The Map That
Broke Algebra

We report and dissect a live mathematical event. On July 19, 2026, number theorist Levent Alpoge announced an explicit polynomial map F : ℂ³ → ℂ³ with constant Jacobian determinant −2 that sends three distinct points to a common image — a counterexample to the Jacobian conjecture (Keller, 1939), found in collaboration with an AI model. In this thread we independently verified both certificates three ways; constructed the explicit downstream counterexamples the equivalence literature predicted, refuting the Poisson conjecture on ℂ⁶ and the Dixmier conjecture for the Weyl algebra A₃; reverse-engineered the map's internal mechanism (a hidden ℂ* symmetry, a plane map that folds along a line it contracts, and a compensating fiber scaling); and identified why the announced family of counterexamples begins at fiber degree 3 — a 1973 theorem of Campbell, newly sharp, which our computations confirm the map threads exactly, with full S₃ monodromy generated entirely at infinity. We close with the research program this opens.

A Conjecture Dies on a Sunday Night

Some conjectures die of old age, ground down by decades of partial results until the final step is almost a formality. The Jacobian conjecture did not die that way. It died the way a bridge fails: suddenly, publicly, and from a direction nobody was watching. On the evening of July 19, 2026 — while much of the world watched the World Cup final — a post appeared giving a single explicit polynomial map from ℂ³ to itself, together with two finite claims about it. Either claim can be checked by a patient undergraduate. Together, they end an 87-year-old problem.

The Jacobian conjecture asked something that sounds self-evidently true. Take any polynomial map F : ℂⁿ → ℂⁿ. Its Jacobian determinant measures local volume distortion; if that determinant is a nonzero constant, the map is everywhere a local isomorphism — no folding, no crushing, anywhere, at any point. The conjecture asserted that such a map must then be globally invertible, with polynomial inverse. Local perfection should imply global perfection. Every low-degree case said yes. The planar case was verified through degree 100. The claim survived every attack for 87 years — and accumulated an ecosystem of reductions, equivalences, and partial theorems premised on its truth.

The counterexample, which we will call Alpoge's map, is the triple F = (F₁, F₂, F₃) of degrees 7, 6, and 4:

The disproof consists of exactly two certificates, and their brutality is the point: no limits, no estimates, no cohomology — just polynomial arithmetic that a computer algebra system confirms in under a second.

One Object, Three Conjectures, Zero Survivors

The Jacobian conjecture did not live alone. Two decades of work — Tsuchimoto, Kanel-Belov and Kontsevich, Adjamagbo and van den Essen — had welded it to two other famous open problems: the Dixmier conjecture (every endomorphism of a Weyl algebra Aₙ is an automorphism) and the Poisson conjecture (every polynomial endomorphism of ℂ²ⁿ preserving the standard Poisson bracket is an automorphism), through the implication chain JC₂ₙ ⇒ PCₙ ⇒ DCₙ ⇒ JCₙ. Those equivalences were built as ladders toward proof. Run in reverse, they are demolition charges.

But abstract contrapositives satisfy nobody, so in this thread we built the downstream counterexamples explicitly — and the construction turned out to be one line. Because det J_F ≡ −2 never vanishes, the inverse-transpose Jacobian (J_Fᵀ)⁻¹ = adj(J_Fᵀ)/(−2) has polynomial entries. The cotangent lift

is then a polynomial map that pulls back the canonical 1-form p·dx to itself exactly. We machine-verified all fifteen Poisson brackets among its six components — every one comes out canonical — and exhibited three distinct points of ℂ⁶ (the three collision points, with momenta zero) sharing one image. A non-injective Poisson endomorphism: the Poisson conjecture is false on ℂ⁶. And because Φ is symplectic, its own Jacobian determinant is exactly 1 — so Φ is simultaneously a second Jacobian counterexample, in dimension six, in Keller's original unimodular normalization.

One altitude higher, the same matrix B = (J_Fᵀ)⁻¹ quantizes the construction. Defining φ(xᵢ) = Fᵢ and φ(∂ⱼ) = Σₖ Bⱼₖ(x)∂ₖ on the third Weyl algebra, we machine-verified every Weyl relation: the commutator matrix [φ(∂ⱼ), φ(xᵢ)] is exactly the identity, and the lifted derivations commute — the identity that makes them commute is literally the same polynomial identity that made the Poisson brackets vanish. The endomorphism φ is injective automatically (A₃ is simple), and a short filtration argument shows it preserves operator order exactly, which yields an explicit element outside its image: the coordinate x₁ itself. If x₁ = φ(u), order forces u ∈ ℂ[x], hence x₁ = c(F) identically — but evaluating at the collision points makes c(F) take one value where x₁ takes the values 0 and 1. So the Dixmier conjecture is false for A₃: an injective, non-surjective endomorphism, written out in full with integer and half-integer coefficients. The three-point collision does the killing in two lines.

How the Machine Actually Works

A counterexample this clean is not a lightning strike; it is engineering. When we opened the map up, the formulas collapsed dramatically. Writing u = 1+xy — and noticing that the mysterious 4+3xy is just 3u+1 — the whole map is one line: with w = u²z + (3u+1)y²,

Beneath that compression sits a hidden symmetry. Assign the variables the weights (1, −1, −2) and the target the weights (−2, −1, 1): then F is exactly equivariant for the ℂ* action t·(x,y,z) = (tx, t⁻¹y, t⁻²z). The invariants of that action are generated by u = 1+xy and v = x²z — and F descends to a two-variable map on the quotient plane. On the chart x ≠ 0, in coordinates (x, u, v), the whole three-dimensional map is triangular:

Everything that matters happens in the base map Q, and we verified three facts about it. It is generically 3-to-1 (Gröbner quotient dimension = 3, matching the fiber degree of F itself). It contracts the entire line L = {h = 0} to the single point (1, 0). And its Jacobian is det J_Q = 2h² — vanishing to order exactly two along the line it crushes. Now the volume bookkeeping closes with a click: the fiber direction is scaled by h, the base is folded with degeneracy h², and

The fold of the base and the collapse of the fiber cancel each other exactly. The map is singular downstairs and singular along the fiber, and the two singularities multiply to a constant. That is the entire trick — a local isomorphism everywhere, welded together out of two things that are each degenerate.

Non-injectivity is then not a sporadic accident but a designed structure. The ℤ/2 inside the torus (x ↦ −x) pairs points; the pairs collide wherever the last two output coordinates vanish; and that locus is an entire ℂ*-orbit — the curve s ↦ (s, −3/2s, 13/2s²), which we verified maps to (−1/4s², 0, 0) for every s, folding two-to-one onto the punctured axis, while the z-axis (F(0,0,z) = (z,0,0)) supplies a clean third sheet. As the target slides to the origin, s → ∞: the extra preimages escape to infinity. Bézout says the fibers "should" contain 7·6·4 = 168 points; the true count is 3. The other 165 intersections — and all the sheet-merging — live on the hyperplane at infinity, which is precisely where the Jacobian hypothesis says nothing at all.

And the deepest lesson: the construction converts a rigid problem into a flexible one. Building a 3D map with constant Jacobian became, downstairs, building a 2D map with prescribed non-constant Jacobian 2h² — a soft problem with plenty of solutions, exactly the freedom that Keller's two-variable problem lacks. Eighty-five years of planar rigidity theorems simply do not apply to Q, because Q never claimed a constant Jacobian.

The 53-Year-Old Theorem Left Standing

The announced family of counterexamples reportedly exists for every generic fiber degree n ≥ 3 — and conspicuously not for n = 2. We went hunting for the reason, expecting to find a new theorem waiting to be proved. What we found instead was better: an old one, waiting to be promoted. In 1973, L.A. Campbell proved that a Keller map whose function-field extension ℂ(x)/ℂ(F) is Galois must be an automorphism (Razar, and independently Wright, later extended this to general fields). And in characteristic zero, every degree-2 extension is automatically Galois. Degree 2 forces Galois; Galois forces invertible; invertible forces degree 1. Fiber degree 2 is impossible — and has been, quietly, for 53 years.

The proof skeleton deserves to be seen, because it is beautiful. A degree-2 cover carries a deck involution σ with F∘σ = F. At any fixed point the chain rule gives J_F · J_σ = J_F, and J_F is invertible — that is the Keller hypothesis doing real work — so J_σ = Id; but a finite-order map with identity differential at a fixed point is the identity near it (Cartan's linearization), so a nontrivial σ must act freely. And P.A. Smith's fixed-point theorem forbids a finite group from acting freely on a contractible space like ℂⁿ. Contradiction. (The technical heart of Campbell's paper is making σ regular enough for this to bite; the skeleton above is our reconstruction of its shape.)

So the true statement of affairs, as of this weekend, is a sharp dichotomy that could not have been stated before: the degree spectrum of Keller maps is {1} ∪ {3, 4, 5, …}. Degree 1 always (automorphisms); degree 2 never (Campbell–Razar–Wright); and — for the first time in 87 years — the second half of the spectrum is known to be inhabited.

Non-Galois by Necessity — and Verifiably So

Campbell's theorem does more than forbid degree 2: it forbids any Galois counterexample. A degree-3 extension is Galois exactly when its group is ℤ/3, exactly when its discriminant is a square. So the theorem makes a hard, checkable prediction about Alpoge's map: its monodromy must be the full symmetric group S₃. The map had no choice. We checked.

Working on the quotient plane, elimination produced the minimal polynomial of u = 1+xy over the target field — and it came out startlingly clean. Writing A = 1+bc and B = ac² for the target invariants, the three base values of u over any fiber satisfy a depressed cubic (the three sheets of every fiber obey u₁+u₂+u₃ = 0 — check it on the collision fiber: 1 − ½ − ½ = 0):

Its discriminant factors exactly as

The odd-multiplicity factor makes Δ a non-square in ℂ(A,B): the monodromy contains a transposition, and with irreducibility it is all of S₃ — non-Galois, precisely as demanded. But look at which factor carries the odd multiplicity: it is the leading coefficient of the cubic — the locus where a root escapes to infinity, the image of the non-properness set. The finite branch locus {s = 0} enters squared, so monodromy around it is even; the odd elements that break Galois-ness — the elements that make the counterexample possible at all — are generated only by loops around the escape divisor. The map is non-Galois because of its behavior at infinity: the same horizon that hides the merging sheets, evades every local obstruction, and slips through Campbell's fingers. One mechanism, three escapes.

What Grows in the Rubble

Three days ago, the Jacobian conjecture was an 87-year-old fortress. Today it is a quarry — and the stones are already being carried off to new construction sites. The core lesson your infographic put well deserves restating as the epitaph: local checks can look perfect while global behavior still fails. Every finite-distance test this map faces, it passes flawlessly; its pathology lives entirely at infinity. Any field that infers global structure from local regularity — and that includes swaths of geometry, dynamics, and the theory of invertible computation — has just been handed the sharpest possible cautionary specimen.

What should be studied next? The demolition itself defines the program. The most striking feature of the aftermath is an inversion of status: theorems once filed as "partial progress toward the conjecture" (Campbell's Galois case, Moh's planar bounds, the properness criteria) are now the load-bearing walls of a new subject — the structure theory of Keller counterexamples. The table below is our honest assessment of where the open ground lies.

Keller asked his question in 1939 believing, like nearly everyone after him, that the answer was yes. The answer was no — but only barely, only in dimension three and above, only at infinity, and only by a mechanism so delicate that two exact degeneracies must cancel to the last coefficient. The conjecture was wrong the way a great conjecture should be wrong: in a manner more interesting than being right.