The Zero-Discount Claim | Abhishek Shankar's Blog

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Here is a claim that does not ask you to trust the mathematician who posted it, the AI lab it credits, or a word of the coverage around it. Take the polynomial map that Levent Alpöge published on July 19, three polynomials in three variables:<br>P=(1+xy)3 z+y2(1+xy)(4+3xy)Q=y+3x(1+xy)2 z+3xy2(4+3xy)R=2x−3x2y−x3 z\begin{aligned}<br>P &= (1+xy)^{3}\,z + y^{2}(1+xy)(4+3xy) \\<br>Q &= y + 3x(1+xy)^{2}\,z + 3xy^{2}(4+3xy) \\<br>R &= 2x - 3x^{2}y - x^{3}\,z<br>\end{aligned}PQR​=(1+xy)3z+y2(1+xy)(4+3xy)=y+3x(1+xy)2z+3xy2(4+3xy)=2x−3x2y−x3z​<br>Differentiate the three coordinates with respect to x, y, and z, assemble the three-by-three Jacobian matrix, and take its determinant. It simplifies to the constant -2, which is never zero. Then evaluate the map at three distinct points, (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2). All three land on the same output, (-1/4, 0, 0). A map whose derivative is invertible at every point, and which nonetheless sends three inputs to one image, so it cannot have a polynomial inverse.<br>If you would rather not do it by hand, this settles it:<br>import sympy as sp<br>x, y, z = sp.symbols("x y z")<br>P = (1 + x*y)**3*z + y**2*(1 + x*y)*(4 + 3*x*y)<br>Q = y + 3*x*(1 + x*y)**2*z + 3*x*y**2*(4 + 3*x*y)<br>R = 2*x - 3*x**2*y - x**3*z<br>F = (P, Q, R)<br>assert sp.factor(sp.Matrix(F).jacobian((x, y, z)).det()) == -2<br>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))]<br>imgs = {tuple(sp.simplify(f.subs({x:a,y:b,z:c})) for f in F) for a,b,c in pts}<br>assert imgs == {(sp.Rational(-1,4), 0, 0)}<br>Run it. The determinant is -2 or it is not. The three points collide or they do not.<br>They do.<br>That is a counterexample to the Jacobian conjecture in dimension three, a problem open since Ott-Heinrich Keller posed it in 1939. Alpöge posted the map and credited Anthropic's Fable model with finding it. Independent symbolic checks appeared within hours. I argued a few days ago that enterprise AI lives in a self-report economy , a market where every load-bearing number about whether the technology pays off was produced by a party with a stake in the answer, so every figure needs a discount before you can use it. This is the inverse of that. The artifact is verifiable on your machine, not theirs. The commercial story labs build around artifacts like it usually is not, and telling the two apart is the whole subject of this piece. When you cannot prove the product pays, you prove a theorem instead.<br>What actually happened, and what actually checks out<br>The Jacobian conjecture is one of those statements that sounds like it has to be true. Take a polynomial map from complex n-dimensional space to itself. If its Jacobian determinant, the quantity measuring how the map stretches space locally, is a nonzero constant everywhere, then the map should have a polynomial inverse. Local invertibility forcing global invertibility. In one variable it is a triviality: a polynomial with a nowhere-vanishing derivative is linear, by the fundamental theorem of algebra, and a linear map is invertible. The conjecture claimed the same rigidity survives into every dimension. For eighty-seven years nobody proved it and nobody broke it.<br>Alpöge broke it in dimension three. His map has a constant Jacobian determinant, which makes it what algebraists call a Keller map, and three distinct points with real coordinates all map to the same place, which makes it not injective. A Keller map that is not injective is exactly the object the conjecture said could not exist. The verification is the entire point, so hold on it. The checks that went up within hours, a SymPy script, a Sage script linked from an answer on MathOverflow, confirm the determinant and the collision as identities of polynomials, not as samples taken at a few coordinates and extrapolated. An identity holds everywhere or fails, and this one holds. The collision uses only real numbers, so it certifies non-injectivity over the reals and the complex numbers at once. Nobody had to convene a committee. Anyone who doubted it could rerun the check before lunch.<br>One correction to the way the result is circulating. This is not a degree-four map. Its three coordinate polynomials have total degrees seven, six, and four, so it is a degree-seven map, and the tidy "degree four" that appeared in some early write-ups, mine included, is wrong. A second correction matters more, because it is the kind of overstatement the moment invites. The counterexample does not invalidate the classical theorems that reduce the general conjecture to cubic or homogeneous forms after adding variables. Those reductions are still correct. What changes is their job. Instead of reducing a conjecture someone hoped to prove, they now transform a known counterexample into other special shapes. Append unchanged coordinates to Alpöge's map and the determinant stays constant and the collision survives, so...