A Fourth-Logarithmic-Derivative Inequality for the Riemann Xi Kernel

We prove a global fourth-logarithmic-derivative inequality for the positive even Riemann Xi kernel \[Φ(x)=∑ₙ₌₁∞π n^2(2π n^2e⁴ˣ-3)exp(5x-π n^2e⁴ˣ),\] extended evenly to the real axis. Writing \[L(x)=logΦ(x),\] the main result is \[{L''''(x)<0(x∈ R).}\] The proof is analytic and does not rely on interval arithmetic. After the substitution \[q=π e⁴ˣ,\] the first three theta modes are isolated and reduced to an exact integer-polynomial inequality. The remaining infinite theta tail is controlled by explicit derivative majorants and elementary rational exponential bounds. Since \(L\) is even and \(L'''(0)=0\), the main theorem implies \[{L'''(x)<0(x>0).}\] This yields a further probabilistic/correlation consequence. Define \[Z(r)=∫RΦ(y+r/2)Φ(y-r/2)\,dy\] and the normalized overlap density \[p_r(y)=Φ(y+r/2)Φ(y-r/2)/Z(r).\] Its variance, \[V(r)=∫Ry^2p_r(y)\,dy,\] is proved to satisfy \[{V'(r)<0(r>0).}\] Thus the transverse width of the overlap of two shifted copies of the Riemann kernel strictly decreases with their separation. The associated second correlation kernel \[ν_2(r)=∫R(r-2s)^2Φ(r-s)Φ(s)\,ds\] admits the exact factorization \[{ν_2(r)=4Z(r)V(r).}\] Its Fourier transform is \[{{ν_2}(t)=1{32}[Ξ'(t/2)^2-Ξ(t/2)Ξ''(t/2)].}\] Using the correlation-kernel characterization of Dimitrov and Xu, the remaining off-axis condition in the present normalization is \[{{(ur)ν_2(r)}(t)>0,∈ R, 0<|u|<1.}\] This condition is equivalent to the Riemann Hypothesis. Accordingly, the fourth-logarithmic-derivative inequality and the strict beat-variance monotonicity proved here are unconditional results, whereas extending them to the above global off-axis Fourier positivity would amount to proving RH. A companion paper establishes the separate finite-order total-positivity result \[Φ_4,\] indeed strict total positivity through order four: H. Umihara, *Strict Total Positivity of Order Four for the de Bruijn--Newman Kernel: An Exact Polynomial--Tail Proof of the Global PF4 Property*, Zenodo (2026), doi:10.5281/zenodo.21924561. That finite-order result is structurally related to the present work but does not imply the remaining RH-equivalent off-axis Fourier-positivity condition. The research was developed with substantial assistance from ChatGPT (OpenAI) for symbolic calculation, proof exploration, algebraic verification, and consistency checks. The human conceptual starting point was the “wave-integer” representation, in which an integer \(a\) is represented through the Fourier-dual pair \[δ(x-a)ⁱᵃᵗ,\] and the subsequent observation that \[eⁱᵃᵗ+eⁱᵇᵗ=2ei(a+b)t/2cos(a-b/2t)\] may be analyzed as a beat phenomenon. This beat-wave viewpoint guided the transition from arithmetic wave representations to correlation kernels and the beat-variance structure studied in the present letter. The paper does not claim a proof of the Riemann Hypothesis.