cert-machine · decided · polynomial maps
Gao's Keller maps and a Markus–Yamabe field, decided
Two papers of August 2026 print polynomial maps: five new counterexamples to the Jacobian conjecture, written with Claude Fable 5's help, and a vector field on ℝ¹⁴ whose Jacobian has every eigenvalue −1 at every point and which still has three zeros. Decided here in exact rational arithmetic from the TeX the papers print: seven of the eight claims hold whole. The largest map's constant determinant rests on the paper's factorisation lemma. The 18-dimensional field's nilpotency, which the paper checks at one sample point, is decided here as an identity.
Sources: S. Gao, arXiv 2608.00222v1, and Á. Castañeda, G. Honorato, F. Valenzuela-Henríquez, arXiv 2608.05392v1, both CC BY 4.0; their TeX mirrored in corpus/polymaps and every map extracted from it by a parser that refuses any text it cannot read. The earlier maps of this story, Alpöge's among them, are decided on the Jacobian-conjecture audit. Nothing has been sent to the authors.
tl;dr
- The finding. 7 CERTIFIED, 1 PARTIAL. Gao's G, F4, F5 and F6 have Jacobian determinants identically 2, −44/9, 160/29 and −290, and each sends two or three distinct rational points to one point. F7 is not injective either, and the determinants of its factors are what the paper says; the determinant of the whole is its lemma. The 14- and 18-dimensional fields have (JX + I)^14 = 0 and (JX + I)^17 = 0 identically, and three zeros each. The cubic Φ on 11 variables, whose first publication says ChatGPT generated it, has det JΦ = −2 and three points with one image.
- The mechanism. Sparse polynomials with Fraction coefficients. Determinants are expanded symbolically, Laplace over column subsets, and powers of JX + I are multiplied out. Collisions are checked by evaluating at points found from each paper's fiber structure; the decider only evaluates them. F6 and F7 are not printed in full, and the ancillary files the paper promises are not on arXiv, so they are rebuilt from the printed construction, and every division the construction makes is checked exact.
- Check it. python3 tools/run-polymaps-ledger.py --check · python3 instruments/polymaps/battery.py — 19 checks, 6 red controls fired.
claims decided
8
Five Keller maps of Gao's, the cubic Φ, and two Markus–Yamabe fields.
certified whole
7
Every fact the claim needs, as an exact identity or an exact evaluation.
partly
1
F7: its non-injectivity and its factors' determinants; the assembly is the paper's lemma.
largest map rebuilt
25,518
Terms in F7's last component; 79,107 in its five.
§1 · the claims
Eight maps, one verdict each
| verdict | claim | source |
|---|
| CERTIFIED | G: a polynomial map of C^3 with det J = 2 that is not injective (a counterexample to the Jacobian conjecture), printed in full | arXiv 2608.00222, Thm 3.5 |
| CERTIFIED | F4: a polynomial map of C^4 with det J = -44/9 that is not injective (a counterexample to the Jacobian conjecture), printed in full | arXiv 2608.00222, Thm 4.3 |
| CERTIFIED | F5: a polynomial map of C^4 with det J = 160/29 that is not injective (a counterexample to the Jacobian conjecture), printed in full | arXiv 2608.00222, Thm 4.4 |
| CERTIFIED | F6: the map the printed construction defines (n = 5), det J = -290, not injective | arXiv 2608.00222, Thm 4.5 (the full map is not printed; the promised ancillary files are not on arXiv) |
| PARTIAL | F7: the map the printed construction defines (n = 5), det J = 119377, not injective | arXiv 2608.00222, Thm 4.13 |
| CERTIFIED | X on R^14 (degree 7): JX has spectrum {-1} at every point and X has three zeros, so the weak Markus–Yamabe conjecture fails in dimension 14 | arXiv 2608.05392, Thm 5.1 |
| CERTIFIED | Phi, a cubic map of C^11 with det J = -2 that is not injective (the gist that first published it says ChatGPT generated it) | arXiv 2608.05392, Prop 6.2 |
| CERTIFIED | X-hat on R^18 (degree 3): spectrum {-1} everywhere, three zeros — decided as a symbolic identity where the paper's appendix checks one sample point | arXiv 2608.05392, Thm 8.1 |
§2 · the Jacobian conjecture
Five maps, and what each one needs
A polynomial map F of ℂⁿ with det JF a nonzero constant is locally invertible everywhere; Keller's conjecture (1939) said it must then be globally invertible. A counterexample needs two facts: the determinant is identically a nonzero constant, and two different points have the same image. Both are finite and exact, and both are decided here.
- G, F4, F5, printed in full. Parsed from the TeX, with 33, 188 and 114 terms. Their determinants are expanded in well under a second. The shared images are G at (1, 5/4, 0); F4 at (1, -1289/270, -113/90, 1453/90); F5 at (1, -89/116, -605/116, -109/58), each shared by distinct rational points.
- F6, printed through its construction. The rebuild reproduces the four printed polynomials X₁…X₄ and the printed first component, then gives terms [6, 342, 421, 507, 904], degrees [7, 38, 40, 42, 44]. Its determinant, −290, is expanded directly in about 45 seconds.
- F7, the largest. The rebuild gives terms [7, 14445, 17727, 21410, 25518], degrees [7, 86, 89, 92, 95] and reproduces the printed first component. Two rational points collide. The determinants of its factors hold: det J(S) = γ, the stage 119377, det J(H₃, H₄) = 1. Expanding the whole 5 × 5 determinant of rows of up to 25,518 terms is out of reach of this pure-Python instrument, so the constant determinant rests on the paper's factorisation lemma.
The paper says its verification scripts are "available from the author", and pp. 24 and 28 promise F6 and F7 "in the ancillary files"; arXiv has none. The verdicts on F6 and F7 are therefore on the map the printed construction defines.
§3 · the weak Markus–Yamabe conjecture
Every eigenvalue −1, and three zeros
A vector field whose Jacobian has eigenvalues with negative real part at every point was conjectured, in the weak form, to have at most one zero. If JX + I is nilpotent as a matrix of polynomials, every eigenvalue of JX is −1 everywhere; so a field with that identity and two zeros refutes it.
- X on ℝ¹⁴, degree 7. The 14 × 14 matrix JX + I, multiplied out: its 13th power is not zero and its 14th is. X vanishes at the three distinct points the paper prints. The weak conjecture fails in dimension 14.
- X̂ on ℝ¹⁸, degree 3. The 17th power of JX̂ + I is zero and the 16th is not. The paper's appendix checks this unipotency "at a sample rational point"; here it is an identity. Three distinct zeros.
- Φ, the cubic behind X̂. det JΦ = −2 by a direct 11 × 11 expansion (the paper uses a Schur complement), and three distinct points share an image. Its 11 components match, term for term, a map published in a gist that says ChatGPT generated it.
§4 · limits
What is not decided here
- F7's whole determinant. Only its factors' determinants are decided; the assembly is the paper's lemma.
- The generic fiber sizes. The papers' counts of preimages (4, 5, 10, 6 and 12 for Gao's maps) and their stratifications need gcd and squarefree certificates at a witness target, which are not built yet.
- "The chain cannot be shortened." The Markus–Yamabe paper says this was verified in exact arithmetic but prints no code or certificate. It is a finite family of variants, not decided here yet.