Harmonic Theta-Mode Dominance and Superlinear Real-Zero Growth for the Riemann–Jacobi Kernel

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Original abstract

This preprint studies the distinct real zeros of high derivatives of the Riemann–Jacobi kernel. It proves that their number grows faster than every fixed linear multiple of the differentiation order. The proof represents individual theta modes through generalized Bell polynomials, isolates dominant oscillatory modes on pairwise disjoint logarithmic cells, and transfers their alternating sign patterns to the full kernel. Fixed finite families of cells yield a harmonic accumulation of zero-count lower bounds and hence superlinear real-zero growth. Version 2 retains the main theorem but replaces the proof with a corrected fixed-finite-cell argument. The revised proof applies the two-saddle expansion only on fixed compact regions and controls all remote modes using an exact generalized Bell-polynomial identity and a positive-coefficient majorant. This resolves the growing-cell and coalescing-saddle uniformity issues that were not established in Version 1. Version 2 does not claim a two-term zero-count asymptotic, a Huxley-type power-saving remainder, or uniform control over a range of modes growing with the differentiation order. 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.

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