r reasoning establishes that the other restriction $\phi$ is a member of ${\prod{\Phi}}$. The unique $\psi \in {\prod{\Psi}}$ and $\phi \in {\prod{\Psi}}$ such that $\upsilon = \psi\phi$ are expressed by the restrictions $\psi = {{\upsilon}\negmedspace\mid\negmedspace{{{\operatorname{dom}{\Psi}}}}}$ and $\phi = {{\upsilon}\negmedspace\mid\negmedspace{{{\operatorname{dom}{\Phi}}}}}$.

Persistent-volatile partition {#S:STATE_EVENT_PRTN}
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Definition \[D:BASIS\] asserts that basis $\langle \Psi, \Phi \rangle$ is comprised of two ensembles satisfying $\Phi \subseteq \Psi$. This allows partitioning terms of $\Psi$ into two sets: those terms that are members of both $\Psi$ and $\Phi$, and those terms that are members of $\Psi$ but not $\Phi$. The ensemble difference terminology of definition \[D:ENSEMBLE\_DIFFERENCE\] poses the minuend, subtrahend, and remainder of this basis as respectively $\Psi$, $\Phi$, and $\Psi \setminus \Phi$.

This partition is important when interpreted as systems theory. The minuend $\Psi$ generates a choice space ${\prod{\Psi}}$ called the *stimulus* space. The subtrahend $\Phi$ generates the *persistent* (alternatively *response*) space ${\prod{\Phi}}$. The remainder $\Psi \setminus \Phi$ generates the *event* space ${\prod{(\Psi \setminus \Phi)}}$.

\[T:STATE\_EVENT\_SPACES\] Let $\langle \Psi, \Phi \rangle$ be a basis. The persistent $(\Phi)$ and volatile $(\Upsilon = \Psi \setminus \Phi)$ generating ensembles are disjoint and complementary with respect to the generating ensemble $\Psi$ of the stimulus space. Expressed in dyadic product, $$\Psi = \Phi\Upsilon$$

Since $\langle \Psi, \Phi \rangle$ is a basis, then $\Phi \subseteq \Psi$ by definition \[D:BASIS\]. With that result and by lemma \[L:DISJOINT\_AND\_COMPLEMENTARY\], the subtrahend $\Phi$ and remainder $\Psi \setminus \Phi$ are disjoint and complementary with respect to $\Psi$.

The dyadic product recapitulates these results. Since $\Psi \setminus \Phi$ and $\Phi$ are disjoint the dyadic product $[\Ps