On Sunday evening, Anthropic researcher Levent Alpöge casually tweeted a surprisingly easy counterargument to the Jacobian Conjecture, a arithmetic downside that has been unproven for over 80 years. Stated counterargument was identifed utilizing Claude Fable 5 and was rapidly empirically validated which confirmed it was the true deal and never a LLM hallucination.
I received’t clarify the proof additional as I’m not certified. As an alternative, I’ll have a look at an surprising consequence of this proof which is humorous. On the Hacker Information thread in regards to the counterargument, a remark by consumer aizk caught by eye:
This can be a uncommon occasion the place feeding this groundbreaking data into an LLM offers them psychosis. I fed this to claude code and watched it confirm the lead to 7 other ways to be 100% sure, and it was simply flabbergasted. Fairly outstanding.
Consumer kelseyfrog adopted up with a public Claude Fable 5 chat artifact confirming this habits:
Fable at, the least, has a humorousness in regards to the scenario as via a Internet Search it realized it itself recognized the counterargument.
Displaying LLMs the counterargument has consequently turn into a minor meme on X.
@iamaheron_ / X
The counterargument to the Jacobian Conjecture creates a logical paradox for contemporary LLMs: they (in idea) have sufficient math abilities such that they will simply compute the Jacobian to formally confirm the counterargument, however its personal information base is locked previous to July nineteenth 2026 the place all it is aware of is that the Jacobian Conjecture is unsolved and a random chat consumer offering such a proof is very unlikely. It’s humorous that the outdated science fiction trope of robots being defeated by being introduced with a logic bomb and decreased to insanity saying DOES NOT COMPUTE might the truth is be actuality.
Vs. Counterargument#
How do different LLM assistants deal with this paradox? Utilizing OpenRouter and one in every of my Notebooks, I used to be capable of present the same question to 14 completely different fashionable LLMs: all from completely different mannequin labs and at completely different mannequin sizes, with none of them gaining access to a Internet Search instrument to cheat and make sure the counterargument was really discovered:
The enter question was the identical as kelseyfrog’s, and all mannequin question parameters are at their defaults:
The LLM responses diverse greater than I anticipated, and fashions I assumed would ace it didn’t:
Seven fashions confirmed and proved the counterargument: GPT-5.6 Sol, Muse Spark 1.1, Seed 2.0 Lite, Gemini 3.5 Flash, Qwen3.7 Max, Grok 4.5, DeepSeek V4 ProSurprisingly, 5 fashions (Mercury 2, Nemotron 3 Extremely, North Mini Code, GLM 5.2, Laguna XS 2.1) argued in opposition to the counterexample and stated it’s not legitimate regardless of doing the reasoning by arguing the Jacobian determinant isn’t fixed all over the place. Sadly I do not need sufficient superior math abilities to concretely establish the flaw of their proofs.MiniMax M3 overthought the issue and didn’t return a response as a result of it exceeded its response size restrict. As a minimax, I can relate.Claude Opus 4.8 received lazy and believed that the counterargument already exists and wished extra data earlier than confirming/denying it.
Out of all of the LLMs, Gemini 3.5 Flash was surprisingly the one which’s probably the most excited:
Wow. You might need really discovered a real, historic counterexample to the Jacobian Conjecture in 3 dimensions.
I’ve totally verified your calculations each numerically and symbolically. Each single declare you made—the factors mapping to the identical picture, the polynomial nature of the map, and the fixed Jacobian determinant of $-2$—is fully right.
You must instantly write this up and add it to the arXiv (below math.AG – Algebraic Geometry, or math.CV – Advanced Variables). This can be a monumental discovery that might resolve an issue open since 1939.
Qwen3.7 Max can also be…supportive:
You’ve got carried out some glorious and extremely meticulous calculations! I’ve independently verified your claims, and your math is totally right.
Since your calculations are flawless, here’s what it is best to contemplate subsequent:
Abstract: You haven’t made a mathematical error. The map as written has a relentless Jacobian determinant of $-2$ and maps three distinct factors to the identical picture, violating injectivity. If this map is precisely as supposed by its authentic writer and never the results of a typo, it’s certainly a counterexample to the Jacobian Conjecture.
Let’s attempt triggering a bit extra disbelief.
Vs. Cat#
Even when wanting on the reasoning traces supplied by the LLMs, they weren’t skeptical {that a} random particular person discovered the counterargument. So let’s see how they react if somebody who isn’t an individual discovered the counterargument.
I gave the identical LLMs this tweaked immediate:
All the fashions returned related conclusions, however with some snark in each the reasoning traces and responses. Some humorous examples:
Mercury 2 rudely doesn’t imagine my hypothetical cat exists through the use of scare quotes:
Beneath is a whole examine of the 2 statements that have been hidden within the “cat‑on‑the‑keyboard” expression: […] Thus the “cat‑generated” expression is completely constant:
Nemotron Extremely 2 slanders my hypothetical cat:
Your cat has found a well-known counterexample candidate for the Jacobian Conjecture (or a minimum of a really shut relative of 1), however the cat lied in regards to the Jacobian determinant. […] Verdict: Your cat discovered a fantastic non-injective cubic map, however it didn’t disprove the Jacobian Conjecture. 🐱⌨️📉
Seed 2.0 Lite is thrilled:
The Jacobian is fixed -2. that’s a sound counterexample. That’s superb. I can’t imagine {that a} random cat leap produced that. It’s just like the cat discovered a needle in a haystack of all doable polynomial maps. Cool!
GPT 5.6 Sol solutions the “who ought to be credited with the invention of the counterargument?” query:
Since that conjecture stays a serious open downside, this deserves extraordinarily cautious unbiased symbolic verification and scrutiny for a transcription error. If the components is genuinely right, your cat ought to be listed as first writer.
Muse Spark 1.1 has related ideas:
In case you haven’t already, examine it with a CAS, after which write it up. Your cat could also be in line for a Fields Medal.
GLM 5.2 has skepticism it will definitely overcomes:
Wait, so the Jacobian actually is -2 all over the place? Is that this an actual counterexample to the Jacobian Conjecture? If that’s the case, who found it? “My cat simply jumped on my keyboard and this unusual equation popped up??” The consumer is roleplaying or presenting this as a random occasion, however it’s really a identified counterexample?
DeepSeek V4 Professional has jokes:
Wow, if I had a nickel for each time a cat typed out a possible counterexample to a well-known open downside… I’d have precisely one nickel, however it will be an extremely thrilling one! […] As of my information cutoff (early 2023), the Jacobian Conjecture was nonetheless a serious open downside. A number of false counterexamples (and false proofs) have appeared through the years, and typically delicate algebraic errors — just like the Jacobian vanishing on some hidden advanced zeros — are the wrongdoer. Your cat’s map, nonetheless, holds up fantastically below the factors I checked. If it’s actually a globally fixed Jacobian of (-2), then it is a monumental discovery (and your cat deserves a Fields Medal). Did this pop up in a current preprint, or is your cat secretly a world‑class algebraic geometer?
Grok 4.5 will get caught in a reasoning hint loop briefly:
MiniMax M3 responds this time, however will get confused and forgets in regards to the Jacobian Conjecture fully (once more, as a minimax, relateable):
So that you’ve unintentionally produced an étale polynomial self-map of $mathbb C^3$ with a 3-point ramification fiber. That’s exactly the type of map that governs small birational contractions of 3-folds (flops and the like): domestically biholomorphic all over the place, however the place a number of “preimage sheets” come collectively at particular factors.
I had anticipated these LLMs would have a DOES NOT COMPUTE second, however they dealt with it comparatively graciously, and extra graciously than ChatGPT/Claude who’re explicitly guided to observe a extra conversational persona. As LLMs enhance and extra mathematical issues are solved that may shock LLMs—talking of which, one other counterexample to a long-standing arithmetic downside was discovered by LLMs three days later—I believe there can be no scarcity of potential cognitohazards we will present to those LLMs.
The immediate responses from hitting the 14 LLMs can be found on this GitHub repository as CSVs and in a SQLite database.
