This repository has two distinct purposes:

1. It gives a self-contained explanation and exact verification of the polynomial map announced by Levent Alpöge on 19 July 2026: its Jacobian determinant is the constant -2, although three distinct real inputs have the same image.
2. It records a follow-on construction derived while analyzing that map. For every integer n >= 3, the construction gives an explicit polynomial map F_n : C^3 -> C^3 with det JF_n = 1 and generic fiber degree n.

The degree-four member is especially useful: generic fiber degree is invariant under polynomial coordinate changes, so this map is not merely a coordinate version of the announced degree-three map. Since four is not a power of three, it is not an iterate of that map either.

Open the interactive mathematical essay.

Unpack the projective-geometric recipe behind the announced map. This companion page translates the short marked-root construction and relates it explicitly to the weighted-lift mechanism used in the essay.

Open the exact counterexample atlas. It instantiates generic fiber degrees 3 through 100 and gives a rational target with all n rational preimages for every row.

• docs/jacobian-explained.html — the self-contained essay and interactive mathematical figures; no build step or runtime dependency is required.
• docs/geometric-construction.html — a translation of the compact projective construction and its exact relationship to the weighted-lift family.
• docs/counterexamples.html — 98 explicit map certificates and 5,047 exact rational preimages, generated and checked in the browser.
• RESEARCH.md — the algebraic derivation, including the uniform all-n construction.
• verify_counterexamples.py — a standard-library exact checker for every map, target, and preimage in the atlas.
• verify.py — exact checks of the principal maps and identities.
• verify_page_facts.py — an exact audit of every mathematical assertion used by the page's live figures.
• verify_independent.py — a separately written check that reconstructs the announced map, the degree-four map, and a degree-five instance of the general construction.

Run without modifying the project environment:

uv run --with sympy python verify.py
uv run --with sympy python verify_page_facts.py
uv run --with sympy python verify_independent.py
uv run python verify_counterexamples.py

The scripts use exact symbolic arithmetic. The browser page also recomputes its plotted fibers at runtime and compares them with exact reference values.

The wrangler.jsonc configuration publishes docs/ with Cloudflare Workers Static Assets and binds the Worker to jacobianfun.org. With Node.js 22 or newer and an authenticated Cloudflare account, deploy with:

just deploy

Deployment is currently manual: pushing to GitHub does not automatically publish to Cloudflare. The recipe runs pinned Wrangler through npx and uses Volta to supply Node.js 22 when Volta is installed.

1. a weighted two-variable representation of the map;
2. an explicit visualization of the two source sheets above the plane C=0;
3. a cubic equation for the hidden variable u = 1 + xy;
4. an exact three-point fiber above (-8/27, 0, 1);
5. a tangent-curve interpretation of inverse images and the branch-at-infinity mechanism;
6. a weighted-lift family with every generic fiber degree at least three, including an explicit generic-degree-four map with component degrees (12, 11, 4) and an exact regular four-point fiber;
7. one closed seed formula producing a map in fixed dimension three for every generic fiber degree n >= 3;
8. a rational-certificate family for degrees 3 through 100, with one complete exact n-point fiber for every map.

The family and degree-four map are exact deductions made during this analysis. No claim of literature priority is made; that comparison remains incomplete. The announced counterexample is credited to Alpöge, whose post credits Fable in its discovery. Alexis Gallagher created and curated this exposition and follow-on analysis, relying mainly on OpenAI's GPT-5.6-sol for mathematical reconstruction, drafting, implementation, and automated checking.

Updated 2026-07-21.