(SAPZ) Quantitative BMO Conditions, Critical Zero Spacing, and Numerical Validation: A Comprehensive Study

This preprint presents a comprehensive study on the relationship between quantitative Bounded Mean Oscillation (BMO) conditions and the distribution of critical zeros of generalized zeta and LLL-functions. We rigorously define a quantitative BMO framework for Dirichlet-type functions, derive explicit lower bounds for consecutive zero spacing on the critical line, and provide proof sketches of key theorems. The study includes numerical estimates of BMO constants for the Riemann zeta function, empirical verification using zero datasets, and statistical validation via Monte Carlo simulations. Graphs, tables, and detailed numerical results are included to support the theoretical findings. Potential applications include extensions to Dirichlet and automorphic LLL-functions, Rankin–Selberg convolutions, and connections with quantum chaos and random matrix theory. This document is intended for academic dissemination and journal submission. The LaTeX source, figures, and supplementary materials are provided for reproducibility.