Exact audit and structural analysis#
On 20 July 2026, Levent Alpöge publicly announced a concrete polynomial map of affine three-space as a counterexample to the Jacobian conjecture [Alpöge, 2026]. Later that day Andy Jiang (@davikrehalt) posted its projective symmetric-product marked-root formulation [Jiang, 2026]. Aaron Lou (@aaron_lou) then published a factorization–resultant derivation and affine chart [Lou, 2026]. This book records an independent exact audit, a short hand proof, and a structural interpretation that turns the example into the forgetful map from a binary cubic with a marked simple root to its coefficients.
Bottom line
The displayed map really does have constant Jacobian determinant \(-2\) and a fiber containing three distinct points. In fact, that fiber is exactly three reduced points and the map has geometric degree three. It refutes the standard Jacobian conjecture in dimension \(3\), and hence in every dimension \(n\geq3\) after adjoining identity coordinates. The dimension-two problem is not affected.
The conclusion rests only on finite polynomial identities over \(\mathbb Q\). It has been checked in SymPy, SageMath, and Magma, but the short proof in Structural determinant certificate does not require trusting a determinant expansion.
The explicit formula, Jiang’s projective restriction, Aaron Lou’s (@aaron_lou) resultant-normalized chart, and the incidence model used in this book are isomorphic presentations of one degree-three etale morphism. They are not literal coordinate identities: Lou’s formula is left–right linearly equivalent to the announced map, while the projective-to-affine identifications require choices. The exact comparison is recorded in the literature audit.
What is new in this audit#
Beyond reproducing the announcement, the notes establish:
a two-line rational-coordinate proof of \(\det JF=-2\);
a global binary-cubic identity explaining all coefficients;
the exact reduced fiber ideal at the announced target;
an integer-coefficient normalization \(H=X+\text{higher terms}\) with \(\det JH=1\);
an explicit inverse branch for every simple root of a cubic;
exact fiber cardinalities \(3,1,0\), an omitted triple-root curve, and generic degree \(3\);
the discriminant hypersurface as the nonproperness set, with explicit escaping families.
exact obstructions to every direct marked-root, polynomial-graph, linear target-slice, and natural equivariant nonlinear descent to dimension two;
an equivariant noninjective plane quotient with Jacobian \(-2c^2\), and an octahedral degree-six cusp cover with Jacobian \(108x^3\);
the general coefficient–resultant determinant law, an exact rejection of the Bring–Jerrard degree-ten factor surface, and a noninjective quadratic-factor \(\mathbb A^2\) scaffold reducing one new route to an explicit Poisson equation;
the identification of that scaffold with Miyanishi’s exceptional pseudo-plane \(S(2,2,1)\), a rational constant-Jacobian near-solution from the Dubouloz–Palka formulas, and exact generator, homogeneous, quadratic, and boundary-preservation no-go theorems.
Reading paths#
For the short synthetic discovery dialogue that joins the geometry to the exact coefficient reconstruction, read Could One Have Found It This Way?. It is explicitly a post-announcement reconstruction, not a historical transcript.
For the shortest decisive argument, read the map, the structural determinant proof, and the exact fiber.
For the conceptual explanation, continue through generic fibers and binary cubics and behavior at infinity.
For historical status and cautious attribution, see the literature audit.
For the active plane attack, start with the current plane status, then read the descent obstructions, the two exact near-models, the factorization–resultant route, the pseudo-plane Poisson obstructions, and the focused program.
For independent reproduction, use reproducibility or run the included SymPy notebook.
Editorial status#
The mathematical identities are stated as theorems because they are proved inside this repository. Public priority and reception are stated more cautiously: the construction was only hours old when this audit was assembled, and no refereed paper for the new map had yet appeared. The web audit is frozen at 2026-07-20, Europe/Warsaw.