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

This research letter establishes a higher-order logarithmic derivativeinequality for the positive even kernel \(Φ\) associated with the RiemannXi 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-seriesrepresentation are isolated and reduced, after the substitution \[q=π e⁴ˣ,\] to an explicit finite exponential-polynomial inequality. The remaininginfinite tail is bounded uniformly and shown to be too small to change thesign. A structural relation with the Jacobi theta function is also exhibited.If \[Θ(t)=θ_3(0,it),=td/dt,=e⁴ˣ,\] then the Riemann kernel satisfies \[{Φ(x)=t1/4(E^2+12E)Θ(t).}\] Equivalently, with \[G(y)=ey/4Θ(e^y),=4x,\] one has \[{Φ( y4)=(d^2/dy^2-1{16})G(y).}\] This identifies the Riemann kernel as a second-order Euler-derived Jacobitheta kernel and connects the main theorem with known logarithmic-scaletheta-function inequalities. The note then applies the higher-order inequality to the standard secondcorrelation 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 gives the factorization \[{ν_2(r)=4Z(r)V(r).}\] An exact covariance formula for \(V'(r)\), together with\((logΦ)'''<0\), yields \[{V'(r)<0(r>0).}\] Thus the normalized overlap, interpreted here as a beat variance, decreasesstrictly with separation. The note also gives a three-channel Krein--Gram decomposition \[{ν_2=R_a+2R_g-R_b,}\] which reduces the Fourier-sign problem to two positive autocorrelationchannels and one negative channel. The arithmetic motivation is provided by Mellin waves \[e-itlog n,\] for which a pair of integers \(m,n\) naturally separates into a logarithmiccarrier \[log(mn)\] and a logarithmic beat \[log(n/m).\] The relation to the Riemann Hypothesis is stated explicitly. Thereal-variable results proved in the note do not establish the requiredFourier positivity. In the normalization used here, the theorem of Dimitrovand Xu, together with Wiener's Tauberian theorem, identifies the remainingRH-equivalent condition as \[{{(ur)ν_2(r)}(t)>0}\] 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, the Krein--Gramdecomposition, and Gaussian heat-flow localization do not by themselvesimply this off-axis Fourier-positivity condition. No proof of the Riemann Hypothesis is claimed. The principal contribution isthe global inequality \[(logΦ)''''<0,\] together with its strict beat-variance consequence, its Euler-derivedtheta-kernel interpretation, and a precise identification of the remainingRH-equivalent Fourier obstruction.