math.PR — Probability
Moment defects and sharp iid Berry–Esseen bounds
The known sharp third-moment inequality supplies an exact defect that identifies the standardized two-point family and controls transport and characteristic functions. We develop the corresponding Fourier reduction and give the qualitative eventual upper-bound argument. A penalized extremal problem yields eventual two-point optimality and a uniform linear deficit, with sharp cubic Wasserstein stability. A fifth-order lattice saddle gives the full oscillatory correction for the optimal constants. A separate effective argument proves for every , by a finite base and a first-failure induction. Exact polynomial certificates give bounds for two and three summands; a complete interval argument treats all three-point laws at four summands. An elementary historical bound is included as an appendix. ChatGPT and Codex assisted the development of the proofs and computational certificates and the preparation of this manuscript, using the GPT-5.6 Sol, GPT-6.1 Sol and GPT-6 Astra models.