How Claude Fable 5 Helped Disprove the Jacobian Conjecture, an 87-Year-Old Mathematical Problem
On July 19, 2026, mathematician Levent Alpöge announced that Anthropic's Claude Fable 5 helped produce a 216 character polynomial map that disproves the Jacobian conjecture for dimensions ≥3. The counterexample is a degree 7 polynomial map F: ℂ³ → ℂ³ with Jacobian determinant identically equal to −2.
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On July 19, 2026, mathematician Levent Alpöge announced that Anthropic's Claude Fable 5 helped produce a 216 character polynomial map that disproves the Jacobian conjecture for dimensions ≥3.
The counterexample is a degree 7 polynomial map F: ℂ³ → ℂ³ with Jacobian determinant identically equal to −2.
The mathematics has been independently verified by multiple mathematicians and a formal Lean 4 proof, but the result has not yet completed conventional peer review or journal publication as of late July 2026.
Search & fact-check with cited sources for What was the Jacobian conjecture, how did mathematician Levent Alpöge use Anthropic's Claude FablThe 216-character polynomial counterexample to the Jacobian conjecture, generated with the help of Anthropic's Claude Fable 5.
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Create a landscape editorial hero image for this Studio Global article: Search & fact-check with cited sources for What was the Jacobian conjecture, how did mathematician Levent Alpöge use Anthropic's Claude Fabl. Article summary: ## The Jacobian Conjecture. Topic tags: general, general web, user generated, academic, education. Style: premium digital editorial illustration, source-backed research mood, clean composition, high detail, modern web publication hero. Use reference image context only for broad subject, composition, and topical grounding; do not copy the exact image. Avoid: logos, brand marks, copyrighted characters, real person likenesses, fake screenshots, UI text, readable text, watermarks, charts with fake numbers, clickbait thumbnails, icons, and tiny thumbnail layouts. Make it useful as an illustrative visual, not as factual evidence.
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The Jacobian Conjecture: A Central Unsolved Problem
The Jacobian conjecture is a foundational problem in algebraic geometry. First stated for two variables by Ludwig Kraus in 1884 and generalized to N variables by Ott-Heinrich Keller in 1939 , it asserts that if a polynomial function F: ℂⁿ → ℂⁿ has a Jacobian determinant — the determinant of its matrix of partial derivatives — that is a nonzero constant everywhere, then F must be a polynomial automorphism. This means F must have a polynomial inverse and therefore be bijective . Such maps are called .
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What is the short answer to "How Claude Fable 5 Helped Disprove the Jacobian Conjecture, an 87-Year-Old Mathematical Problem"?
On July 19, 2026, mathematician Levent Alpöge announced that Anthropic's Claude Fable 5 helped produce a 216 character polynomial map that disproves the Jacobian conjecture for dimensions ≥3.
What are the key points to validate first?
On July 19, 2026, mathematician Levent Alpöge announced that Anthropic's Claude Fable 5 helped produce a 216 character polynomial map that disproves the Jacobian conjecture for dimensions ≥3. The counterexample is a degree 7 polynomial map F: ℂ³ → ℂ³ with Jacobian determinant identically equal to −2.
What should I do next in practice?
The mathematics has been independently verified by multiple mathematicians and a formal Lean 4 proof, but the result has not yet completed conventional peer review or journal publication as of late July 2026.
How Claude Fable 5 Helped Disprove the Jacobian Conjecture, an 87-Year-Old Mathematical Problem
On July 19, 2026, mathematician Levent Alpöge announced that Anthropic's Claude Fable 5 helped produce a 216 character polynomial map that disproves the Jacobian conjecture for dimensions ≥3. The counterexample is a degree 7 polynomial map F: ℂ³ → ℂ³ with Jacobian determinant identically equal to −2.
Published byEdited with DeepSeek-V4-FlashImages generated with GPT Image 1.5
On July 19, 2026, mathematician Levent Alpöge announced that Anthropic's Claude Fable 5 helped produce a 216 character polynomial map that disproves the Jacobian conjecture for dimensions ≥3.
The counterexample is a degree 7 polynomial map F: ℂ³ → ℂ³ with Jacobian determinant identically equal to −2.
The mathematics has been independently verified by multiple mathematicians and a formal Lean 4 proof, but the result has not yet completed conventional peer review or journal publication as of late July 2026.
Search & fact-check with cited sources for What was the Jacobian conjecture, how did mathematician Levent Alpöge use Anthropic's Claude FablThe 216-character polynomial counterexample to the Jacobian conjecture, generated with the help of Anthropic's Claude Fable 5.
AI Prompt
Create a landscape editorial hero image for this Studio Global article: Search & fact-check with cited sources for What was the Jacobian conjecture, how did mathematician Levent Alpöge use Anthropic's Claude Fabl. Article summary: ## The Jacobian Conjecture. Topic tags: general, general web, user generated, academic, education. Style: premium digital editorial illustration, source-backed research mood, clean composition, high detail, modern web publication hero. Use reference image context only for broad subject, composition, and topical grounding; do not copy the exact image. Avoid: logos, brand marks, copyrighted characters, real person likenesses, fake screenshots, UI text, readable text, watermarks, charts with fake numbers, clickbait thumbnails, icons, and tiny thumbnail layouts. Make it useful as an illustrative visual, not as factual evidence.
openai.com
The Jacobian Conjecture: A Central Unsolved Problem
The Jacobian conjecture is a foundational problem in algebraic geometry. First stated for two variables by Ludwig Kraus in 1884 and generalized to N variables by Ott-Heinrich Keller in 1939 , it asserts that if a polynomial function F: ℂⁿ → ℂⁿ has a Jacobian determinant — the determinant of its matrix of partial derivatives — that is a nonzero constant everywhere, then F must be a polynomial automorphism. This means F must have a polynomial inverse and therefore be bijective . Such maps are called .
Studio Global AI
Continue your research
This page includes a source-backed answer you can continue inside Studio Global.
What is the short answer to "How Claude Fable 5 Helped Disprove the Jacobian Conjecture, an 87-Year-Old Mathematical Problem"?
On July 19, 2026, mathematician Levent Alpöge announced that Anthropic's Claude Fable 5 helped produce a 216 character polynomial map that disproves the Jacobian conjecture for dimensions ≥3.
What are the key points to validate first?
On July 19, 2026, mathematician Levent Alpöge announced that Anthropic's Claude Fable 5 helped produce a 216 character polynomial map that disproves the Jacobian conjecture for dimensions ≥3. The counterexample is a degree 7 polynomial map F: ℂ³ → ℂ³ with Jacobian determinant identically equal to −2.
What should I do next in practice?
The mathematics has been independently verified by multiple mathematicians and a formal Lean 4 proof, but the result has not yet completed conventional peer review or journal publication as of late July 2026.
The conjecture was listed as number 16 on Stephen Smale's 1998 list of Mathematical Problems for the Next Century and was notorious for attracting many flawed proofs over decades .
How Levent Alpöge Used Claude Fable 5
On the night of July 19–20, 2026, while watching the FIFA Club World Cup final, mathematician Levent Alpöge — a number theorist who works at Anthropic after holding a Junior Fellowship at Harvard's Society of Fellows — was asked about the Jacobian conjecture by a friend named Akhil Mathew . Alpöge asked Anthropic's Claude Fable 5 model to work on the problem. The model returned a concrete 216-character polynomial map that satisfied the condition of having a constant nonzero Jacobian determinant but was not injective, thereby violating the conjecture's conclusion .
Alpöge posted the result on X on July 19, 2026, writing: "hello there the jacobian conjecture is false thanx to my close friend akhil for asking about it and my other close friend fable for working during the world cup final".
The Specific Counterexample
The counterexample is a polynomial map F: ℂ³ → ℂ³ defined explicitly as :
P(x, y, z) = (1 + xy)³z + y²(1 + xy)(4 + 3xy)
Q(x, y, z) = y + 3x(1 + xy)²z + 3xy²(4 + 3xy)
R(x, y, z) = 2x − 3x²y − x³z
Two key properties hold :
Jacobian determinant: det Jac F = −2 (a nonzero constant), so F is a Keller map.
Non-injectivity: Three distinct input points all map to the same output (-1/4, 0, 0):
(0, 0, -1/4)
(1, -3/2, 13/2)
(-1, 3/2, 13/2)
Because a non-injective map cannot have a polynomial (or any) inverse, F contradicts the Jacobian conjecture. The construction works over any field of characteristic zero since all coefficients and witness points are rational .
A companion preprint (dated July 20, 2026) provides a full algebraic verification, determines the global geometry of F, and proves that the nonproperness set of F is a hypersurface — the mechanism by which the map fails to be a polynomial automorphism . The paper further constructs infinite families of such counterexamples .
Status of the Two-Variable Case
The Jacobian conjecture for n = 2 (two complex variables) remains an open problem as of July 2026. The counterexample only disproves the conjecture for dimensions n ≥ 3.
Current Status of Verification and Peer Review
As of July 22, 2026, the following is known:
Mathematically credible: The counterexample is explicit and mechanically checkable. Mathematicians are taking it seriously . A formal verification in Lean 4 has been produced and passes continuous integration builds . Mathematician Kevin Buzzard called the result "a big day" .
No peer-reviewed publication yet: There is no published peer-reviewed paper or arXiv preprint as of the latest reports . The material exists as a social media post and a companion PDF on ulam.ai, but has not been through formal journal review.
Wikipedia has updated: The Wikipedia entry for the Jacobian conjecture now lists the counterexample as disproving the conjecture for N > 2 .
Wolfram MathWorld updated: The MathWorld entry now also lists the counterexample .
Community assessment: The consensus among commentators is that the mathematics is almost certainly correct, but formal acceptance and citation in the official record are still pending as of this date . The result is credited to a human–AI collaboration, not to Claude acting alone .
Bottom line: The Jacobian conjecture appears to be genuinely false in dimensions ≥ 3. The two-variable case remains open. The counterexample is mathematically sound and has been verified independently, but has not yet completed formal peer review.
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Did Claude Fable 5 disprove the Jacobian Conjecture? | PacketNebula
The conjecture was listed as number 16 on Stephen Smale's 1998 list of Mathematical Problems for the Next Century and was notorious for attracting many flawed proofs over decades .
How Levent Alpöge Used Claude Fable 5
On the night of July 19–20, 2026, while watching the FIFA Club World Cup final, mathematician Levent Alpöge — a number theorist who works at Anthropic after holding a Junior Fellowship at Harvard's Society of Fellows — was asked about the Jacobian conjecture by a friend named Akhil Mathew . Alpöge asked Anthropic's Claude Fable 5 model to work on the problem. The model returned a concrete 216-character polynomial map that satisfied the condition of having a constant nonzero Jacobian determinant but was not injective, thereby violating the conjecture's conclusion .
Alpöge posted the result on X on July 19, 2026, writing: "hello there the jacobian conjecture is false thanx to my close friend akhil for asking about it and my other close friend fable for working during the world cup final".
The Specific Counterexample
The counterexample is a polynomial map F: ℂ³ → ℂ³ defined explicitly as :
P(x, y, z) = (1 + xy)³z + y²(1 + xy)(4 + 3xy)
Q(x, y, z) = y + 3x(1 + xy)²z + 3xy²(4 + 3xy)
R(x, y, z) = 2x − 3x²y − x³z
Two key properties hold :
Jacobian determinant: det Jac F = −2 (a nonzero constant), so F is a Keller map.
Non-injectivity: Three distinct input points all map to the same output (-1/4, 0, 0):
(0, 0, -1/4)
(1, -3/2, 13/2)
(-1, 3/2, 13/2)
Because a non-injective map cannot have a polynomial (or any) inverse, F contradicts the Jacobian conjecture. The construction works over any field of characteristic zero since all coefficients and witness points are rational .
A companion preprint (dated July 20, 2026) provides a full algebraic verification, determines the global geometry of F, and proves that the nonproperness set of F is a hypersurface — the mechanism by which the map fails to be a polynomial automorphism . The paper further constructs infinite families of such counterexamples .
Status of the Two-Variable Case
The Jacobian conjecture for n = 2 (two complex variables) remains an open problem as of July 2026. The counterexample only disproves the conjecture for dimensions n ≥ 3.
Current Status of Verification and Peer Review
As of July 22, 2026, the following is known:
Mathematically credible: The counterexample is explicit and mechanically checkable. Mathematicians are taking it seriously . A formal verification in Lean 4 has been produced and passes continuous integration builds . Mathematician Kevin Buzzard called the result "a big day" .
No peer-reviewed publication yet: There is no published peer-reviewed paper or arXiv preprint as of the latest reports . The material exists as a social media post and a companion PDF on ulam.ai, but has not been through formal journal review.
Wikipedia has updated: The Wikipedia entry for the Jacobian conjecture now lists the counterexample as disproving the conjecture for N > 2 .
Wolfram MathWorld updated: The MathWorld entry now also lists the counterexample .
Community assessment: The consensus among commentators is that the mathematics is almost certainly correct, but formal acceptance and citation in the official record are still pending as of this date . The result is credited to a human–AI collaboration, not to Claude acting alone .
Bottom line: The Jacobian conjecture appears to be genuinely false in dimensions ≥ 3. The two-variable case remains open. The counterexample is mathematically sound and has been verified independently, but has not yet completed formal peer review.
packetnebula.com
Did Claude Fable 5 disprove the Jacobian Conjecture? | PacketNebula
nocode.techClaude Fable Just Produced a Counterexample to a 87-Year-Old Math Conjecture — What AI Breakthroughs Mean for Every Builder Using Agent Tools
en.wikipedia.orgJacobian conjecture - Wikipedia
glitchwire.comA Mathematician Used Claude Fable to Disprove the 87-Year-Old ...
arxiv.orgarXiv:1705.10921v1 [math.CV] 31 May 2017