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) 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