Curvilinear Geometry, Primary Structure, and Gröbner Degenerations of Conductor Fiber Cones in a Huneke–Wiegand Family

For every integer p>=4, a previously constructed symmetric numerical semigroup ring R_p and rigid two-generated monomial ideal determine a conductor ideal T_p whose special fiber C_p = F(T_p) is one-dimensional Cohen-Macaulay, generated in degree one by 10p monomials, and has multiplicity 24p. We prove that this fiber cone admits the explicit standard-graded parametrization C_p ≅ 𝕜[xy^a : a ∈ G_p] ⊆ 𝕜[x,y]/(y^24p), with deg(xy^a) = 1, where G_p is the exact degree-one offset set. If P_p = 𝕜[X_a : a ∈ G_p] and J_p is the defining ideal, then √J_p = L_p = (X_a : a > 0), J_p is L_p-primary. Thus the complete primary decomposition has one component. The nilradical of C_p has sharp nilpotency index 24p: its (24p-1)st power is nonzero, while its 24p-th power vanishes. Dehomogenization at X_0 gives C_p/(X_0-1) ≅ 𝕜[y]/(y^24p). Consequently Proj(C_p) is a saturated length-24p curvilinear fat point with one-dimensional Zariski tangent space. It is locally Gorenstein, although its homogeneous coordinate ring has Cohen-Macaulay type 10p+1 and is neither level nor Gorenstein. We also compute its affine Kähler differential module, including the exact characteristic split. The proof is deductive from previously proved offset-basis and Cohen-Macaulay input theorems. We further give the complete reduced Gröbner basis for graded reverse lexicographic order with X_0 last. Its degree profile is (50p^2-17p, 5p-1, p-2) in degrees two, three, and four, with no later elements. No minimal leading monomial contains X_0, yielding a flat Cohen-Macaulay monomial degeneration. Exact parameter sweeps, an independently encoded clique-based recomputation, and an all-parameter Presburger boundary certificate validate the formulas and run negative controls, but do not replace the deductive arguments. The results concern the conductor fiber cones of this explicit family; they do not assert that arbitrary monomial fiber cones are primary, curvilinear, Cohen-Macaulay or locally Gorenstein. Source code, compact artifacts, premise hashes and proofs: https://github.com/fsantibanezleal/CAOS_RESEARCH .