Spectral Central Radicals and Filtered Deformations of Hochschild Cohomolog

Let \(k\) be a field and let \(A\) be a finite-dimensional unital \(k\)-algebra equipped with a finite decreasing multiplicative filtration \(F\). For \(G=\operatorname{gr}_F A\), we construct the filtration-degree spectral sequence of the normalized Hochschild cochain complex and define a tower of spectral central radicals \[ \mathfrak p_r(A,F)=\operatorname{rad}Z(G)\cap E_r^0. \] We prove that \(\mathfrak p_r(A,F)=\operatorname{rad}(E_r^0)\), that the tower stabilizes after finitely many pages, and that its successive losses determine well-defined internal-degree death multiplicities. The powers \(\mathfrak p_r(A,F)^sE_r\) define pagewise radical-image filtrations, for which \[ d_r\bigl(\mathfrak p_r^sE_r\bigr) \subseteq \mathfrak p_r^{s-1}E_r \qquad (s\geq 1). \] The induced differential on the radical-order associated graded object is square-zero and has bidegree \((-1,1)\) with respect to radical order and total Hochschild degree. After choosing a unital filtration splitting, we express the transported multiplication as \(\mu=\sum_{i\geq0}\mu_i\). If a homogeneous element \(z_0\in Z(G)^p\) admits corrections \(z_a\in G^{p+a}\) through order \(r-1\), then the \(r\)-th spectral differential is represented by the recursive central-lifting obstruction \[ \mathcal O_r(z_0;z_1,\ldots,z_{r-1}) =\sum_{i=1}^{r}[\mu_i,z_{r-i}]_{\mathrm G}, \qquad d_r([z_0]_r)=[-\mathcal O_r]_r. \] At the limit page, we obtain a canonical isomorphism \[ \mathfrak p_\infty(A,F) \cong \operatorname{gr}_{F\cap\operatorname{rad}Z(A)} \operatorname{rad}Z(A). \] Analogous comparisons for radical powers and their actions on Hochschild cohomology are characterized by strictness of explicit filtered multiplication maps. Finally, we compute a four-dimensional filtered deformation in which two generators of \(\operatorname{rad}Z(G)\) support nonzero \(d_1\)-differentials, whereas the remaining generator recovers the associated graded actual central radical. **Keywords** Hochschild cohomology; filtered algebra; spectral sequence; central radical; Rees algebra; Gerstenhaber bracket; filtered deformation.