A quotient-Jacobian identity for the full (1,−1,−2)-equivariant class containing the July 2026 counterexample to the Jacobian conjecture, with an unconditional classification through degree four

This deposit contains two related notes and their complete computational reproduction package. An apparently new explicit quotient-Jacobian identity for the equivariant class containing Alpöge's July 2026 counterexample to the Jacobian conjecture. Let C* act on A³ with coordinate weights (1,−1,−2), and consider equivariant polynomial maps whose components have weights (−2,−1,1), as in the counterexample. Writing u = xy and v = x²z, every such map has the normal form F = (zP + y²Q, yR + xzS, xT), with P, Q, R, S, T in C[u,v], subject to the natural module relations. The induced map G on the categorical quotient plane satisfies the polynomial identity det DG = −T²·det DF, and under the stated dominance hypotheses the generic field-extension degrees agree: μ(F) = μ(G). The deposit additionally contains a computer-assisted classification of low-degree maps in this equivariance class: every (1,−1,−2)-equivariant Keller map of polynomial degree at most four is shown to be a polynomial automorphism, supported by explicit polynomial inverses, verification of both compositions, and an exhaustive decomposition of the Keller ideal independently reproduced in Singular and Macaulay2. To our knowledge, this is the first classification of Keller maps of degree at most four in this specific equivariance class. All certificates, solver inputs and transcripts, verification scripts, checksums, and reproduction materials are included. Preliminary degree-five computations — modular decompositions and exactly verified candidate families — are also included but are explicitly marked as non-exhaustive. No degree-five classification is claimed. Licensing: documents CC BY 4.0; code and solver inputs MIT (see LICENSE.md in the archive). The deposit is pseudonymous. A SHA-256 commitment to the authors' identity is included in COMMITMENT.txt. Correspondence: minustsquared@gmail.com. The formulation and exploratory computations were carried out with assistance from Claude Fable by Anthropic. The mathematical arguments are presented explicitly, and the computational claims are supported by reproducible exact-arithmetic certificates that do not depend on accepting an AI system's reasoning. Files: quotient-jacobian-identity-note.pdf — normal form, quotient-Jacobian identity, and generic-degree proposition. deg4-classification-note.pdf — computer-assisted degree-at-most-four classification theorem. keller-quotient-deg4-artifacts.zip — complete reproducibility archive containing certificates, solver inputs and transcripts, verification scripts, and checksums.