Third-Level Log-Concavity of the Riemann Xi Kernel: A Computer-Assisted Proof

We present a computer-assisted proof of the third-level log-concavity of the Riemann Xi kernel. With s(t)=Phi(sqrt(t)), L[g]=g'^2-g g'', f1=L[s], and f2=L[f1], the result establishes (log f2(t))''<0 for every t>0. The proof combines an exact structural reduction, a Taylor-Bernstein certificate with a Cauchy remainder bound near the modular point, a directed-interval certificate on a compact region, and an asymptotic first-theta-summand plus rigorous tail argument. The identity f3=-tau2(tau3^2+s^2 tau4) is verified symbolically. The numerical proof is supplied with a reproducibility package and a separately rewritten cross-validation path. No assertion of the Riemann Hypothesis is made.