Curvilinear geometry and primary structure of conductor fiber cones in a Huneke-Wiegand family

For every integer p>=4, a previously constructed symmetric numerical semigroup ring and rigid two-generated monomial ideal determine a conductor ideal whose special fiber is a one-dimensional Cohen-Macaulay algebra of multiplicity 24p. This companion preprint proves the explicit standard-graded parametrization C_p = k[x y^a : a in G_p] inside k[x,y]/(y^(24p)). If P_p/J_p is the presentation in its 10p degree-one variables and L_p is generated by all positive-offset coordinates, then radical(J_p)=L_p and J_p is L_p-primary. Thus the complete primary decomposition has one component. The nilradical has sharp nilpotency index 24p. Dehomogenization at the reduction variable gives k[y]/(y^(24p)). Consequently Proj(C_p) is a saturated length-24p curvilinear fat point with one-dimensional tangent space. It is locally Gorenstein, although its homogeneous coordinate ring has Cohen-Macaulay type 10p+1 and is neither level nor Gorenstein. The affine Kahler differential module is computed exactly, including the distinction between characteristics dividing 24p and all other characteristics. The proof is deductive from frozen offset-basis and Cohen-Macaulay input theorems. An exact campaign for p=4,...,300 and an independently encoded all-row audit validate the implementation and adversarial controls but do not replace the symbolic argument. The scope is confined to the explicit conductor family and does not assert analogous properties for arbitrary fiber cones or arbitrary Huneke-Wiegand counterexamples. Code, compact artifacts, premise hashes, proof, and verdict: https://github.com/fsantibanezleal/CAOS_RESEARCH.