Breaking · 2026-07-19
On July 19, 2026, mathematician Levent Alpoge announced an explicit 3-dimensional counterexample to the Jacobian Conjecture — built with help from an AI model. If it holds up, it doesn't just settle one problem. It ripples through a web of 22 connected theorems and conjectures.
Every node below is real data — 23 nodes, 33 edges, drawn from the literature. Scroll to walk the story, or jump straight to the full "what falls?" breakdown.
1 · The conjecture
First posed by Ludwig Kraus in 1884 for two variables, then generalized by Ott-Heinrich Keller in 1939. It asks: if a polynomial map has a Jacobian determinant that's a nonzero constant everywhere, must it be reversible by another polynomial map?
It's Problem #16 on Stephen Smale's list of great math problems for the 21st century. For 87 years, no one has resolved it in 3 or more dimensions.
2 · The new counterexample
Alpoge's map F: ℂ³ → ℂ³ has Jacobian determinant identically −2 everywhere — constant and nonzero, exactly the JC hypothesis. Yet three distinct points, (0,0,−¼), (1,−3/2,13/2), and (−1,3/2,13/2), all map to the same output. That's non-injectivity: the map can't be inverted. If the arithmetic checks out under review, JC is false for n≥3.
3 · What falls with it
Dixmier and Poisson are mathematically proved equivalent to JC — not just related, literally the same statement in different clothing. If JC falls, they fall. Separately, Mathieu's Conjecture and Zhao's Image/Vanishing Conjecture program were proved to imply JC — if JC is false, these open conjectures can no longer be true either.
4 · What survives untouched
Wang's theorem (degree ≤2), Moh's verification (2 variables, degree ≤100), and the Jung–van der Kulk theorem (all 2-variable automorphisms are tame) are proved facts about restricted cases. A 3-dimensional counterexample doesn't retroactively unprove a 2-variable theorem — these stay exactly as true as they were.
5 · Already false, for unrelated reasons
The Markus–Yamabe conjecture and the naive Real Jacobian Conjecture were both disproved years ago, by explicit counterexamples unrelated to Alpoge's. They're structurally similar stories — dimension 3 is where things break — but their falsity isn't news, and isn't caused by the 2026 announcement.
6 · The implications explorer
This isn't hand-picked — it's computed live from the 33 edges in the dataset, walked outward from the new counterexample node. Tap any bucket to expand it, tap any node for full detail and sources.
7 · Why it matters outside pure math
JC has stayed open for so long that some adjacent fields quietly treat it as "probably true." A confirmed counterexample doesn't break any deployed system — nothing was built assuming JC as a security proof — but it does close off JC-based reasoning in multivariate cryptography design discussions, computer-algebra inversion heuristics, and the Markus–Yamabe-adjacent stability literature in control theory.
7.5 · Why should you care?
8 · Explore the full graph
Search or tap any node below for its full statement, status, applications, and sources.
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