On the Log-Concavity of the Riemann Xi Kernel

The Riemann Xi function admits the representation \( \Xi(t) = \int_0^\infty \Phi(u)\cos(tu)\,du \) where \( \Phi \) is a positive, even, integrable function. By a classical theorem of P\'olya (1927), if \( \log\Phi \) is concave on \( [0,\infty) \), then \( \Xi \) has only real zeros, which is equivalent to the Riemann Hypothesis. We prove that the dominant term of \( \Phi \) has strictly negative second logarithmic derivative for all \( u \geq 0 \), reducing the full log-concavity to a quantitative tail estimate. We verify this estimate by rigorous interval arithmetic (5000 certified subintervals on \( [0, 1/2] \) at 80-digit precision, with the complement handled analytically). The entire argument is formalised in the Lean~4 proof assistant with the Mathlib library.