We establish two complementary local rigidity results for quasi–Lie brackets on quaternionic Banach right modules under quantitative control of antisymmetry and Jacobi defects. Unconditional result (Theorem 4.12): In the homogeneous case (Jacobi defect homogeneous of degree 2 with exact antisymmetry), we prove rigidity with no cohomological hypothesis required. The inclusion \(\textrm{Im}(\Pi _{\textrm{cone}}) \subset \textrm{im}(d)\) is established constructively via an explicit primitive (Lemma B.1 ). Conditional result (Theorem 4.13 ): In the general non-homogeneous case, rigidity holds under the cohomological hypothesis \(F \subset \textrm{im}(d)\) , where F is a canonical finite-dimensional subspace and d is the Chevalley–Eilenberg differential (Definition 3.5). This hypothesis is: proven in the homogeneous case (Lemma B.1),
conjectured for high-regularity Sobolev spaces \(H^s(\mathbb {R}^n, \mathbb {H})\) with \(s > \frac{n}{2} + 1\) (Remark 5.5),
open in general abstract settings,
fails for \(s \le \frac{n}{2} + 1\) (Example 5.6).
The proof combines a radial homotopy operator, controlled Neumann-series inversion, and finite-rank adjustment, all with explicit operator estimates. Applications to quaternionic nonlinear PDEs (local well-posedness, Beale–Kato–Majda continuation criteria) are established under the same cohomological hypothesis. We provide numerical evidence supporting the conjecture but emphasize that a complete analytical proof remains an open problem.
