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Claim analyzed
Science“The Jacobian conjecture is true.”
Submitted by Keen Crane bc3e
The conclusion
The evidence does not support treating the Jacobian conjecture as a proved fact. Current authoritative sources describe it as an open problem, and the cited pro-proof material is an unreviewed preprint on a related/generalized statement rather than an accepted proof of the standard conjecture. Special-case results also do not establish the conjecture in full.
Caveats
- An unreviewed preprint is not sufficient to overturn the published consensus that the conjecture remains open.
- Proofs for restricted cases, such as low-degree maps, cannot be generalized to the full Jacobian conjecture without additional argument.
- The claim omits the crucial fact that, as of 2026, no accepted general proof exists in the mathematical literature.
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Sources
Sources used in the analysis
We improve the algebraic methods of Abhyankar for the Jacobian Conjecture in dimension two and describe the shape of possible counterexamples. The paper does not present a counterexample; instead it derives constraints on what a counterexample would have to look like.
“The Jacobian Conjecture appeared as Problem 16 on a list of 18 famous open problems in the paper by Steve Smale. The Jacobian Conjecture has been intensively studied, but it remains open in general.” The article works on a weak form of the conjecture and explicitly states that the full Jacobian conjecture is still open.
The abstract claims: 'We completely prove the Generalized Jacobian conjecture in the field of real numbers, which implies the Generalized complex Jacobian conjecture.' This is a preprint claim of proof, but it is not an endorsed acceptance by a journal or community consensus statement.
The preprint addresses the two-variable case and situates itself among partial results. Its framing reflects that the two-dimensional Jacobian conjecture has long been studied, but a universally accepted proof has not been established in the literature.
The Jacobian conjecture is one of the most famous open problems in mathematics. In general, the conjecture remains unsolved, although it has been proved in special cases such as dimension two.
Despite this, no proof for mappings of degree 3 has been found, so the conjecture remains unproven. Thus we have that the Jacobian Conjecture is true for polynomial mappings of degree at most 2, and we need only show that it is also true for mappings of degree at most 3 to prove that it is true for every mapping.
In the session description: “In 1939, Keller conjectured that every polynomial map F whose Jacobian determinant is a nonzero constant has a compositional inverse F^{-1} that is itself a polynomial map. This hypothesis, known as the *Jacobian conjecture*, is still one of the greatest unsolved problems of mathematics, and appears in Smale’s list of 18 open mathematical problems for the 21st century.”
The Jacobian conjecture in the real plane is an open problem that consists in determining whether any polynomial map F with nonzero constant determinant of the derivative is globally invertible. The article presents new characterizations of this open problem rather than a proof that it is solved.
The paper discusses special cases and partial results toward the Jacobian conjecture. The claim that the conjecture is fully proved is not established here; instead, the document treats it as an ongoing research problem with conditional results and historical context.
Discussion from mathematicians notes that the conjecture is not known in general. The standard summary is that the Jacobian conjecture remains open for all dimensions greater than one, although several special cases have been proved. This is a community discussion, not a formal proof or official status statement.
A frequently cited answer by a professional mathematician explains that the Jacobian conjecture “remains open,” noting that many partial results exist (such as special cases and degree bounds) but “no general proof or counterexample is known.” The discussion emphasizes that several claimed proofs have turned out to be incorrect.
The Jacobian conjecture remains an open problem in mathematics as of 2026. It is proved in some special cases, including dimension two, but no accepted general proof is known.
This post reports a rumored proof of the Jacobian conjecture and then notes that a later update says the proof had a hole in it. The page is evidence of an attempted proof, not a verified resolution of the conjecture.
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Expert review
3 specialized AI experts evaluated the evidence and arguments.
Expert 1 — The Logic Examiner
The evidence overwhelmingly supports only the meta-claim that the Jacobian conjecture remains open (e.g., Sources 2, 5, 7, 8, 11), while the sole “support” (Source 3) is an unvalidated preprint about a generalized variant and the partial-case results (Source 6) do not entail the full conjecture, so the inference to “the conjecture is true” does not follow. Therefore the claim is false as stated, because “true” asserts a settled mathematical fact whereas the provided evidence (and standard status summaries) indicate it is not established/accepted as proved.
Expert 2 — The Context Analyst
The claim 'The Jacobian conjecture is true' omits the critical context that this is one of mathematics' most famous unsolved problems: as of 2026, no accepted general proof exists, it appears on Smale's list of 18 open problems for the 21st century, and multiple authoritative sources (Sources 2, 5, 7, 8, 11, 12) explicitly state it 'remains open.' The only supporting source (Source 3) is an unreviewed arXiv preprint about a generalized variant with no community validation, and Source 13 documents a prior claimed proof that turned out to have a hole. While the conjecture has never been disproven and is proven in special cases (degree ≤ 2), asserting it as simply 'true' without qualification creates a fundamentally false impression — the claim presents as settled fact what is universally recognized in the mathematical community as an open conjecture.
Expert 3 — The Source Auditor
The most reliable sources in this pool — including a peer-reviewed 2024 article in AIMS Mathematics (Source 2), Wolfram MathWorld (Source 5), a 2025 academic conference description (Source 7), and multiple arXiv preprints framing the problem as open (Sources 1, 4) — consistently and explicitly state that the Jacobian conjecture remains an unsolved open problem with no accepted general proof. The sole supporting source (Source 3) is an unreviewed arXiv preprint claiming a proof of a generalized variant without community validation or journal acceptance, which is far too weak to overturn the overwhelming consensus from higher-authority independent sources that the conjecture is not established as true.
Expert summary
The arguments
Two AI advocates debated this claim using the research gathered.
Argument for
Source 3 presents a preprint on arXiv claiming a complete proof of the Generalized Jacobian Conjecture over real and complex numbers, and Source 6 confirms the conjecture is already proven true for polynomial mappings of degree at most 2, establishing a meaningful partial truth that supports the broader claim. While the full general proof remains contested, the conjecture has never been disproven — no counterexample exists as confirmed by Source 1, which only describes constraints on what a counterexample would look like — meaning the claim that it is true is entirely consistent with all available evidence.
The Proponent's argument commits an argument-from-ignorance fallacy: the absence of a known counterexample in Source 1 does not constitute evidence that the conjecture is true, and Source 1 explicitly frames the problem as unresolved by merely constraining hypothetical counterexamples. More importantly, the Proponent treats an unrefereed arXiv claim about a generalized variant (Source 3) plus limited special-case results (Source 6) as if they settle the full conjecture, while ignoring that multiple higher-authority status references explicitly state the Jacobian conjecture “remains open in general” or “unsolved” (Sources 2 and 5, echoed by Sources 7, 8, and 11).
Argument against
Multiple independent, higher-authority references explicitly state the Jacobian conjecture remains open—e.g., AIMS Mathematics (Source 2) says it “remains open in general,” MathWorld (Source 5) calls it “unsolved,” and MathOverflow status discussions (Source 11) emphasize that no accepted general proof or counterexample is known and that past claimed proofs have failed. The lone “support” item is an unrefereed arXiv preprint asserting a proof of a generalized variant (Source 3) without community validation, which is far too weak to overturn the consistent consensus across Sources 2, 5, 7, 8, and 11 that the conjecture is not established as true.
The Opponent conflates 'unproven' with 'false,' committing a classic argument from ignorance fallacy — the absence of a universally accepted proof does not establish that the conjecture is untrue, and Source 1 explicitly confirms no counterexample exists, meaning the claim that the conjecture 'is true' remains logically consistent with all available evidence. Furthermore, the Opponent dismisses Source 3 as unvalidated while simultaneously treating community discussion forums such as Source 11 as authoritative consensus, applying an inconsistent evidentiary standard that selectively elevates informal MathOverflow threads over a formal mathematical preprint.