Abstract: Value-based Task Selection (VTS) asks whether a real-time system can select tasks whose total value reaches a required target while preserving every deadline. We study its threshold decision form for finite collections of independent sporadic tasks scheduled preemptively on one processor. We prove that this problem is $Σ_2^P$-complete for both constrained and arbitrary deadlines, even when all numerical parameters are encoded in unary. The existential choice is the selected task set, while the universal condition requires every processor-demand window to remain feasible. Our hardness reduction already uses constrained-deadline tasks. It represents universal assignments by Chinese Remainder Theorem timestamps and uses polynomial-weight local selector locks to enforce a valid existential assignment. The reduction produces polynomially many tasks, and every generated execution time, deadline, period, task value, and target is polynomially bounded in the source-instance size.