Constrained degree-five Pell equations over characteristic-zero fields and a sharp Gaussian specialization minimum

We present candidate results classifying polynomial triples (x,D,C) of exact degrees (5,8,9) over characteristic-zero fields for which C^2 - 8Cx^3 + 6CxD + D^3 is a nonzero constant. Over an algebraically closed field, the constrained Pell problem has thirteen reflection-symmetric classes and one asymmetric class. The symmetric residual algebra is a reduced Q-algebra of dimension fifteen: two points have zero norm and the other thirteen lift to original polynomial sources. All exceptional elimination fibers are retained. Descent is described using the roots of a degree-twelve polynomial and the coefficient field's squareclasses; the two-shape classification over Q(i) extends exactly to fields in which that polynomial has no root. For genuine Gaussian specializations of the Q(i) class, with exact radical orders 1,2,3 and modulus height at most B, the sharp restricted minimum is (2^29 3^3)^(-1/5) B^(-21/10) + O(B^(-9/4)), uniformly as models, norms and presentations vary. Eventually every minimizer has unit norm and belongs to the symmetric class. The results concern this complete polynomial class; they do not determine the unrestricted Gaussian mixed-radical minimum or Gaussian outputs from the additional number-field sources. The full residual chart has seventeen distinct geometric points, but no full-scheme length or reducedness claim is made.