Generic Multiplicities of Hilbert Schemes on Thickened Lines

Let T_(n,beta) be the Hilbert scheme of n points on the thickened line defined by y^beta=0 over C. For the component indexed by a partition lambda of n with largest part at most beta, we compute its ambient generic multiplicity as the product of binomial(beta,lambda_i). The proof uses disjoint-support factorization and a cyclic chart in which a monic polynomial divides t^beta; the remainder equations form a weighted complete intersection. In particular, a chart of T_(2,3) has generic local ring K[u]/(u^3), whereas imposing a repeated-root discriminant produces K[u]/(u^2). This distinguishes ambient multiplicity from the multiplicity of a support stratum. We compare with the known multinomial multiplicity of the characteristic-polynomial nilpotent fiber, and describe reduced Jordan strata and their closure obstruction. The stated geometric theorem is complete; the broad AIM categorical cell question remains unresolved. This preprint is unrefereed and makes no absolute-priority or formal-verification claim.