The stochastic noise of splitting, defined initially on the (basic) algebra of finite unions of intervals of the real line, is extended to a largest class of domains. The $σ$-fields of this largest extension constitute the completion, in the sense of noise-type Boolean algebras, of the range of the unextended (basic) noise. The basic noise extends to a given measurable domain precisely when a certain stability property is met: the times at which a Brownian motion has local maxima which fall inside the domain must remain unaffected under resampling of the Brownian increments outside the domain; together with the same being true for the complement of the domain. A set that is equal to an open set modulo a Lebesgue negligible one, with the same holding of its complement, has this stability property, but others have it too: the extension is non-trivial. Some domains are totally unstable with respect to the indicated resampling, and to them the extension cannot be made.