The Jacobian Conjecture for N = 2

There is a folder in one of my repositories that I keep coming back to. It holds a research workspace for a single open problem: the two-dimensional complex Jacobian conjecture, which everyone in the area writes as JC(2). Thirty files, a dozen exact verifier scripts, and two ledgers whose only job is to stop me from believing my own conclusions.

In July 2026 that folder had to be rewritten. On 20 July the Jacobian conjecture — open since 1939 — was disproved in three variables by a 216-character polynomial map posted to X. It did not touch the plane. What had been “the last unresolved dimension of a conjecture most people expected to be true” became “the only surviving fragment of a conjecture that is now known to be false almost everywhere.”

Updated 8 August 2026. The complex plane case remains open. Since this article was first published, the three-dimensional example has acquired an independent formal verification and a self-contained arXiv treatment that produces whole new families above the plane. A separate two-variable counterexample has appeared in characteristic two; it does not address the characteristic-zero conjecture discussed here. The new results and their evidence boundaries are folded into the account below.

I want to walk through all of it: what the conjecture claims, why it is harder than it looks, exactly how three variables fell, why that trick provably cannot be pushed down to two, and what is actually known about the plane today. Most of what follows is other people’s mathematics, carefully sourced. The parts that are ours are marked as such, and the parts that failed are kept in the open where they can keep teaching.

What the conjecture actually claims

Take a field $k$ of characteristic zero — think $\mathbb C$, or $\mathbb Q$, or $\mathbb R$ — and a polynomial map

meaning each coordinate $F_i$ is an honest polynomial in $x_1,\ldots,x_n$. No square roots, no fractions, no infinite series.

Its Jacobian matrix is the matrix of first partial derivatives. In two variables, writing $F=(P,Q)$:

The entries are themselves polynomials, so $\det JF$ is a polynomial too. The Keller condition demands that this polynomial be a nonzero constant:

That is a much stronger demand than it first sounds. A polynomial is a nonzero constant only if it has no roots at all, anywhere in $k^n$. So the Keller condition says the derivative of $F$ is invertible at every point of the space, with no degeneration in any direction, no matter how far out you go.

Now, where does that condition come from? It is not arbitrary — it is forced. Suppose $F$ really is a polynomial automorphism: invertible, with an inverse $G$ that is also a polynomial map. Differentiating $G\circ F=\mathrm{id}$ by the chain rule gives

so $\det JF$ is a polynomial that divides $1$ in the polynomial ring, hence a nonzero constant. The Keller condition is therefore a necessary condition for polynomial invertibility, obtainable in one line.

The conjecture asked whether it is also sufficient:

Jacobian conjecture. If $F:k^n\to k^n$ is a polynomial map with $\det JF\in k^*$, then $F$ is a polynomial automorphism.

One variable is a one-liner

In $n=1$ the whole thing collapses immediately. $f’(x)=c\ne0$ is a constant, so $f$ has degree $1$:

Done. JC(1) is true, and the proof fits on a napkin.

Two variables already has interesting maps

It helps to see that the hypothesis is not vacuous — there are plenty of nontrivial planar automorphisms. Take

Its Jacobian matrix is triangular with ones on the diagonal, so $\det JF\equiv1$, and its inverse is visibly polynomial:

More generally, every triangular map $(x+g(y),\,y)$ and every invertible linear map satisfies the Keller condition, and so does any composition of them. Jung and van der Kulk proved in the plane that these generate everything: every polynomial automorphism of $k^2$ is a finite composition of linear and triangular maps. So we know exactly what the automorphisms look like. What nobody could rule out was a Keller map sitting outside that list.

Why this is not just the inverse function theorem

The natural first reaction is that calculus already answers this. Nonzero derivative everywhere implies locally invertible everywhere — that is the inverse function theorem — so surely $F$ is invertible.

It is not, and the gap is instructive. Local invertibility at every point gives you a map that is a local homeomorphism. To upgrade that to a global bijection you need one more ingredient: properness, meaning that preimages of bounded sets stay bounded. A proper local homeomorphism is a covering map, and a covering of the simply connected space $\mathbb C^n$ is a bijection. So the Keller condition plus properness really does give invertibility.

Everything therefore hinges on non-properness. A Keller map can fail to be injective only if points run off to infinity while their images stay put. Two preimages cannot merge at a finite point — merging would create a critical point and kill the determinant — but a sheet can drift out to infinity and reappear elsewhere. That escape hatch is invisible to any purely local argument, and it is precisely the hatch the counterexample uses.

There is a second warning sign that had been sitting in plain view for decades. Weaken “nonzero constant” to “nowhere zero” and the plane statement is already false over $\mathbb R$: Pinchuk constructed in 1994 an explicit polynomial map $\mathbb R^2\to\mathbb R^2$ with everywhere-positive Jacobian determinant that is not injective. The complex conjecture survived only because $\det JF$ nowhere zero over $\mathbb C$ forces it to be constant, which the real case does not.

A short history of not proving it

The statement is usually credited to Ott-Heinrich Keller, who posed it in 1939 for polynomial maps with integer coefficients. It has an older ancestor: a paper of L. Kraus from 1884 states essentially the same result, with a flawed final step, and — as Rodríguez Díaz’s recent reconstruction shows — ideas that only resurfaced a century later.

Since then it has accumulated a reputation. It landed on Smale’s list of problems for the twenty-first century. At least five published proofs have been wrong, and the count of unpublished attempts is not worth estimating. The problem is unusual in that the community never converged on an expected answer: unlike the Riemann hypothesis, where almost everyone bets one way, informed opinion on the Jacobian conjecture was genuinely split.

What the field did produce was a remarkable set of reductions, all of which say “you only have to handle this restricted shape”:

• Wang. If every coordinate of $F$ has degree at most $2$, the conjecture holds — in every dimension.
• Yagzhev (1980), Bass–Connell–Wright (1982). If the conjecture holds for every map of the form $F(x)=x-H(x)$ with $H$ homogeneous of degree exactly $3$, it holds for all degrees and all dimensions.
• Drużkowski. It suffices to treat cubic-linear maps, where each component of $H$ is the cube of a linear form.
• Connell–van den Dries. If a counterexample exists at all, one exists with integer coefficients and $\det JF=1$.

Each of these is a real theorem, and every one of them has the same catch: the reductions raise the dimension. Cutting the degree to three costs you extra variables. So none of them ever said anything about a fixed $n$, and in particular none of them ever said anything about $n=2$. That footnote was a curiosity for forty years. In July 2026 it became the whole story.

July 2026: dimension three falls

The sequence of events is short enough to state exactly.

Akhil Mathew asked Levent Alpöge about the problem. Alpöge put it to Fable, Anthropic’s frontier model, over the weekend of the World Cup final. On 20 July 2026 he posted the result to X, in its entirety:

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

Followed by the map. Define $F=(A,B,C):\mathbb C^3\to\mathbb C^3$ by

Two facts about it, and they are logically independent. First, exact symbolic differentiation gives a global polynomial identity:

Second, a finite collision certificate:

Three distinct source points, one image. A Keller map that is not injective. The conjecture is false in dimension three.

You can check it yourself in ten lines

This is the part I most want to emphasize, because it is what makes the result unusual. The claim is not a hundred-page argument requiring three referees and two years. It is a finite computation in a polynomial ring, and anyone with a computer algebra system can replay it in a few seconds:

import sympy as sp

x, y, z = sp.symbols("x y z")
A = (1+x*y)**3*z + y**2*(1+x*y)*(4+3*x*y)
B = y + 3*x*(1+x*y)**2*z + 3*x*y**2*(4+3*x*y)
C = 2*x - 3*x**2*y - x**3*z
F = sp.Matrix([A, B, C])
print(sp.factor(F.jacobian([x, y, z]).det()))  # -2

pts = [
    (0, 0, -sp.Rational(1, 4)),
    (1, -sp.Rational(3, 2), sp.Rational(13, 2)),
    (-1, sp.Rational(3, 2), sp.Rational(13, 2)),
]
for p in pts:
    print(tuple(sp.simplify(v.subs(dict(zip((x, y, z), p)))) for v in F))

Note what checking the three points does not do. It certifies noninjectivity and nothing else. The Keller condition is a statement about every point of $\mathbb C^3$, so it has to be verified as an identity in the polynomial ring, not sampled. Restricting to $x=0$, for instance, makes the matrix small enough to read:

Because the coefficients are integers and the colliding points are rational, the same map also refutes the corresponding statement over $\mathbb R$ and over $\mathbb Q$.

What happened next

Within a day, Terence Tao published a digestion of the construction; the Secret Blogging Seminar posted an independent analysis; Zihan Zhang wrote up the direct consequences. The arithmetic held everywhere it was checked, and an Archive of Formal Proofs entry now verifies the determinant identity and the three-point collision independently in Isabelle/HOL.

Those consequences reach past the conjecture itself: the Mathieu conjecture for $\mathrm{SU}(3)$ falls with it, and Zhao’s vanishing conjecture and the image conjecture fail in some finite dimension. Meng and Yang have since used a Schur descent of the map to give a five-variable counterexample to the Hessian conjecture, leaving only the four-variable Hessian case open and linking it to JC(2). An August preprint on the weak Markus–Yamabe conjecture turns the same collision into a degree-seven Hurwitz vector field in fourteen real variables with three rational singularities. These are real mathematical aftershocks, not progress on the complex plane case.

A reported infinite family $F_m$, one for each $m\ge2$, with

preimages over a collision point, has also circulated. My corpus keeps its formulas as a research lead and has not independently verified every member.

The largest follow-up arrived on 31 July. Shuhong Gao’s self-contained arXiv preprint identifies the mechanism as a tangent sweep and generalizes it from tangent lines of plane curves to direction fields on hypersurfaces. It produces counterexamples in every dimension greater than two with arbitrarily large geometric degree, and works out five new maps: a three-dimensional map of degree four, two four-dimensional maps of degrees five and ten, and two five-dimensional maps of degrees six and twelve. The paper checks its identities in exact rational arithmetic and uses Gröbner bases to determine exact fiber structures.

That changes one interpretation of the first example. Its three-sheet fiber was not a low-multiplicity accident marking the edge of what counterexamples can do: above the plane, the same geometry supports as many generic sheets as one asks for. But Gao’s construction still starts in dimension three and explicitly applies only in dimensions greater than two. It expands the world on the far side of the wall; it does not lower the wall.

Two caveats still belong in the same breath as the result. There is now an arXiv treatment and an independent formal verification, but no journal peer review. Gao’s paper is a mathematical account of the construction, not a publication of the original discovery session, and the exact model, prompts, and transcript behind that session remain undisclosed. The mathematics is checkable; the provenance story is still testimony.

How the counterexample actually works

A map with three coordinates and a hundred monomials sounds like a lucky collision among hundreds of cancelling terms. It is nothing of the kind. In the right coordinates it is almost embarrassingly transparent, and understanding why it works is exactly what tells you it cannot be pushed into the plane.

On the dense chart $x\ne0$, substitute the root coordinates

In these variables the target collapses to

which is a disguised way of saying two things at once. First, $t$ is a root of the cubic

and second,

is the derivative of that cubic evaluated at the marked root.

So the map is: take a cubic, and remember one of its three roots. That is where noninjectivity comes from for free — a generic cubic has three roots, so three source points sit over each target point.

Now watch the determinant. In root coordinates the two Jacobians factor as

and since $r=2/x$ these are reciprocal:

That is the entire mechanism. Marking a root of a cubic naturally produces a determinant factor equal to the root-derivative $r$ — which vanishes exactly where roots collide, and would ordinarily destroy the Keller condition. The construction survives by routing that factor through an independent scale coordinate $x=2/r$, which contributes the exact reciprocal. The obstruction is not removed; it is paid for out of a separate account.

Tao’s reformulation makes the same point coordinate-free. Identify linear polynomials in two variables with a $2$-dimensional space, quadratics with a $3$-dimensional space, and cubics with a $4$-dimensional space, and consider multiplication

When a cubic splits into three independent linear factors $L_1L_2L_3$, the three pairs $(L_1,L_2L_3)$, $(L_2,L_1L_3)$, $(L_3,L_1L_2)$ all multiply to it, and they are not related by any rescaling symmetry. That is the degree-$3$ noninjectivity. Imposing the resultant normalization

kills the rescaling ambiguity and removes the ramification divisor at once, leaving a map that is étale everywhere on an affine source.

Finally, the escape hatch. The cubic $cT^3-2T^2+BT-2A$ drops degree when $c=0$. In the collision fiber two roots stay finite and the third is naturally read as a root that has gone to infinity. That is how noninjectivity coexists with a nowhere-vanishing finite Jacobian: the sheets never meet in the affine plane; they are reconciled at the boundary.

Why three variables is not two

Everything above used a resource that the plane does not have, and the failure is structural rather than a matter of insufficient cleverness.

Look at the symmetry. The source of the three-variable map carries the multiplicative action

whose invariants are $v=xy$ and $t_0=x^2z$, with $x$ playing the role of a pure scale coordinate. So dimension three decomposes as

with the cubic’s multi-sheeted geometry living on the two shape directions and the reciprocal derivative factor riding on the scale direction.

In the plane, the analogous action

leaves only the single invariant $xy$. Spend a direction on scale and you have one shape direction left, which is not enough for interesting sheet geometry. Keep both directions for shape and there is nowhere to put the derivative factor.

That is the trilemma the planar shadow of this construction keeps running into:

1. remove the resultant divisor — the map becomes étale, but the source acquires a nonconstant unit and stops looking like $\mathbb A^2$;
2. fill the divisor — the source becomes plane-like, but the determinant vanishes on the filled curve;
3. normalize the divisor — in dimension three the factor-rescaling direction absorbs this without costing a dimension; in dimension two there is no spare direction.

The projective construction confirms the same thing from above. The analysis in the Secret Blogging Seminar discussion computes the relevant class-group obstruction and finds it unimodular only at $d=3$; in dimension $d\ge4$ a $\mathbb Z/(d-2)\mathbb Z$ Picard obstruction prevents the complement from being affine space. In other words the geometry threads the needle exactly once. The counterexamples in dimensions $n\ge4$ are not new geometry at all; they come from padding:

You can pad upward. There is no operation that pads downward.

In dimension three, a derivative factor can be routed through an independent scale direction while two shape variables keep their multi-sheet geometry. In dimension two, every attempted routing has to be reconciled on a boundary lattice or on a high-genus punctured fiber.

A proof of JC(2) must show that reconciliation is impossible in every degree. A counterexample must exhibit the first boundary architecture where it succeeds. Everything below is one of those two errands.

What is left: the plane

So the fixed-dimensional scoreboard now reads

Exactly one case remains:

There is no loss in normalizing the constant: if $J(P,Q)=c$, replace $Q$ by $Q/c$ and assume $J(P,Q)=1$. In differential-form language the same condition reads $dP\wedge dQ=dx\wedge dy$ — the map is area-preserving in the algebraic sense — and in Poisson-bracket language it is simply

Two variables. One equation. Open since 1939, and now alone.

What would actually count as an answer

It is worth being precise about the finish line, because a lot of near-misses fail here rather than in the mathematics.

A counterexample is finitely certifiable — this is exactly what dimension three just demonstrated. The ideal artifact is explicit $P,Q\in\mathbb Q[x,y]$ with $J(P,Q)\equiv1$, plus explicit distinct points $p\ne q$ with $(P,Q)(p)=(P,Q)(q)$ — or a rigorous field-degree or non-surjectivity certificate in place of the collision.

A proof is harder to certify honestly. It has to be uniform in degree and cover arbitrary planar Keller maps. Eliminating weighted, symmetric, cubic-root, or degree-bounded families is not enough unless a universality theorem forces every hypothetical counterexample into the eliminated class. That missing universality step is where most of the interesting partial results in this folder actually sit, and saying so out loud is most of what keeps the corpus useful.

Counting sheets: the field-degree ladder

The single most organizing invariant for the plane case is the generic field degree. A warning about notation first, since the two meanings of “$N$” collide constantly in this subject: from here on $n$ is the number of variables — fixed at $2$ — and $N$ counts sheets.

Write $K=\mathbb C(P,Q)$ inside $L=\mathbb C(x,y)$ and set

Concretely, this is the number of points in a generic fiber, counted before ramification — how many sheets the map has. $N=1$ is the birational case, and there the Keller condition forces invertibility. So a counterexample needs $N\ge2$, and the last forty years have been a slow climb up that ladder:

• $N=2$ falls to the finite-normalization and Galois argument.
• $N=3$ is excluded by Orevkov’s three-sheet theorem.
• $N=4$ is excluded by the Domrina–Orevkov boundary-diagram theorem, in two parts.
• $N=5$ is excluded by Żołądek.

So any counterexample satisfies

For calibration: the three-variable counterexample has $N=3$. The mechanism that just broke the conjecture in dimension three sits two full rungs below where a planar counterexample would have to start.

That last rung is also where this corpus learned a lesson about itself. For a long stretch the workspace was organized around degree $5$ as the first open case, with two entire files and a long chain of numbered findings devoted to the dihedral $D_5$ frontier. An independent audit — run by an AI model, over the whole corpus, with instructions to check citations rather than extend them — found the Żołądek result in a published reference list and reported the consequence plainly: the frontier was $N\ge6$, and those two files were beautiful mathematics about an empty case.

Nothing in them was wrong. They were simply aimed at a case that no longer existed. That correction was worth more than the new lemmas that came with it, and it is the reason the ledger discipline described at the end of this post exists.

The normalization picture

Here is the structure almost every serious attack uses. Put

and let $S$ be the integral closure of $A$ inside $L$ — the finite normalization. Geometrically, $X=\operatorname{Spec} S$ is a normal affine surface carrying a finite map $\pi:X\to\mathbb A^2$ of degree $N$, and the original plane sits inside it as an open subset whose complement is a boundary divisor $D$. In fact JC(2) is equivalent to the statement that no such $X$ exists with $N\ge2$ and an open embedding $\mathbb A^2\hookrightarrow X$ making $\pi$ étale on that open part.

This is a good trade. A Keller map is not a finite morphism — that is the whole problem — but $S$ is finite and flat over $A$, free of rank $N$ by Quillen–Suslin, and normality lets codimension-one valuations control everything. All the failure gets pushed into the boundary $D$, where it becomes divisor combinatorics.

Two structural facts survive the trade. Since the map is étale on the affine part, every branch curve downstairs must be met by both an unramified affine sheet and a ramified boundary sheet — the mixed-sheet condition, which is the real bridge from the Keller hypothesis into finite group theory. And the normalization of a counterexample cannot be globally monogenic, cannot have a principal different, and cannot be Galois: each of those would force invertibility outright.

The Galois closure of $L/K$ carries a monodromy group $G$ acting transitively on $N$ sheets, generated normally by the inertia of the boundary divisors. At $N=6$, exactly $6$ of the $16$ transitive subgroups of $S_6$ survive the inertia filter — an exact computation, not a sample.

A budget at the boundary

The sharpest external input is a corollary of Orevkov’s that the corpus had cited for years without using. If $e_i$ is the ramification index along boundary divisor $D_i$, then

That single inequality cascades. With $k$ the number of boundary components and $b$ the number of ramified ones:

Every ramified boundary divisor costs at least one unit of a budget that never grows with degree — and the class group of the normalization is free on the boundary components, so the same bound caps its rank.

Then comes the honest limitation, which was found by trying to break the result rather than to extend it. The obvious hope is that high degree needs more ramification than the budget allows. It does not. A transposition costs exactly $2$ units for every $N$, so a hypothetical counterexample can keep its ramification bounded while its degree runs away. The budget is real and it is not asymptotically binding. Recording that as a negative result rather than leaving it as an open direction is, I think, the single highest-value habit in the whole workspace.

A residue-weighted refinement sharpens the same idea, with $f_i$ the residue degree and $\kappa\ge0$ an Orevkov defect:

and a collision bound says that if a point of an image curve has $q$ normalization preimages then

In degree six with an irreducible nonproper set, those two together cut the monodromy group down to $A_6$ or $S_6$. That is a very small target, and it still has not been closed.

Where the map fails to be proper

This is the section that the three-variable counterexample retroactively promoted, because non-properness turned out to be the entire mechanism there.

A noninvertible Keller map cannot be proper, and its failure is visible in the target as the nonproper-value set $A_F$: the points approached by images of source points escaping to infinity. A counterexample exists exactly when this set is nonempty, which makes it the most Keller-specific object available.

The corpus’s own contribution here is a clean identification, $A_F=\pi(D)$ — the nonproper set is precisely the image of the boundary. Combined with the budget above:

Externally, $A_F$ is known to have all of its branches meeting the line at infinity at one point, and a polynomial Vitushkin argument shows that a counterexample’s nonproper set cannot be homeomorphic to $\mathbb C$. So the curve must have a self-intersection or several branches — for an irreducible $A_F$, its parameterization is provably non-injective, a genuine self-crossing rather than a cusp.

On a generic fiber the bookkeeping becomes an identity. With $\Lambda$ the total ramification, $g$ the genus of the fiber’s smooth completion and $r$ its number of punctures,

proved three independent ways in the corpus, and later sharpened: separating finite-valued punctures from poles on an adapted fiber gives

which forces $r\ge3$ unconditionally.

One component at a time

Everything so far constrains the pair $(P,Q)$ or the surface $X$. The most recent turn constrains one component on its own, and it is my favorite thing in the folder.

Fix a Keller component $P$ and consider the Gelfand–Leray form on a fiber $Y$ of $P$:

The Keller condition says exactly that $\omega$ is regular and nowhere zero on every component of every fiber — and that the partner $Q$ restricts to a regular primitive of it, $dQ|_Y=\omega|_Y$. In other words, $Q$ is the flow time of a form determined by $P$ alone.

That has real teeth. If $P$ is a Keller component then $\omega$ must be exact on every fiber component: all residues and all periods vanish. And the degree itself turns out to be computable from $P$ without ever mentioning $Q$:

so total pole order $1$ means automorphism, and a counterexample needs pole order at least $6$. Applied to a family that looks perfectly innocent — nowhere-vanishing gradient, smooth fibers, irreducible generic fiber — it kills all of it at once:

The pleasant irony is that the corpus’s own long-standing “hard counterexample”, $x(1+xy)$, turns out not to be a Keller component either. It had been blocking a line of attack for months by pure misclassification.

Degree six, braids, and models that saturate everything

The current frontier is where I find the honest character of the problem clearest. In the last remaining sextic branches, the surviving arithmetic is so tight that people can write down curves matching every numerical budget at once — plane degree, genus, conductor, ramification $5$, defect $3$, pole band, collision type — and then watch them die for a reason no counter could see.

A degree-ten rational curve realizes the complete local, plane, and pole ledger; its auxiliary degree-five projection then fixes no transitive marking, and it is gone. An exact rational sextic saturates both the genus and the ramification–defect ledger; its infinity braid fixes none of the complete transitive Nielsen classes of sizes $15625$, $20480$, and $21870$, and it is gone. Exhausting all $17$ genus-allowed one-place sextic types leaves only smooth infinity, and a complete signed census of self-carousels closes that too.

The lesson repeats with unusual regularity: the scalar budgets are sharp, and sharp is not the same as sufficient. Every time the numbers alone are made to fit, the obstruction moves one level up — into full braid factorization, or into whether the configuration is realizable by an actual plane curve at all.

The coefficient campaign

A parallel line attacks coefficients directly. The first vertical profile not removed by the audited degree screen is

Writing the leading coefficients as $a_6=\alpha h^2$ and $b_9=\beta h^3$, the first negative fractional-root coefficient forces an exactness condition on a cube root of $h$:

Squarefree nonconstant $h$ dies immediately. Repeated $h$ collapses into a classical Moh/Formanek–Stothers polynomial differential equation plus a finite Belyi passport problem, and the full bracket system over $\mathbb C(x)$ reduces to

Polynomiality and $3\mid\deg H$ force $H$ to be a cube, which removes the entire noncube branch — including all nine enumerated low-degree passports — without assuming polynomial descent of the approximate roots. Inside the surviving cube branch, rationality forces a nonconstant leading cube root to be a single-root power $g=\lambda(x-b)^s$ with $s\ge2$, and a long sequence of tropical and weighted-boundary arguments closes fiber after fiber of the coefficient tail.

These are real exclusions, verified by exact enumerations running into tens of thousands of cells. They are also bounded: no theorem yet places every hypothetical Keller map into the vertical $(6,9)$ profile. Which is the same missing universality step as before, wearing different clothes.

The degree wall, and the certificate that is not here

There is one more ladder, this one in ordinary polynomial degree rather than sheet count. Moh proved in 1983 that no planar counterexample exists below maximum degree $100$; the published Newton-polygon program has since pushed that to $108$, and below maximum degree $125$ it leaves exactly one surviving degree pair and its transpose:

An external note reports exact ideal-membership certificates excluding the two residual systems, which would raise the bound to $125$. The corpus records the equations, the construction dimensions, the file hashes, and the reported output — and states plainly that the roughly 89 MB certificate archive is not present locally and cannot be replayed.

I like that entry more than most of the proofs. It is a result recorded at exactly the confidence its evidence supports, with the boundary of that evidence written down next to it.

It is also a useful contrast with what just happened one dimension up. The three-variable counterexample fits in a tweet. If a planar one exists, the degree wall says its coordinates have at least seventy-two terms’ worth of degree before you start counting monomials. Nobody is going to stumble onto it.

Reading the news

Since 20 July the signal-to-noise ratio around this problem has been poor, and separating the two is now part of the work. Here is what a careful sweep of arXiv, X, MathOverflow, formal-proof archives, and linked research records found through 8 August 2026.

No disclosed proof or counterexample for the plane. Repeated indexed searches on X for “JC(2)”, “plane Jacobian conjecture”, “Jacobian conjecture in dimension two”, and “proof” return the three-dimensional announcement, unrelated uses of the string JC2, or nothing. X’s own trend summary for the counterexample says the plane case remains open.

The near-misses are real theorems that say so themselves. A terse jc₂ link circulating on 24 July resolves to a cyclic-rigidity report proving a theorem for maps of the restricted shape $F(x,y)=\bigl(H(x^m,y),xg(x^m,y)\bigr)$; the report explicitly states that it does not solve unrestricted JC(2). A viral claim that someone had “accidentally solved” the conjecture during a long chat session points, through its own shared link, to an explanation of the three-dimensional counterexample.

The July literature is symmetry results, not a solution. Shaska’s new theorem proves every nontrivially $\mathbb G_m$-equivariant plane Keller map is invertible; Jelonek’s paper constrains components of the bounded-degree Keller locus. Neither claims unrestricted JC(2). A MathOverflow thread and a dated Zenodo record report eliminating the last residual supports at the degree wall, but the answer itself says the write-up is in preparation and the research record calls its own campaign strong machine evidence rather than a complete proof.

The new complex construction still begins above the plane. Gao’s preprint replaces one isolated example with a general tangent-sweep mechanism, unbounded geometric degree, and explicit maps in dimensions three through five. It is the strongest new structural result in this update, but its stated range is dimensions greater than two. It supplies no unrestricted planar map and no reduction of JC(2) to the sweep class.

There is a plane counterexample in characteristic two — to a different conjecture. Romy Mondello’s 29 July preprint gives a two-variable map over the algebraic closure of the two-element field with Jacobian determinant one, generic degree three, a separable function-field extension, and three colliding points. That refutes the separable Jacobian conjecture in characteristic two. The paper explicitly says it does not address the classical characteristic-zero conjecture. This is why the phrase “characteristic zero” near the top of this article is load-bearing, not decoration.

The real and neighboring problems have moved too. A 30 July revision by Piotr Migus classifies the possible generic degrees of real polynomial Keller maps with non-dense Euclidean image above the plane and proves that any two-dimensional example of that type would have even degree at least six; it neither constructs such a planar map nor settles the complex case. The new Hessian and weak Markus–Yamabe counterexamples described above are further consequences of the three-dimensional map, not evidence of a hidden JC(2) solution.

One honest caveat on all of that: X search is login-gated and incompletely indexed. Private, deleted, reply-only, image-only, and very recent material is invisible to this kind of sweep. “I did not find it” is a negative search result, not a theorem.

The part I actually reuse

Strip away the algebraic geometry and there is a working method in that folder that I have since used elsewhere, in projects with nothing to do with polynomials.

Two ledgers, not one. A claim ledger says which mathematical statements may be used as premises, with statuses — established, proved here, computationally verified, plausible, open blocker, disproved, dormant, superseded. An idea registry separately says which research mechanisms are active, dormant, superseded, or withdrawn. They answer different questions. Conflating “this statement is false” with “this direction is not worth funding” is how programs quietly lose ideas that were only mis-executed. Anything unmapped into either ledger is unclassified, and unclassified never gets treated as a theorem.

A circularity filter. Some appealing targets have the form: if condition $\mathcal C$ holds, the map is invertible. Before spending months on one, check the converse. If invertible also implies $\mathcal C$, then the theorem is a kill switch, not a reduction — its hypothesis is just the conjecture again, in costume. Kill switches are perfectly good counterexample filters and useless as proof targets. Applying that one test reclassified several of the corpus’s own “active closure targets” in an afternoon.

Negative results are results. The budget’s failure to bind asymptotically, the numerical system that turned out feasible with $5304$ integer solutions, the Euler-characteristic obstruction that reduced to $0=0$ — each is written down as a finding with its own identifier. They are the cheapest thing in the folder and they save the most time.

Retractions stay visible. When a lemma named the wrong pencil and inferred a false degree, the fix was not to quietly edit it. The wrong version stays, marked, with the correction beside it and a note on which downstream results survive. Anyone who reads it learns the failure mode too.

Every finite claim gets an executable check. Eleven verifier scripts, each printing a single pass line, covering identities, budgets, monodromy over all transitive subgroups of $S_4$, $S_5$, $S_6$, fibration and projection facts, and the Gelfand–Leray arithmetic. A claim with a finite symbolic component and no verifier does not get promoted.

Where it stands

As of 8 August 2026, JC(2) is not proved and not disproved. A counterexample needs $N\ge6$ sheets, non-Galois monodromy, a nonprincipal different at the deleted boundary, mixed affine and boundary sheets over every branch curve, at most $N-2$ boundary components, a nonproper curve that is not homeomorphic to $\mathbb C$ with all branches through one point at infinity, a generic fiber of genus at least $2$ under the standard reduction, and coordinate degrees past the $(72,108)$ wall. That is a remarkably specific ghost. Nobody has found it, and nobody has proved it cannot exist.

The past three weeks cut both ways on the odds. On one hand, the conjecture is now known to be false in every dimension above the plane, and Gao has shown that those counterexamples are neither isolated nor bounded in sheet count. On the other, every known characteristic-zero construction still spends a dimension the plane does not possess. The old numerical comfort — that the first example had only three sheets, below the planar floor — is gone. The surviving distinction is structural, not numerical, and I do not think anyone is entitled to be confident yet.

What strikes me, after enough time with this material, is how much of the work is not deduction at all. It is bookkeeping under uncertainty: keeping the established separate from the plausible, the empty case separate from the open one, the sharp bound separate from the sufficient one, the checkable identity separate from the reported one. The mathematics is out of my depth on most pages. The discipline is not, and it turned out to be the transferable part.

The next real move is unglamorous and clearly stated: get the two remaining braid frontiers to full nonabelian factorization, extend the coefficient exclusions past their current fibers, and — above all — find the coordinate-invariant bridge that carries an arbitrary Keller map into one of the controlled models. Without that bridge, every exclusion in the folder stays exactly what it is: true, exact, and bounded.

Sources

All links open in a new window.

The July 2026 counterexample and its aftershocks

• Levent Alpöge’s announcement — the original post, 20 July 2026, with the map, the determinant, and the three colliding points.
• Shuhong Gao, “Counterexamples to the Jacobian conjecture in dimensions greater than two” — the self-contained tangent-sweep construction, arbitrary geometric degree above the plane, and five new explicit maps.
• Archive of Formal Proofs, “Formal Verification of an Explicit Counterexample” — an independent Isabelle/HOL verification of the determinant identity and collision.
• Terence Tao, “A digestion of the Jacobian conjecture counterexample” — the multiplication-map reformulation and the marked-root explanation.
• Secret Blogging Seminar, “The new counterexample to the Jacobian conjecture” — discriminant structure and the class-group argument for why the construction is special to dimension three.
• Zihan Zhang, “Direct Consequences of the Three-Dimensional Counterexample” — what else falls, and what does not.
• jacobianfun.org — a plain-language walkthrough of the non-properness mechanism.
• John D. Cook, “Locally everywhere does not imply everywhere” — the shortest statement of why the inverse function theorem was never enough.
• Meng–Yang, on the five-variable Hessian counterexample, Migus, on generic degrees of real non-dense Keller maps, and Castañeda–Honorato–Valenzuela-Henríquez, on weak Markus–Yamabe — follow-up consequences and classifications that do not settle JC(2).

The plane case

• S. Yu. Orevkov, on the three-sheet exclusion and boundary multiplicities — also the source of the $\sum_i e_i\le N-1$ budget.
• Domrina–Orevkov, degree four, part I and part II.
• H. Żołądek, Newton–Puiseux and topological degree five — the publisher record that moves the frontier to $N\ge6$.
• Nguyen Van Chau, on the non-proper value set and the polynomial Vitushkin obstruction.
• T. T. Moh, “On the Jacobian conjecture and the configurations of roots” (1983) — the degree-$100$ bound.
• “Increasing the degree of a possible counterexample from 100 to 108” — the current published wall.
• T. Shaska, on $\mathbb G_m$-equivariant plane Keller maps and Z. Jelonek, on components of the Keller locus — July 2026 partial results.
• “The simplest case of Jacobian conjecture” on MathOverflow and its Zenodo research record — the claimed degree-$125$ update, write-up in preparation.
• Cyclic Rigidity in Dimension Two, from this post — a restricted-shape theorem that says of itself that it does not settle JC(2).
• R. Mondello, “A Dimension-Two Counterexample to the Separable Jacobian Conjecture in Characteristic Two” — a genuine planar counterexample over characteristic two whose statement explicitly excludes the classical characteristic-zero problem.

Background

• A. Rodríguez Díaz, “On the origin of the Jacobian conjecture” — Kraus 1884, and what his flawed final step anticipated.
• Bass, Connell, and Wright, “The Jacobian conjecture: reduction of degree and formal expansion of the inverse” — the cubic-homogeneous reduction, and the dimension increase that comes with it.
• S. Pinchuk, “A counterexample to the strong real Jacobian conjecture” (1994) — the plane, over $\mathbb R$, with the constancy hypothesis dropped.