Moment defects and sharp iid Berry–Esseen bounds
Starting from Shevtsova’s sharp third-moment inequality, we develop an exact defect from the standardized two-point family and derive quantitative transport and characteristic-function bounds. A penalized extremal problem yields eventual two-point optimality, \(C_n=C_Q^*(n)<C_{\mathrm E}\), together with a uniform linear deficit and sharp cubic Wasserstein stability. A fifth-order lattice saddle-point expansion gives the full oscillatory \(n^{-2}\) correction to the optimal constants. A separate effective argument proves \(C_n<C_{\mathrm E}\) for every \(n\ge31\), using finite-index…
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