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Updated Mermaid diagram

Consequence diagram for the explicit three-variable Jacobian counterexample. It yields explicit counterexamples to the Jacobian, Dixmier, and Poisson conjectures in every dimension at least three. The Yagzhev-form box is now marked explicit in dimension 19; symmetric or gradient and Druzkowski-form extractions remain open. Other open boxes include the dimension-two Jacobian conjecture and the Kontsevich-BKK automorphism conjecture.

Construction card

Sparse Segre-Yagzhev lift. The posted 11-variable degree-three map is normalized as Phi prime equals X plus Q plus C. Only seven components of the cubic part are nonzero. The displayed lift uses seven auxiliary coordinates and one Segre variable, giving 11 plus 7 plus 1 equals 19.

Exact claim card

There is an explicit map G-hat equals X plus H from complex 19-space to itself. Every nonzero component of H is cubic homogeneous, JH to the eighteenth power is zero but JH to the seventeenth is nonzero, and three distinct rational points share an image. Therefore the Jacobian determinant is identically one, while the map is not injective.

Exact certificate card

The dimension-19 map has JH to the eighteenth power equal to zero, JH to the seventeenth power nonzero, and three distinct rational points colliding. One displayed nonzero entry equals minus 18 times t to the sixteenth, x to the tenth, y to the sixth, and z squared.

Verification card

Two complementary exact checks. SymPy reconstructs the 19-variable map from the posted 11-variable input. A separate pure-Python verifier checks the canonical JSON using exact rational dictionary polynomials. No floating-point arithmetic or numerical sampling is used.

Comparison card

Dimension comparison among public constructions located by July 21, 2026. Demirci gives an explicit Yagzhev realization in dimension 79. Distinct-Soft-3991 and Atkins-Turkish give intermediate degree-three maps in dimensions 17 and 11. The present note gives an explicit sparse Yagzhev lift in dimension 19.