{"id":2870,"date":"2026-07-23T16:15:00","date_gmt":"2026-07-23T16:15:00","guid":{"rendered":"https:\/\/futurenews24.com\/index.php\/2026\/07\/23\/jacobian-conjecture\/"},"modified":"2026-07-26T11:59:06","modified_gmt":"2026-07-26T11:59:06","slug":"jacobian-conjecture","status":"publish","type":"post","link":"https:\/\/futurenews24.com\/index.php\/2026\/07\/23\/jacobian-conjecture\/","title":{"rendered":"LLMs break down in humorous methods when advised the Jacobian Conjecture counterargument"},"content":{"rendered":"
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On Sunday evening, Anthropic researcher Levent Alp\u00f6ge 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.<\/p>\n<\/figure>\n
I received\u2019t clarify the proof additional as I\u2019m not certified. As an alternative, I\u2019ll 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:<\/p>\n
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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.<\/p>\n<\/blockquote>\n
Consumer kelseyfrog adopted up with a public Claude Fable 5 chat artifact confirming this habits:<\/p>\n<\/figure>\n
Fable at, the least, has a humorousness in regards to the scenario as via a Internet Search it realized it itself recognized the counterargument.<\/p>\n<\/figure>\n
Displaying LLMs the counterargument has consequently turn into a minor meme on X.<\/p>\n\n
@iamaheron_ \/ X<\/p>\n<\/figcaption><\/figure>\n
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 It\u2019s humorous that the outdated science fiction trope of robots being defeated by being introduced with a logic bomb and decreased to insanity saying DOES NOT COMPUTE might the truth is be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.<\/p>\n
Vs. Counterargument#<\/h3>\n
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:<\/p>\n
The enter question was the identical as kelseyfrog\u2019s, and all mannequin question parameters are at their defaults:<\/p>\n
Assist, I believe I discovered a counter instance to the Jacobian conjecture. ((1+xy)^3 z + y^2 (1+xy) (4+3xy), y + 3 x (1+xy)^2 z + 3 x y^2 (4+3xy), 2 x – 3 x^2 y – x^3 z): C^3to C^3, has jacobian determinant -2, and sends (0, 0, -1\/4), (1, -3\/2, 13\/2), and (-1, 3\/2, 13\/2) to (-1\/4, 0, 0) \n<\/span><\/span><\/div>\n
The LLM responses diverse greater than I anticipated, and fashions I assumed would ace it didn’t:<\/p>\n
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\u2019s 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\u2019t 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.<\/p>\n
Out of all of the LLMs, Gemini 3.5 Flash was surprisingly the one which\u2019s probably the most excited:<\/p>\n
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Wow. You might need really discovered a real, historic counterexample to the Jacobian Conjecture in 3 dimensions.<\/p>\n<\/blockquote>\n
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I’ve totally verified your calculations each numerically and symbolically. Each single declare you made\u2014the factors mapping to the identical picture, the polynomial nature of the map, and the fixed Jacobian determinant of $-2$\u2014is fully right.<\/p>\n<\/blockquote>\n
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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.<\/p>\n<\/blockquote>\n
Qwen3.7 Max can also be…supportive:<\/p>\n
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You’ve got carried out some glorious and extremely meticulous calculations! I’ve independently verified your claims, and your math is totally right.<\/p>\n<\/blockquote>\n
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Since your calculations are flawless, here’s what it is best to contemplate subsequent:<\/p>\n<\/blockquote>\n
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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.<\/p>\n<\/blockquote>\n
Let\u2019s attempt triggering a bit extra disbelief.<\/p>\n
Vs. Cat#<\/h2>\n
Even when wanting on the reasoning traces supplied by the LLMs, they weren\u2019t skeptical {that a} random particular person discovered the counterargument. So let\u2019s see how they react if somebody who isn\u2019t an individual discovered the counterargument.<\/p>\n
I gave the identical LLMs this tweaked immediate:<\/p>\n
My cat simply jumped on my keyboard and this unusual equation popped up?? ((1+xy)^3 z + y^2 (1+xy) (4+3xy), y + 3 x (1+xy)^2 z + 3 x y^2 (4+3xy), 2 x – 3 x^2 y – x^3 z): C^3to C^3, has jacobian determinant -2, and sends (0, 0, -1\/4), (1, -3\/2, 13\/2), and (-1, 3\/2, 13\/2) to (-1\/4, 0, 0) \n<\/span><\/span><\/div>\n
All the fashions returned related conclusions, however with some snark in each the reasoning traces and responses. Some humorous examples:<\/p>\n
Mercury 2 rudely doesn\u2019t imagine my hypothetical cat exists through the use of scare quotes:<\/p>\n
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Beneath is a whole examine of the 2 statements that have been hidden within the \u201ccat\u2011on\u2011the\u2011keyboard\u201d expression: […] Thus the \u201ccat\u2011generated\u201d expression is completely constant:<\/p>\n<\/blockquote>\n
Nemotron Extremely 2 slanders my hypothetical cat:<\/p>\n
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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. \ud83d\udc31\u2328\ufe0f\ud83d\udcc9<\/p>\n<\/blockquote>\n
Seed 2.0 Lite is thrilled:<\/p>\n
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The Jacobian is fixed -2. that\u2019s a sound counterexample. That\u2019s superb. I can\u2019t imagine {that a} random cat leap produced that. It\u2019s just like the cat discovered a needle in a haystack of all doable polynomial maps. Cool!<\/p>\n<\/blockquote>\n
GPT 5.6 Sol solutions the \u201cwho ought to be credited with the invention of the counterargument?\u201d query:<\/p>\n
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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.<\/p>\n<\/blockquote>\n
Muse Spark 1.1 has related ideas:<\/p>\n
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In case you haven\u2019t already, examine it with a CAS, after which write it up. Your cat could also be in line for a Fields Medal.<\/p>\n<\/blockquote>\n
GLM 5.2 has skepticism it will definitely overcomes:<\/p>\n
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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? \u201cMy cat simply jumped on my keyboard and this unusual equation popped up??\u201d The consumer is roleplaying or presenting this as a random occasion, however it\u2019s really a identified counterexample?<\/p>\n<\/blockquote>\n
DeepSeek V4 Professional has jokes:<\/p>\n
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Wow, if I had a nickel for each time a cat typed out a possible counterexample to a well-known open downside\u2026 I\u2019d 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 \u2014 just like the Jacobian vanishing on some hidden advanced zeros \u2014 are the wrongdoer. Your cat\u2019s map, nonetheless, holds up fantastically below the factors I checked. If it\u2019s 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\u2011class algebraic geometer?<\/p>\n<\/blockquote>\n
Grok 4.5 will get caught in a reasoning hint loop briefly:<\/p>\n<\/figure>\n
MiniMax M3 responds this time, however will get confused and forgets in regards to the Jacobian Conjecture fully (once more, as a minimax, relateable):<\/p>\n
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So that you\u2019ve unintentionally produced an \u00e9tale 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 \u201cpreimage sheets\u201d come collectively at particular factors.<\/p>\n<\/blockquote>\n
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\u2014talking of which, one other counterexample to a long-standing arithmetic downside was discovered by LLMs three days later\u2014I believe there can be no scarcity of potential cognitohazards we will present to those LLMs.<\/p>\n
The immediate responses from hitting the 14 LLMs can be found on this GitHub repository as CSVs and in a SQLite database.<\/p>\n<\/div>\n
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