  Much recent work has been devoted to the study of information scrambling in
quantum systems. In this paper, we study the long-time properties of the
algebraic out-of-time-order-correlator ("$\mathcal{A}$-OTOC") and derive an
analytical expression for its long-time average under the non-resonance
condition. The $\mathcal{A}$-OTOC quantifies quantum scrambling with respect to
degrees of freedom described by an operator subalgebra $\mathcal{A}$, which is
associated with a partitioning of the corresponding system into a generalized
tensor product structure. Recently, the short-time growth of the
$\mathcal{A}$-OTOC was proposed as a criterion to determine which partition
arises naturally from the system's unitary dynamics. In this paper, we extend
this program to the long-time regime where the long-time average of the
$\mathcal{A}$-OTOC serves as the metric of subsystem emergence. Under this
framework, natural system partitions are characterized by the tendency to
minimally scramble information over long time scales. We consider several
physical examples, ranging from quantum many-body systems and stabilizer codes
to quantum reference frames, and perform the minimization of the
$\mathcal{A}$-OTOC long-time average both analytically and numerically over
relevant families of algebras. For simple cases subject to the non-resonant
condition, minimal $\mathcal{A}$-OTOC long-time average is shown to be related
to minimal entanglement of the Hamiltonian eigenstates across the emergent
system partition. Finally, we conjecture and provide evidence for a general
structure of the algebra that minimizes the average for non-resonant
Hamiltonians.
