Unbounded Sign-Change Excess in Derivatives of the Riemann–Jacobi Kernel

Abstract

This preprint studies the real-zero geometry of the derivatives of the standard Riemann–Jacobi kernel associated with the Riemann Xi function. The main result shows that every failure of the derivative Laguerre inequality at the symmetry point forces two additional sign changes beyond those supplied by the usual Rolle-theorem mechanism. Consequently, the accumulated center failures produce an unbounded excess in the number of real sign changes of higher derivatives. Combining this transfer principle with previously established asymptotic estimates for the number of center failures gives a quantitative lower bound for the growth of the sign-change excess. In particular, the excess grows at least on the square-root scale. This shows that the Riemann–Jacobi kernel is not bell-shaped in the classical sense, and that the failure becomes increasingly pronounced at high derivative orders rather than remaining a finite-order phenomenon. The paper also studies failures away from the symmetry point through a ratio-flow formulation. This distinguishes productive failures, which create additional real zeros, from sterile failures, which do not. Explicit finite-order examples illustrate both behaviors. The results concern derivative inequalities and the real-zero geometry of the Riemann–Jacobi kernel. They do not prove or disprove the Riemann hypothesis.