We prove that for all integers n ≥ 0, the degree-4 Jensen polynomial K_{4,n}(X) of the completed Riemann xi function has all four roots real and negative. This extends the theorem of Dimitrov and Lucas (Proc. AMS 139, 2011) from degree 3 to degree 4 in the Jensen polynomial chain for the Riemann xi function. The proof proceeds through two discriminant conditions: a Turán-type inequality (C2) established via the Brascamp-Lieb inequality applied to Phi moment ratios, and a monotonicity argument (C1) for the factorial-normalized discriminant. This result does not prove the Riemann Hypothesis, which requires the analogous statement for all degrees simultaneously.