Local Rigidity of Quasi--Lie Brackets on Quaternionic Banach Modules and Applications to Nonlinear PDEs

We establish a local rigidity theorem for quasi--Lie brackets on quaternionic Banach right modules. Under quantitative control of antisymmetry and Jacobi defects, we construct an explicit bilinear correction that preserves right $\mathbb{H}$--linearity and restores the exact Lie property. The approach combines a radial homotopy operator, a controlled Neumann-series inversion, and a finite-rank adjustment, all with explicit operator estimates. This constructive framework bridges quaternionic functional analysis with rigidity theory and yields concrete applications to nonlinear PDEs, including local well-posedness and Beale--Kato--Majda continuation criteria with explicit thresholds.