A Fourth-Derivative Inequality for the Riemann Xi Kernel (Letter)

This research letter establishes a higher-order differential inequality for the positive even kernel \(Φ\) associated with the Riemann Xi function. Writing \[L(x)=logΦ(x),\] the main theorem proves \[L''''(x)<0 (x≥0),\] and therefore, by evenness, \[L'''(x)<0 (x>0).\] The proof is analytic. The first three terms of the theta-series representation are isolated and reduced, after the substitution \[q=π e⁴ˣ,\] to an explicit finite exponential-polynomial inequality. The remaining infinite tail is bounded uniformly and shown to be too small to change the sign. The note then applies this higher-order logarithmic inequality to the standard second correlation kernel \[ν_2(r)=∫R(r-2s)^2Φ(r-s)Φ(s)\,ds.\] Introducing the ordinary overlap autocorrelation \(Z(r)\) and the variance \(V(r)\) of the normalized overlap distribution, one obtains the factorization \[ν_2(r)=4Z(r)V(r).\] An exact covariance formula for \(V'(r)\), together with \(L'''<0\), yields the strict monotonicity theorem \[V'(r)<0(r>0).\] Thus the normalized overlap, interpreted here as a beat variance, becomes strictly narrower as the separation increases. The note also gives a three-channel Krein--Gram decomposition \[ν_2=R_a+2R_g-R_b,\] which isolates a single negative Fourier channel in the first Laguerre-sign problem. The arithmetic motivation is provided by Mellin waves \[e-itlog n,\] for which a pair of integers \(m,n\) naturally separates into a logarithmic carrier \(log(mn)\) and a logarithmic beat \(log(n/m)\). The relation to the Riemann Hypothesis is stated explicitly. The real-variable results proved here do not establish the required Fourier positivity. In the normalization used in the note, the remaining RH-equivalent condition, via the theorem of Dimitrov and Xu and Wiener's Tauberian theorem, is the strict positivity \[{(ur)ν_2(r)}(t)>0\] for all \[t∈ R,0<|u|<1.\] The note explains why positivity and strict decrease of the beat variance, positive definiteness of the ordinary autocorrelation, and the three-channel decomposition do not by themselves imply this off-axis Fourier-positivity condition. No proof of the Riemann Hypothesis is claimed. The principal contribution is the global inequality \[(logΦ)''''<0,\] together with its strict beat-variance consequence and a precise identification of the remaining RH-equivalent Fourier obstruction.