Abstract
Starting from the recently announced three-dimensional noninjective Keller map, a rank-compressed homogenization gives a 22-variable cubic homogeneous noninjective Keller map. The de Bondt–van den Essen symmetric reduction then gives a homogeneous quartic Hessian-nilpotent polynomial in 44 variables, with 538 monomials and an exact collision. This was the consolidated paper for Explorations 01 and 02; Discovery 07 now incorporates its cubic-to-quartic construction into the canonical inverse-series consequence theorem.
1. The six-variable symmetric map
Let \(\Phi:\mathbb C^3\to\mathbb C^3\) be the identity-linear normalization of the announced map. For \(A,B\in\mathbb C^3\), set
Theorem A. The gradient \(\nabla\mathcal S:\mathbb C^6\to\mathbb C^6\) has symmetric Jacobian, identity linear part, and Jacobian determinant one. It maps the three distinct points
to \((0,0,-1/8,0,0,i/8)\). The potential has degree eight and 204 expanded monomials.
This is a normalized instance of Meng's classical gradient lift. Cassidy's repository had already executed the equivalent six-dimensional symmetric transport and the same lifted fiber on 20 July 2026, before this note. No novelty is claimed for Theorem A. Its Hessian has constant determinant one by a two-block determinant formula followed by a complex linear congruence of determinant \(i\).
2. Rank-compressed homogenization
Lemma. Suppose \(\Psi=X+H_2+H_3:\mathbb C^n\to\mathbb C^n\) has determinant one and \(H_3=BK\), where \(B\) is a constant \(n\times r\) matrix and \(K\) is cubic homogeneous. Then
is cubic homogeneous in \(n+r+1\) variables, \(Jh\) is nilpotent, and every collision of \(\Psi\) lifts to one for \(W\mapsto W+h(W)\).
In the certified 13-variable stable model, the thirteen cubic components span eight dimensions. Thus \(13+8+1=22\). The resulting cubic map has 72 monomials across 21 active nonlinear coordinates and an exact rational three-point fiber.
3. The 44-variable quartic
For that 22-variable cubic map, define
Theorem B. The polynomial \(\mathcal P\) is a homogeneous quartic in 44 variables, has 538 expanded monomials, and has nilpotent Hessian. The Keller map \(Z\mapsto Z-\nabla\mathcal P(Z)\) is noninjective. Therefore \(\mathcal P\) explicitly witnesses failure of Zhao's Vanishing Conjecture.
The de Bondt–van den Essen identity transfers nilpotence of \(Jh\) to nilpotence of \(\operatorname{Hess}(\mathcal P)\). The exact colliding 44-tuples have coordinates in \(\mathbb Q(i)\), with numerator at most 261 in absolute value and denominator at most 16.
4. Exact certificates
The proof-side verifier rebuilds both maps and checks the structural identities. A Python standard-library checker and a separate Node.js BigInt checker read only the four JSON files and differentiate the sparse polynomials independently.
symmetric_potential_sparse.json— 204-term potential.
SHA-256:1e0c97e1c4965c3ef7d85cdfb115d468f79d8b5195a7f34f498015c3c3f5fdd4symmetric_collision.json— exact three-point fiber.
SHA-256:6b5b546f24e839a10ab330ae9b05d1d03d23a6fbbbff8cfa6d1ce742768f7169potential_sparse.json— 538-term quartic.
SHA-256:2a912728161888849e77d607ea1f635233576543ed12d5fe8b2a65e0751789f4collision.json— exact colliding 44-tuples.
SHA-256:aeab7adb021c07dea396d2c0eca0cc7880b93dc7b09b74f60289936a711addd0
5. Scope and priority
The six-variable map was already public in equivalent form: Cassidy's commit 40e1e20…, authored at 14:46:10 UTC on 20 July 2026, executes the de Bondt–van den Essen/Meng transport, checks symmetry, and gives the same lifted three-point fiber. Our determinant-one, identity-linear formula is a normalized presentation, not an independent discovery. The first version of this page missed that source.
Exploration 02's 27/54-variable construction is now an archival derivation subsumed by this paper. William Thompson posted a 24-variable cubic homogeneous reduction at 03:29:42 UTC on 21 July 2026 and has priority for the rank-compression idea. His map has 54 monomials across 23 active nonlinear coordinates; ours has 72 across 21. Both cubic maps are over \(\mathbb Q\) with rational collisions. Ours uses fewer ambient variables (22 versus 24), while Thompson's is sparser. The residual candidate contribution is the different \(13+8+1=22\) construction and its executed 44-variable quartic.
Source-specific searches found no earlier public 22-variable cubic certificate or executed 44-variable quartic certificate. This is not an exhaustive worldwide-priority claim; all novelty statements are provisional. Read the full correction and source comparison.
Appendix A. A uniform rational collision
Exploration 01's full-\(S_n\) theorem was already available in stronger form for the entire Gallagher weighted-lift family in Mikhail Szh's earlier commit. We therefore preserve only its explicit specialization and rational collision, without a monodromy or deck-group novelty claim.
For \(n\ge3\), let \(s_n=(4-2^n)/(n-2)\), \(g_1=(n+2-2^n)/(n-2)\), and \(g_2=2^{n-1}\). The appendix defines an explicit weighted-lift map \(F_n\) and points
for which \(F_n(X_{n,1})=F_n(X_{n,2})=(s_n,s_n,1)\) exactly. At \(n=3\), this is \(F_3(-1/3,4,-54)=F_3(1/4,-2,36)=(-4,-4,1)\). The archived checker verifies the identities in exact rational arithmetic.
AI-assistance and verification disclosure
The constructions, searches, proof organization, verification programs, website, and drafts were developed with heavy assistance from ChatGPT 5.6 Sol. 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, “An explicit 44-variable vanishing witness from a 22-variable cubic Keller map,” provisional research note, first posted 21 July 2026; priority correction 21 July 2026. Permanent page.