An Explicit Rational Homotopy from the Faran Map to a Linear Map in Target Dimension Four

We construct a homotopy through proper rational holomorphic maps from the unit ball in \(\mathbb{C}^2\) to the unit ball in \(\mathbb{C}^4\) , joining \((z,w)\mapsto(z^3,\sqrt{3}zw,w^3,0)\) to \((z,w)\mapsto(z,w,0,0)\) . The construction answers the explicit target-four question in the AIM list on the Cauchy–Riemann equations. A mixed-term polynomial family of coefficient rank four joins a unitary transform of the Faran map to a partially tensored quadratic map. An explicit rational segment then degenerates to a Whitney map, which is joined to the linear embedding. The full family is jointly continuous on the closed source ball, and each slice extends holomorphically past that ball. All slices have rational degree at most three. We prove a uniform estimate at the degenerating endpoint and explain why this particular path is not continuous in the \(C^1\) boundary topology. No classification of arbitrary ball-map homotopies is asserted.