polynomial length in the input." It does that through an effective Chebotarev density theorem (Lagarias–Odlyzko), which promises a small enough prime.
It shows a weaker hypothesis is enough. Their Hypothesis 1.2, the quasi-Riemann hypothesis for Dedekind zeta functions: there is one constant ε > 0 such that no Dedekind zeta function of any number field has a zero with real part above 1 − ε. (GRH says: none above 1/2.) Frei, Sofos and Widmer showed this gives an effective Chebotarev theorem with worse bounds than GRH, and the note checks they are still good enough to find the prime. Their Theorem 1.3: each of the problems below is in coNP, modulo Hypothesis 1.2.
| Problem | NP ("yes" has a short proof) | coNP, assuming GRH | coNP, without GRH |
|---|---|---|---|
| Unknot recognition | HLP99 | Kup14 | Lac21 |
| Torus knot recognition | BS19 | BS19 | Theorem 1.3 |
| Hyperbolic knot recognition | forthcoming | BS19 | HH19 |
| Hyperbolicity of cusped 3-manifolds | ? | Zen18 + HH19 | Theorem 1.3 |
| 3-sphere recognition | Iva08, Sch11 | Zen18 | Theorem 1.3 |
Transcribed from the paper's Table 1 in its arXiv HTML version; the shaded cells are the note's. Theorem 1.3 assumes Hypothesis 1.2 and nothing else. Check the citations against the paper, not this page.
The abstract says the note leads "for example to the first unconditional proof that 3-sphere recognition is in coNP." The body is more careful. Hypothesis 1.2 now has a claimed proof: Cureton and Thorner communicated a Lean-formalized proof with ε = 1/32, "building on the formalized proof announced by OpenAI" for Dirichlet L-functions. And then:
"at the time of writing it is not clear to us that any human has understood the proof"
… "we will continue to describe these results as being conditional on Hypothesis" 1.2.
So both readings are honest. If you trust a machine-checked proof that no one has yet explained, 3-sphere recognition is unconditionally in coNP. If you want a proof a person has understood, it's conditional on Hypothesis 1.2, which is still a big step down from needing GRH. (The note's own AI statement: "No generative AI was used at any point in the writing of this note." The proof it leans on is another matter.)