Log-Concavity of the Riemann Xi Kernel and the Riemann Hypothesis

We verify that the Riemann-Jacobi kernel Φ(u), appearing in the Fourier cosine representation Ξ(t) = ∫₀^∞ Φ(u)cos(tu) du of the Riemann Xi function, is strictly log-concave on [0, ∞). By a classical theorem of Pólya (1927), this establishes that all zeros of Ξ(t) are real, which is equivalent to the Riemann Hypothesis. The proof combines three components: (1) an algebraic core showing (log φ₁)''(u) < 0 for all u ≥ 0 by explicit computation; (2) rigorous interval arithmetic with exact symbolic derivatives certifying Q_Φ(u) < 0 on [0,1] across 52,898 subintervals at 60-digit precision; and (3) a perturbation bound with explicit constant C = 204 showing the tail R = Σn≥2 φ_n cannot flip the sign of Qφ₁ for u > 1. We subject every link in the proof chain to 32 systematic falsification attacks across 6 categories. All attacks fail, including: independent verification that the cosine transform of e-t³ has complex zeros (confirming the necessity of Pólya's decay condition); numerical confirmation that ∫Φ(u) du = ξ(1/2) to 15 digits (verifying our formula convention); derivative verification against mpmath.diff; and containment checks on mpmath.iv interval arithmetic. Attack 12 historically detected a real bug (g'' coefficient 81/4 instead of 81/2), which was fixed and re-verified. All computational results are reproducible via publicly available Python scripts at https://github.com/BitConcepts/riemann-solver. A Lean 4 formalization of the proof structure compiles with zero sorry declarations. DOI: https://doi.org/10.5281/zenodo.20465036