---
title: Exact audit and structural analysis
tags: [jacobian-conjecture, keller-map, counterexample, index]
---

# 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
{cite}`alpoge2026x`.  Later that day Andy Jiang (@davikrehalt) posted its
projective symmetric-product marked-root formulation {cite}`jiang2026x`.
Aaron Lou ([@aaron_lou](https://x.com/aaron_lou)) then published a
factorization--resultant derivation and affine chart {cite}`lou2026note`.
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.

:::{admonition} Bottom line
:class: important
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](structural-certificate.md)
does not require trusting a determinant expansion.

The explicit formula, Jiang's projective restriction, Aaron Lou's
([@aaron_lou](https://x.com/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](literature-audit.md).

## What is new in this audit

Beyond reproducing the announcement, the notes establish:

1. a two-line rational-coordinate proof of $\det JF=-2$;
2. a global binary-cubic identity explaining all coefficients;
3. the exact reduced fiber ideal at the announced target;
4. an integer-coefficient normalization $H=X+\text{higher terms}$ with
   $\det JH=1$;
5. an explicit inverse branch for every simple root of a cubic;
6. exact fiber cardinalities $3,1,0$, an omitted triple-root curve, and
   generic degree $3$;
7. the discriminant hypersurface as the nonproperness set, with explicit
   escaping families.
8. exact obstructions to every direct marked-root, polynomial-graph, linear
   target-slice, and natural equivariant nonlinear descent to dimension two;
9. an equivariant noninjective plane quotient with Jacobian $-2c^2$, and an
   octahedral degree-six cusp cover with Jacobian $108x^3$;
10. 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;
11. 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?](../output/pdf/six_questions_keller_map.pdf).
  It is explicitly a post-announcement reconstruction, not a historical
  transcript.
- For the shortest decisive argument, read [the map](map-definition.md),
  [the structural determinant proof](structural-certificate.md), and
  [the exact fiber](fiber-scheme.md).
- For the conceptual explanation, continue through
  [generic fibers and binary cubics](generic-fiber.md) and
  [behavior at infinity](infinity.md).
- For historical status and cautious attribution, see the
  [literature audit](literature-audit.md).
- For the active plane attack, start with [the current plane status](plane-status.md),
  then read [the descent obstructions](plane-descent-obstructions.md),
  [the two exact near-models](plane-near-models.md),
  [the factorization--resultant route](resultant-factor-route.md),
  [the pseudo-plane Poisson obstructions](pseudoplane-poisson-obstructions.md), and the
  [focused program](plane-program.md).
- For independent reproduction, use [reproducibility](reproducibility.md) or
  run the included [SymPy notebook](notebooks/sympy-verification.ipynb).

## 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.