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 \(G=gr_F A\), we construct the filtration-degree spectral sequence of the normalized Hochschild cochain complex and define a tower of spectral central radicals \[ p_r(A,F)=radZ(G)∩ E_r^0. \] We prove that \( p_r(A,F)=rad(E_r^0)\), that the tower stabilizes after finitely many pages, and that its successive losses determine well-defined internal-degree death multiplicities. The powers \( p_r(A,F)^sE_r\) define pagewise radical-image filtrations, for which \[ d_r( p_r^sE_r) ⊆ p_rˢ⁻¹E_r (s≥ 1). \] 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 \(μ=∑i≥0μ_i\). If a homogeneous element \(z_0∈ Z(G)^p\) admits corrections \(z_a∈ Gᵖ⁺ᵃ\) through order \(r-1\), then the \(r\)-th spectral differential is represented by the recursive central-lifting obstruction \[ O_r(z_0;z_1,…,zᵣ₋₁) =∑ᵢ₌₁ʳ[μ_i,zᵣ₋ᵢ]G, d_r([z_0]_r)=[- O_r]_r. \] At the limit page, we obtain a canonical isomorphism \[ p_∞(A,F) grF(A) radZ(A). \] 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 \(radZ(G)\) 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.