This paper studies the single-machine problem of minimizing the weighted completion time variance (WCTV) and proposes a polynomially computable lower-bounding framework that generalizes the benchmark introduced by Nessah and Chu (2010). We first develop a generalized augmented-sequence decomposition that mathematically connects the weighted problem with the classical unweighted variance setting. Using this decomposition, we derive a family of polynomially computable lower bounds for WCTV. The proposed framework exploits three decision parameters: the choice of the first job, the free processing-time parameter assigned to that job through an invariance property of WCTV, and the split level used in the generalized augmented-sequence representation. We show how these parameters can be incorporated into an improved lower-bounding scheme, establish the validity of the resulting bounds, and characterize their optimization structure with respect to the free processing-time parameter and the split level. We also provide an asymptotically motivated approximation for selecting the free processing-time parameter, which can help circumvent numerical stability issues in large-scale applications. We test our approach on publicly available scheduling instances and provide numerical evidence of the effectiveness of the proposed polynomial lower bounds.