Sharp Energy Inequalities, Exact One-Moment Bounds and Optimal Wasserstein Stability for Expanding Circle Maps

Preliminary Research Results — Work in Progress This preprint presents preliminary mathematical research conducted across Labs 10–19, focusing on sharp energy inequalities for exponential sums, exact complex moment inequalities, and quantitative Wasserstein stability for expanding circle maps. The research investigates optimal coefficients, equality cases, stability bounds, and counterexamples involving finite Fourier moment constraints. It also examines the relationship between these results and problems arising in the study of the digits of pi. The work combines mathematical analysis, AI-assisted research, and computer-assisted verification. Research status: Preliminary results. This manuscript has not undergone peer review. Some computer-assisted results require further independent verification, and the complete original computational certificates are not included in this release. Academic novelty and priority have not been conclusively established. No proof of the normality of pi or a new law governing its decimal digits is claimed. Author: Lee Chae Hun Researcher status: Independent high school student researcher, South Korea.