A 14-variable unipotent Keller map and every-order image and vanishing obstructions

Alec Kriebel

Abstract

This draft gives one self-contained formal-inverse mechanism behind several consequences of the announced three-variable Jacobian counterexample. Its flagship is an exact 24-term rational vector in 14 variables whose determinant pencil is one, whose nonlinear Jacobian is generically regular nilpotent, and whose identity-linear map has exactly three reduced rational points in one fiber. Closed inverse coefficients prove failure of the Special Image Conjecture at every exponent and, after exact homogeneous and symmetric transforms, failure of the Hessian-nilpotent Vanishing Conjecture at every order.

Flagship map14 variables · 24 monomials · type \((14)\)

Image obstruction\(\operatorname{SIC}(14)\) fails for every exponent

Vanishing endpoints30D degree 8 · 44D degree 4 · every order

1. The unifying mechanism

Let \(Q_t\) be the formal inverse of a Keller pencil \(I+tH\), and let \(\lambda\) be a linear observable. The paper proves two coefficient transfers.

Image transfer. For \(A_H=-\xi\mathbin{\cdot}H\),

\[\lambda(Q_t(Z))=\sum_{m\ge0}\frac{t^m}{m!}E_n(\lambda A_H^m)(Z).\]

Symmetric transfer. For \(P_H(A,B)=iH(A+iB)\mathbin{\cdot}B\), every inverse coefficient becomes

\[D_v\!\left(\Delta^mP_H^{m+1}\right)=2^m m!(m+1)!\,[t^{m+1}]\lambda(Q_t).\]

These identities—not a bare appeal to equivalence with the Jacobian conjecture—are the logical spine. One cubic series supplies closed nonzero coefficients in all three residue classes modulo three.

2. The exact 14-variable map

The nonlinear vector \(g\in\mathbb Q[Z]^{14}\) has 24 displayed monomials and satisfies

\[\det(I+sJg)=1,\qquad (Jg)^{14}=0,\qquad (Jg)^{13}\ne0.\]

The special fiber is certified scheme-theoretically by the lexicographic basis

\[4z+1-27x^2,\qquad 2y+3x,\qquad x(x-1)(x+1),\]

so it contains exactly three reduced rational points. The same map has generic degree three and geometric monodromy \(S_3\). A weighted dilation conjugates every nonzero member of its pencil.

3. Every-exponent Special Image failure

For \(A=-\xi\mathbin{\cdot}g\) and one fixed linear multiplier \(b\), the draft proves for every positive integer \(m\)

\[E_{14}(A^m)=0,\qquad E_{14}(bA^m)\ne0.\]

This makes \(\ker E_{14}\) non-Mathieu and gives an explicit counterexample to \(\operatorname{SIC}(14)\). Padding propagates the statement to every dimension at least 14.

4. Every-order Hessian endpoints

The same 14D inverse series gives a 30-variable homogeneous degree-eight Hessian-nilpotent polynomial. A complementary rank-compressed cubic route gives a 44-variable homogeneous quartic. For each headline homogeneous polynomial \(P\),

\[\Delta^mP^m=0\quad(m\ge1),\qquad \Delta^mP^{m+1}\ne0\quad(m\ge0).\]

The 30D result optimizes ambient dimension within this package; the 44D result addresses the classical homogeneous-quartic formulation. The nonvanishing is proved by closed formulas at every order, not extrapolated from a finite computation.

5. Named consequences without double counting

The exact real three-point fiber and constant spectrum also refute the stated Campbell, full Chamberland–Meisters, and Kulikov injectivity or fixed-point formulations. They arise from direct sign, translation, and fixed-point transforms of one map and are presented as one consequence cascade. Older Markus–Yamabe counterexamples are expressly acknowledged.

6. Exact certificates

verify_symbolic.py — exact fiber, conjugacy, homogeneous Jordan type, inverse formulas, symmetric gradient identity, term counts, and Zhao normalization.
unified_every_order.json — expanded 178-term and 608-term companions, targets, formulas, fiber basis, and pinned precursor hashes.
verify_exported_stdlib.py and verify_exported_node.mjs — independent Python and Node.js/BigInt checks.
MANIFEST.md — claim-to-check inventory and explicit dependency boundaries.

Unified certificate SHA-256: deb01a83cea8543b17c13e8849cead0159d5d8feac07ae18034fd880a274495c

The reference symbolic run peaked at approximately 74 MiB RSS. The verification uses sparse structural identities and never expands high powers of the 30D or 44D polynomial.

7. Scope, priority, and archival policy

The general reductions, symmetric transport, Vanishing formulation, and dimension-14 degree bookkeeping are classical. The narrow candidate contribution is the sparse explicit realization and its coefficient-complete obstructions. No global minimum is claimed. The literature search is provisional and cannot establish worldwide priority.

Discoveries 03, 05, and 06 remain public as timestamped technical precursors and exact certificate sources; they are not counted as three additional current papers. Discovery 04 remains separate because its iterated-monodromy theorem studies different mathematics.

AI-assistance and verification disclosure

The exploratory algebra, theorem architecture, proof drafting, verification programs, audits, and website were developed with heavy assistance from ChatGPT 5.6 Sol under Alec Kriebel's direction. Alec Kriebel is the named human author and cannot independently validate the claims. The work has not been peer reviewed.

Suggested citation

Alec Kriebel, with heavy assistance from ChatGPT 5.6 Sol, “A 14-variable unipotent Keller map and every-order image and vanishing obstructions,” provisional research note, 22 July 2026. Permanent page.