Sharp n log n Real-Zero Asymptotics and Clock Spacing for High Derivatives of the Riemann–Jacobi Kernel

This preprint determines the asymptotic growth of the distinct real zeros and sign-changing real zeros of high derivatives of the Riemann–Jacobi kernel. It proves that both zero counts grow on the scale of the differentiation order multiplied by its logarithm, with an explicit leading constant and a linear error term. The result also shows that all but a lower-order number of the real zeros are simple sign changes. After logarithmic rescaling of the real axis, the zeros become uniformly distributed on a fixed interval. At the microscopic scale, almost all neighboring real zeros exhibit the same asymptotic spacing. The proof represents each theta mode through a common family of generalized Bell polynomials and analyzes their saddle-point geometry. Exact crossing points between neighboring mode envelopes produce a tessellation of the relevant region. Uniform dominance estimates transfer the oscillations of the leading mode to the full theta sum over a growing family of cells. Complex-analytic zero bounds control the transition regions, while an exact generalized-Bell root estimate excludes zeros in the remaining tail. The accompanying computation package independently verifies the algebraic identities, saddle geometry, cell-mass calculations, and selected finite-order zero counts used as consistency checks. The numerical computations are not used as substitutes for the analytic proof. Research methodology and AI assistance:This work was developed using the CARMA-Math research workflow, a cumulative AI-assisted mathematical research methodology using persistent research archives, literature and prior-art investigation, iterative proof exploration, and verification procedures. Generative AI (ChatGPT) was used extensively for mathematical exploration, proof development, computational reasoning, literature research, and manuscript preparation.