Building on the definition of knowability, we must now account for how knowability is preserved when systems are combined. In the Corpus, composition is not merely the union of sets, but the creation of a new system structure that must maintain the identity of its constituents.
1. The Composite System
Consider two systems \(S_1, S_2 \subseteq U\). We define their composition \(S_3\) as the system formed by their union:
\[ S_3 := S(S_1 \cup S_2) \]
This composite system possesses its own constraints \(C_3 = C(S_3)\) and state space \(\Omega_3 = \Omega(S_3)\).
2. Subsystem Distinguishability
A critical requirement for reasoning about composite systems is Subsystem Distinguishability. We say that \(S_1\) and \(S_2\) are distinguishable within \(S_3\) (denoted \(S_1 \perp_{dist} S_2 \text{ in } S_3\)) if and only if there exist projection mappings that preserve the uniqueness of the composite state.
Definition (Subsystem Distinguishability): \(S_1 \perp_{dist} S_2 \text{ in } S_3\) iff \[ { \exists \pi_1: \Omega_3 \to \Omega_1 \wedge \exists \pi_2: \Omega_3 \to \Omega_2 \mid \phi(\omega_3) = (\pi_1(\omega_3), \pi_2(\omega_3)) \text{ is injective} } \]
3. Product Embedding
The formal consequence of distinguishability is that the state space of the composite system can be embedded into the product of the subsystem state spaces.
Lemma (Product Embedding under Distinguishability): If \(S_1 \perp_{dist} S_2 \text{ in } S_3\), then \(\Omega_3 \hookrightarrow \Omega_1 \times \Omega_2\).
Proof: By the injectivity of the mapping \(\phi\), for every composite state \(\omega \in \Omega_3\), there exists a unique pair of subsystem states \((\omega_1, \omega_2) \in \Omega_1 \times \Omega_2\). Therefore, \(\phi\) is an embedding. \[ \forall \omega \in \Omega_3, \exists! (\omega_1, \omega_2) \in \Omega_1 \times \Omega_2 \] Q.E.D.
This formalization ensures that a composite system remains a product of its parts, a necessary condition for knowability under composition.
More to follow on the algebra of invariant space.