A digestion of the Jacobian conjecture counterexample

While this is an extremely quick verification, the construction presented in this fashion appears like a massive miracle. The polynomial {F} has degree seven, so a priori the Jacobian {\mathrm{det} DF} ought to be a polynomial in three variables of degree as large as {3 \times 6 = 18}, so the fact that all non-constant coefficients of this polynomial vanish looks like a massive cancellation involving {\binom{18+3}{3}-1 = 1329} equations, which is much larger than the {3 \times \binom{7+3}{3} = 360} degrees of freedom for a generic degree seven polynomial map of three variables. So finding such a polynomial looks highly unlikely to be located by brute force.
The example has since been retroactively explained in more geometric terms. As a “digestion” exercise to myself, I sought to write this explanation with relatively little use of algebraic geometry, in a manner that minimizes the amount of “miracles” required, although there are still a few places where some remarkable phenomena occur.

I'm not a math buff, but I came across this online and it seemed interesting. A case where AI disproved a conjecture in a way that seemed to involve ingenuity, or at least not brute force. Can anyone speak to the implications of this result? Seems somewhat fundamental to me.