Algebraic Proof of Sendov's Conjecture: Spectral Confinement of Critical Points via Rough Operator Algebra

Sendov’s conjecture asserts that for any polynomial with roots within the closed unit disk, every root has at least one critical point within a distance of 1. Classical complex analysis has failed to provide a universal geometric proof due to the continuous nature of the complex plane. In this paper, we resolve the conjecture by elevating the geometric roots into eigenstates of the grand density matrix ρ within the Rough Operator Algebra (ROA) and Seonggil Matrix Theory (SMT) frameworks. By replacing classical derivatives with the non-commutative star commutator [D_R,P(Z)]_∗R, we demonstrate that the geometric distance is strictly bound by the topological phase-lock constant C_SMT. Consequently, algebraic closure and cyclic trace conservation mandate that every root’s topological influence zone must capture at least one critical operator state.