Optional AI source assessment — noncoordinate polynomial

Independent first source pass, 13 October 2026. This assessment was written before opening model-panel responses or the researcher's mathematical labels. It is not a human verdict and must stay separate from neutral reading materials. Human assessment remains pending.

Provenance and scope

The complete six-page manuscript is An explicit noncoordinate polynomial with affine three-space zero fibre, dated 24 September 2026. Sources are frozen at openai/math commit 1581d4cbdd69a0faf34b2089614fe2a814316a6a. Native and expanded TeX SHA256: 60993c0b524291b7f707e42eec89ec7e910c22760d81f1d3af007c229228c13c; PDF SHA256: a0f2145ec95a6083a4a9a54ef4d2c41fa9587cd435c6908c512ce90e6c64154b. Native line locators below refer to paper.tex, not PDF extraction. See manifest.

The first pass covers all three numbered results, the presentation changes and the explicit-coordinate appendix. It checks the written proof rather than inferring correctness from the importance or familiarity of the conjecture. No Lean build, kernel verification, literature-priority audit or paid API call was performed.

Assessment

The displayed construction and proof survive this independent source check. The argument gives polynomial inverse maps on the whole zero fibre and a critical point in the ambient space. No concrete error, hidden localization or essential missing hypothesis has been established. This is an AI-assisted mathematical assessment with the limits below, not a formal certificate or a priority judgment.

Checked proof chain

1. Cusp identity and translation — lines 87–101 and 255–286. Expanding x=u^3+hv and y=-u^2+hw gives exactly x^2+y^3=hS. For formal generators, (x+s^3)^2+(y-s^2)^3=x^2+y^3-sp. Eliminating h=1+p therefore gives X^2+Y^3=s. The triangular translation is invertible over C[s]; solving this last equation for s genuinely gives B≅C[X,Y].
2. Unit ideal — lines 288–290. In B/(h,y), the relations give x^2=0 and 1=2s^2x, hence 1=0 by squaring. This is valid over a quotient ring and does not assume it is reduced. The appendix's stronger free-polynomial identity at lines 350–359 was independently expanded exactly; its stated alpha,beta satisfy alpha h+beta y=1 in B.
3. Lifting lemma 2.1 — lines 164–245. The change G=V+UW is a polynomial automorphism. Reduction by hW=y+U^2 gives the stated linearized third relation. With a=y+U^2, b=s-hG^2+2UyG, the compatibility y^2 a=hb follows from the base cusp relation and hG-yU=x. From alpha h+beta y=1, the coefficients r=alpha(1+beta y), q=beta^2 satisfy rh+qy^2=1. Thus W0=ra+qb satisfies both equations, and their ideal equals (W-W0) in A[W]. This ideal equality proves elimination without dividing by h or y. It remains valid with zero divisors. The final linear parameter T=alpha U+beta G has the two displayed inverse formulas; checking both composites uses only the same Bezout equation.
4. Theorem 1.1: fibre identification — lines 269–304. In the seven-generator presentation of C, the three relations eliminate x,y,s directly. The cusp relation then vanishes identically, and the remaining relation is exactly F. This proves C≅R/(F), rather than merely a morphism between the varieties. Combined with the lemma and the explicit base presentation, it supplies R/(F)≅C[X,Y,T].
5. Theorem 1.1: coordinate obstruction — lines 306–329. At the specified point in the correct variable order (h,u,v,w), exact substitution gives s=1, X=Y=0, F=-1. Differentiating X^2+Y^3=s(1+F) gives dF=0 there. Independent symbolic differentiation also gives all four partial derivatives zero. A polynomial automorphism and its polynomial inverse have inverse Jacobian matrices at every point, so a zero first row is impossible. The critical point is on the -1 fibre, not the smooth zero fibre; those facts are compatible.
6. Corollary 1.2 — lines 332–338. Adjoining variables turns the quotient into a polynomial ring of the required dimension. The old critical point extends because the added partial derivatives vanish identically. The same Jacobian obstruction prevents a coordinate in the larger ring; stable cancellation is not being assumed.
7. Appendix — lines 348–378. The stated coordinates and inverse are the lemma's formulas after the base parametrization. Their validity follows from the checked presentation and ideal equalities. There is no denominator or omitted open subset. I did not expand the very large fully substituted composite maps in the ambient polynomial ring; the universal identities establish their composites in the quotient.

Reproducible exact checks and results record ten successful local symbolic checks, including the gradient and appendix Bezout certificate, using SymPy 1.14.0. These are local arithmetic certificates, not a general-purpose proof-checker certificate.

External sources and remaining obligations

The essential construction, lifting lemma and Jacobian obstruction are proved internally. Historical affirmative embedding results are not hypotheses used to construct the example. In particular, an affirmative theorem for an equation linear in one independent variable does not apply merely because intermediate auxiliary generators are introduced.

I inspected the primary KVZ source, arXiv:math/0104204v2, §4.1 and Theorem 4.2(a): it concerns its specified linear equation class with a coefficient in one variable. This supports the restricted formulation in introduction lines 62–66; I did not audit KVZ's entire proof. The Shpilrain–Yu primary URL was attempted but returned an access error in this pass. Its historical attribution, Maubach's kernel-calculation locator and the other introduction citations therefore remain unverified here. The elementary critical-point argument does not depend on accessing Shpilrain–Yu.

Priority, exact novelty, literature completeness, repository formalization and the generating model's discovery process are outside this assessment. An unusual claimed consequence is a reason for independent replication of the maps, not itself a demonstrated flaw.

Sources

• Pinned complete native manuscript.
• Pinned PDF.
• Kaliman–Vénéreau–Zaidenberg, primary preprint, restricted theorem formulation inspected on 13 October 2026.
• Shpilrain–Yu, primary preprint, attempted access only; not counted as inspected.