Explicit Baker--Campbell--Hausdorff Radii in Special Banach--Malcev Algebras of Shifts

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 $\mathrm{\parallel }[x,y]\mathrm{\parallel }\le B\mathrm{\parallel }x\mathrm{\parallel }\mathrm{\parallel }y\mathrm{\parallel }$, the series converges absolutely whenever $B(\mathrm{\parallel }x\mathrm{\parallel }+\mathrm{\parallel }y\mathrm{\parallel })<1\mathrm{/}(4K)$, where $K\ge 1$ bounds the absolute BCH coefficients. The constant $1\mathrm{/}(4K)$ 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 ($B=2$, $\rho =1\mathrm{/}(8K)$). All results require the speciality assumption; the framework does not apply to general Malcev algebras. Geometrically, $\rho =1\mathrm{/}(4KB)$ is the analyticity radius of the induced Moufang loop; numerically, it governs stability of BCH-type integrators.