Abstract
This draft gives an exact rational polynomial vector \(g\in\mathbb Q[Z]^{14}\), with 24 monomials, for which \(\det(I+sJg)=1\), while \(I+g\) has three displayed rational points in one fiber. Generically, \(Jg\) is a single nilpotent Jordan block of size 14. The same object gives an explicit counterexample to the Special Image Conjecture in dimension 14 whose multiplier obstruction is nonzero at every positive exponent.
1. Explicit unipotent map
The construction lifts the announced three-variable degree-three Keller map by eleven constant nilpotent state variables. In the coordinate order \(Z=(x,y,z,U)\), its nonlinear part has exactly 24 monomials and satisfies the exact polynomial identity
Cayley–Hamilton gives \((Jg)^{14}=0\), while the checker obtains the nonzero entry
Thus the generic nilpotency index is 14 and the generic Jordan type is \((14)\).
2. A rational three-point fiber
Three explicit rational points lift from the source map and have the common image
The lifted map has generic degree three and geometric monodromy \(S_3\). The construction therefore supplies a concrete noninjective map whose derivative is everywhere unipotent, together with fixed-point consequences for the nilpotent-Jacobian formulation discussed in the paper.
3. Every-exponent failure of SIC(14)
For the image operator \(E_{14}\), set
Provisional theorem. For every positive integer \(m\),
The nonzero value is given by closed binomial-coefficient formulas in all three residue classes modulo three, so this is not an extrapolation from a finite computation.
4. Scoped optimality and companions
Exact coefficient-Hankel ranks \(11,8,5,3,1,0\), including a displayed nonzero minor of determinant \(275562\), prove that eleven state variables and 24 terms are minimal inside the precisely stated constant-state realization ansatz. No global minimality claim is made. The paper also records a 15-dimensional homogeneous companion and the standard 30-dimensional Hessian-nilpotent symmetrization.
5. Exact certificates
verify_symbolic.py— determinant pencil, collision, generic nilpotency, every-exponent formulas, scoped minimality, and homogeneous companion.unipotent14_sparse.json— deterministic sparse map and collision certificate.verify_exported_stdlib.py— dependency-free rational verification of the exported certificate.MANIFEST.md— complete artifact and claim inventory.
Certificate SHA-256: ce6ca33b38c808a973b18da3d5f4a1f5a647c7836c2fbd78889fa7ffb3ba746c
6. Scope and priority
The 14-dimensional degree bookkeeping is classical and is not claimed as new. The narrow candidate contribution was the explicit sparse instantiation, its regular-nilpotent and rational three-point certificates, and the every-exponent SIC(14) formula. The refreshed audit located no earlier public source for that exact object or an explicit SIC witness in dimension at most 14 before this page's 22 July release. Roy van Rijn's four-term SIC(3) counterexample, published 28 July 2026, is later rather than prior art, but it supersedes the quantitative SIC dimension headline.
AI-assistance and verification disclosure
The exploratory algebra, proof drafting, verification programs, and website were developed with heavy assistance from ChatGPT 5.6 Sol under Alec Kriebel's direction. Alec Kriebel is a complete amateur and cannot independently verify the claims. The note has not been peer reviewed.
Suggested citation
Alec Kriebel, with heavy assistance from ChatGPT 5.6 Sol, “A 14-variable polynomial map with everywhere unipotent Jacobian and a three-point fiber,” provisional research note, 22 July 2026.