Explicit Baker--Campbell--Hausdorff Radii in \textit{Special} Banach--Malcev Algebras of Shifts

Abstract: We establish explicit convergence radii for the Baker--Campbell--Hausdorff (BCH) series in special Banach--Malcev algebras of shifts-those embeddable into a Banach alternative algebra. Under the continuity estimate $|[x,y]|\leq B|x||y|$, the series converges absolutely whenever $B(|x|+|y|)<1/(4K)$, where $K\geq1$ bounds the absolute BCH coefficients. The constant $1/(4K)$ stems from a Catalan-number majorization and is sharp in the exponential-weight model. We compute $B$ explicitly for operator, exponential, polynomial, damped, and tree-like shift algebras, including the non-Lie split-octonionic (Zorn) algebra ($B=2$, $\rho=1/(8K)$). All results require the speciality assumption; the framework does not apply to general Malcev algebras. Geometrically, $\rho=1/(4KB)$ is the analyticity radius of the induced Moufang loop; numerically, it governs stability of BCH-type integrators.