Every year somebody announces a proof of a famous open problem — how do mathematicians decide it is wrong so fast?

There is a triage, it is quite reliable, and the key insight is that you do not have to find the error to know one exists.

The signals that get a manuscript set aside quickly:

• It is too short and too elementary. Famous problems have been attacked for a century by people with every available tool. A short argument using only undergraduate techniques would have been found. Not impossible, but the prior is extremely low.
• It does not engage with why previous approaches failed. Serious work on a hard problem knows the graveyard: which strategies were tried, what obstruction killed each. A manuscript that shows no awareness of that is usually walking into a known wall.
• It proves too much. If the same argument, applied slightly differently, would also prove something already known to be false, the argument is wrong and you do not need to locate the step.
• The claim outruns the machinery. A technique with known limits is being used past them without explaining why the limit does not apply.

That third one is the workhorse. It converts "find the mistake in eighty pages" into "exhibit one counterexample", which takes an afternoon.

On whether a real proof has been dismissed: the honest answer is that the historical failure mode is different from what you are imagining.

What has genuinely happened is that unconventional work from outside the mainstream took a long time to be read, sometimes years, and was eventually recognised. That is a delay rather than a permanent dismissal, and it is a real cost of the system.

What has essentially not happened is a correct proof of a famous problem being rejected on the merits and then vanishing. The incentive to be the person who validates a landmark result is enormous, and correctness is checkable. The bottleneck is attention, not willingness.

The practical advice given to anyone who believes they have one is unchanged and reasonable: write it up properly, state precisely where the new idea enters, and show what your argument does on a simpler case that is already known. That last one is persuasive in a way that assertion never is.

Worth adding the asymmetry that drives the whole thing: verifying a proof is far more expensive than producing a claim.

A careful read of a substantial manuscript is weeks of expert time. Claims arrive constantly. Without triage the field would spend all of its attention on the incoming queue and none on research, so some filter is unavoidable — and a filter based on structural signals is a good deal better than one based on who the author is.

The other half is a social process that outsiders often mistake for gatekeeping.

A claimed proof of something major gets looked at, informally and quickly, by a lot of people at once. Somebody notices something questionable in one lemma. That gets circulated, checked, and either resolved or confirmed within days. No individual read the whole thing; collectively the community did, in parallel.

And when the result is plausible, this process runs the other way just as fast — experts convene, seminars are organised, and the work gets read in enormous detail over months. The famous recent proofs of major results went through exactly that, and several took years to be fully accepted.

So the speed of dismissal is not the same mechanism as the depth of acceptance. Fast rejection is triage; slow acceptance is verification.