A genus-zero obstruction for a marked polynomial Pell equation over Q(i)

Let K = Q(i), and let x, D and C be polynomials over K with degrees (3s, 5s, 6s), where s is a positive integer. This candidate paper studies the equation C² − 8Cx³ + 6CxD + D³ = q for a nonzero constant q. It gives an argument that A = x² − D is not a square and that the smooth quadratic curve determined by the squarefree part of A cannot have genus zero. The norm identity yields a polynomial Pell equation with the additional square condition 3x² = 4V − A, where V = x² − D/4. In genus zero, Laurent parametrization forces a narrow support interval. A local argument at 3 treats coefficient poles; in the ramified critical case a uniquely determined comparison polynomial has incompatible support. A separate differential identity bounds the support of gcd(x, C) and gives a noncube-norm consequence in degrees (6, 10, 12). The results concern these polynomial equations and degree hypotheses. They do not assert that arbitrary radical expressions enter this model or determine an unrestricted radical-approximation exponent. The matching support archive includes article source, proof maps, exact proof records and artifact-identity checks.