Hadamard Rigidity and Sharp Stability in the Completely Bounded Bohnenblust--Hille Inequality

The completely bounded Bohnenblust--Hille inequality of Arunachalam, Dutt, Escudero Gutiérrez and Palazuelos controls coefficient summability with optimal constant one, independently of the ambient dimension. We classify all nonzero equality cases over both the real and complex fields: they are precisely scalar multiples of Hadamard chains supported on equal-sided Cartesian boxes. Thus equality determines both the support and the factorization of the coefficient tensor. For each fixed degree $d\ge2$, we also establish sharp structural stability. A form with completely bounded norm one and deficit $\varepsilon$ lies within $C_d\sqrt\varepsilon$, in the critical coefficient $\ell_{2d/(d+1)}$ norm, of both a flat unimodular path and a normalized unitary chain on the same box. The corresponding unimodular and scaled unitary edges satisfy a matching normalized Frobenius estimate. At zero deficit both approximants coincide with the original Hadamard chain. All constants are independent of the ambient dimension, and the exponent $1/2$ is optimal for both approximations over either field.