Resolves on a scale of 0% to 100% where 0% is "no new ideas" and 100% is "essentially proved the theorem as stated". Resolves when the mathematical community has formed a consensus or at the end of 2023, whichever is earlier. Resolution will be based on a subjective judgement of the mathematical community's assessment of the paper - think of this as a sort of poll, but with subjective weighting of people's opinions, with a former mathematician deciding how to do the judging.
Obviously this is highly subjective, I think it's unavoidable if we want to capture the question we care about. Reasons for this resolution method are that the proof may have flaws that can be fixed, or that may end up proving a weaker bound which achieves some but not all of the important consequences.
Preprint: https://arxiv.org/abs/2211.02515
Discussions:
Nov 7, 11:13pm: Is Yitang Zhang's preprint on Landau-Siegel zeros correct? → How correct is Yitang Zhang's preprint on Landau-Siegel zeros?
1,000
3.00Resolving this to 5%, because it appears to most likely be "serious flaws and there's nothing interesting that works" with some possibility of "introduces a useful tool" (none is apparent as of yet, so it's only a possibility) and some possibility of "a major flaw that is not easy to fix". Guesstimating the expected value across those cases as 5%.
We aren't aware of any updates on this so it seems it's probably basically wrong. Anyone have info to share before we resolve?
Check out this post for some useful info: https://manifold.markets/Loppukilpailija/is-yitang-zhangs-preprint-on-landau-fd18d6951407#DEk2P7e5VjgjEeIAhFwv
I think this market is too high, based on a combination of:
There has been some rumours about some computational/numerical issues with the work. My understanding is that the author has attempted to fix those, but...
it's been a while and there has been no recent attention to the paper (from outside or updates to the arXiv version), so it's starting to look like the issues are serious.
Interestingly, the author's mathematical history is mixed: there's on one hand the sensational work Bounded gaps between primes, and on the other hand there's previous work on Landau-Siegel zeros that appears to be flawed (see discussion here). This made predicting initially harder for me, but given the above two bullet points it now seems that this is, unfortunately, not a breakthrough but a failed attempt of one.
The resolution criteria is, to be blunt, horrible. How good of a contribution does 30%, say, correspond to? Is "provides novel new ideas, but far from proving the stated result" get more like 10% or 90%? What does the "crank" part in the description "crank, no new ideas" mean? Does "not crank, serious attempt, but the proof just does not work" get a strictly positive resolution, as it does not fit the description for 0%? And so on.
This made me very reluctant to predict on the market (until now, after I've obtained strong information about the case).
@Loppukilpailija It's subjective, trade at your own risk. Do you have suggestions for a better resolution criteria? I brainstormed the question with a couple people and all the possibilities that captured the intent of the question were highly subjective, I think that's unavoidable. An objective question could be whether he publishes a result meeting some specific criteria in a journal, but like I mentioned above, it's very easy for the work to prove something useful and interesting but not the claimed result.
@Loppukilpailija To answer your questions: I expect novel ideas but far from proving the stated result would be closer to 0% than 100%. The "crank, no new ideas" basically just means the same thing as "no new ideas", I guess "crank" is there for example/illustrative purposes. I think any serious attempt will almost surely have some new idea (even if it's that something doesn't work), and therefore get a positive resolution.
Actually, let me edit that slightly. Novel ideas but far from proving a result with the important consequences would likely be <50%. If it can prove a result that still achieves the important consequences, but the stated result turns out to be much harder to achieve, that's more like 90%. All of this is very subjective of course.
@jack (I noticed that my comment came out more aggressive than I intended, sorry.)
A binary criteria such as "the work is generally recognized to prove the result (or broadly the same result) stated in the paper" would in my opinion be better. Here "broadly the same result" could mean e.g. that the exponent 2022 needs to be changed with a larger constant, but not "makes progress on an adjacent problem on Siegel zeros".
More generally, I would like there to be examples of what kind of situations receive what kind of resolutions. So, for example, list things like
"The same result except that 2022 needs to be replaced by 10^100 (or some large effective constant) receives 99%",
"The same result, except that 2022 may need to be replaced by some constant and the result is ineffective, receives 80%",
"Actually only provides a proof of the bound 1 - C_{\epsilon}/q^{\epsilon} [the famous Siegel's theorem, best previous result on the problem] but with the constant C_{\epsilon} effective [this famously has not been achieved yet] receives 40%",
"The paper introduces a new tool that's perhaps useful for problems on Dirichlet L-functions or improves on previous tools, but which does not directly lead to any results as above without substantial new work, receives 10%".
(The numbers above are something I could find reasonable with a couple of minutes of thought, though someone could reasonably give quite different numbers.) But the main point is that some quantitative criteria are given: it both helps making the predictions and, in the somewhat unlikely case of gray area situations, aid the resolution process.
@jack I'm slightly hesitant about getting to too much detail about the information as it's based on private conversations with other number theorists. However, the proof relies (among other things) on choosing numerical values suitably. Unfortunately, there seems to be a (meaningful) mistake in numerics in the proof. Hopefully the mistake can be fixed, but I do think there's a fair chance that it is critical. [Needless to say, it is too premature to do any final resolutions at this point.]
@Loppukilpailija Thanks for the detailed commentary. Let me get back to you on the cases you suggested - at a glance that seems reasonable. Yeah, if I were a number theorist myself and resolving by my own judgement I would definitely want to include some examples like that, this question is much less well operationalized because I'm not sufficiently familiar with the math.
The binary question was how I had originally framed it - YES if it has proven the claimed result (or one with largely the same consequences). But after some discussion I realized this also had a problematic amount of subjectivity. I'm not a number theorist and I don't remember the details of the discussion very clearly, but I think it was similar to the examples you suggested above. You are probably right that trying to set the boundary between yes and no is still cleaner than trying to score all the potential outcomes, though. I could open a second version of this question based on that if people are interested.
@jack Your first paragraph suggests a possible new feature for Manifold Markets: a place where one can request for a market on topic X. I would guess that there sometimes there are occasions where one would like to have a topic on X, but is itself a non-expert on X and thus might not be able to set up a good market on the topic. Then a more knowledgeable person on the topic could create the market with better resolution criteria that a non-expert could. (I might overestimate how common such a situation is, in which case this feature would not be that valuable.)
I think a binary market would be good. Would you like for me to create it / write the resolution criteria, or do it yourself? I would expect there to still be some gray area situations with positive probability, so the resolution may still rely on subjective assessments - I could provide the assessment (of course meaning that I won't bet on the market).
@Loppukilpailija Sorry it took a little while, but here are the responses.
Firstly the resolution source of this market is intended to be essentially based on asking people with informed opinions about it, and then subjectively aggregating them together somehow - that's what was meant by "judge the mathematical community's assessment of the paper". So it wasn't possible to write down specific examples ahead of time without polling people about a bunch of hypothetical scenarios (which would be a lot of work). And of course, trying to figure out how to score things on a numeric scale is always hard.
In this sense, it's similar to markets predicting the result of a poll, like https://manifold.markets/SneakySly/at-the-end-of-2023-will-manifold-us or https://manifold.markets/jack/will-we-believe-sbf-committed-willf. But using a subjective aggregation instead of a simple average to give more flexibility to do things like weight opinions of different people, etc.
The rough numbers I got for the hypothetical scenarios you proposed were 99%, 90%, 50%, 10% - so pretty similar (that was also with a couple minutes of thought, from someone who isn't an expert in the subfield).
Feel free to write a market! It seems like you know a lot more about the specific implications of the work. I'll probably make a slightly different market too and then we can try out variants with slightly different resolution criteria.
Also, I agree on the feature suggestion, I think more generally a feature that allows us to draft markets collaboratively would be great and help with these sorts of cases.
@jack I created the market at https://manifold.markets/Loppukilpailija/is-yitang-zhangs-preprint-on-landau-fd18d6951407
@alwaysrinse I made some changes to the resolution method after talking with a mathematician who pointed out that it's quite possible for it to prove something important but substantially weaker than the original claim. If you want to undo your trade, let me know.
Original text for reference:
Resolves YES if the mathematical community's consensus is Yitang Zhang has proven the claimed result (or one with largely the same consequences) by the end of 2023. Resolves NO if the consensus is that he has not. Resolves N/A if there is no consensus.
So, if the proof is flawed, but such flaws are not serious, or any serious flaws are corrected by the end of 2023, then that would be a YES. If the claimed bound is incorrect but it can be corrected to another bound that yields largely the same consequences, that would be a YES.)
