Appendix C · Why the effective network preserves the laws

C Strong quotient and composition proofs

C.1 The finite refinement construction

Let $\Pi_0$ be the partition into equal protected marks. Given $\Pi_t$, assign to $x$ the signature consisting of its current block and, for every $u,o$, the vector

Let $\Pi_{t+1}$ be the partition into equal signatures. The partitions only split. There are at most $|X|-|\Pi_0|$ strict refinement rounds because every strict round increases the block count. Once a round does not split, the signatures already have equal joint transition sums within every block; the partition satisfies Equation 6.1 and is stable.

To prove coarseness, let $\Lambda$ be any stable mark-preserving partition. It refines $\Pi_0$. If it refines $\Pi_t$, each $\Pi_t$-block is a union of $\Lambda$-blocks. Stability of $\Lambda$ makes the probabilities into each such union identical for states in one $\Lambda$-block. Their $\Pi_{t+1}$ signatures therefore agree. Induction proves that $\Lambda$ refines the terminating partition.

This construction uses the full joint record-and-class matrix, not only transition probabilities after records have been summed out. A protected mark may already distinguish two otherwise observationally identical states because their physical resource obligations differ. Its inclusion is part of the promised quotient semantics, not a discovery from boundary observations.

The proof also fixes the domain. On a full closed domain it considers every supplied state. On a reachable subdomain it first removes states unreachable under the nominated initial supports and admitted inputs, then requires closure. Mixing full-domain minimization on one side of a composition identity with reachable-only minimization on the other can change the answer.

C.2 The composed row calculation

Write $x=(x_1,\ldots,x_n)$, $z=(q_1(x_1),\ldots,q_n(x_n))$, and let $v$ be a current exterior action. A typical elementary event selects $(i,u)$ by a context rule $\Gamma_v(i,u\mid z,c)$, updates module $i$ using $K^i_{u,o}(x_i,y_i)$, and updates its retained context and exterior record $r$ by a kernel $\Lambda_v(r,c'\mid c,i,u,o,z,q_i(y_i))$. Dependence on $z$ or $q_i(y_i)$ is permitted only insofar as those variables are supplied by the declared ports. These expressions are an algebraic statement of fibre-constant access, not authorization of a new observation.

For this event the fine probability is a sum of terms

Sum over the global fibre $Q(y,c')=(z',c')$. In each term the only nontrivial sum is

Every other factor is constant on the fibre. Thus the summed row is exactly the event generated by the effective module and the same context. The statement is preserved by summing over $i,u,o$ and over common context randomization. This proves strong quotient closure of each elementary event and hence of the sequentially composed joint instrument.

Initial correlations cause no difficulty if the original law on $(x,c)$ is pushed forward by $Q$ as a whole. Replacing it by a product of marginal preparations would be a different model. Similarly, independent simultaneous calls with frozen inputs can be serialized without altering their joint product row. A common noise event is not generally such a product; it requires its own full joint-row certificate. Delayed feedback is represented by queues in $c$, not by an assumed solution to instantaneous feedback equations.

Induction on events proves complete-history equality. Induction on a finite policy tree gives the same result for adaptive experiments, since the next action is selected from the same retained history. A finite stopping rule merely collects probabilities at selected leaves. This establishes Theorem 6.2 in precisely the causal context class specified.

C.3 Why minimization commutes in this domain

Let $F$ be the fine composed instrument and $H$ the composed instrument after the local quotients. The product map $Q:F\to H$ is strong by the preceding calculation. Let $\Pi$ be the coarsest strong marked partition of $F$. Since the fibre partition of $Q$ is itself strong and mark-preserving, $\Pi$ is coarser than it. Hence

is well defined. A block of $\Pi$ is a union of $Q$-fibres, and summing the effective rows over that union proves stability of the induced partition on $H$.

Conversely, the inverse image under $Q$ of any stable marked partition of $H$ is stable in $F$: each transition sum into a pulled-back block is the corresponding effective transition sum. By coarseness it refines $\Pi$. Thus the induced partition is the coarsest strong marked partition of $H$. Mapping corresponding final blocks yields the instrument isomorphism in Equation 6.3; the joint initial laws agree by pushforward. The proof does not minimize over unrelated stochastic generators or alternative physical implementations.

C.4 Exact word laws and port-only contexts

Proposition C.1 (Behavioral substitution from complete word equality)

Fix matched preparations, timing, allowed action words and joint boundary records. Any initially correlated reference accessible to the context is included in the compared joint word/reference laws. If those complete laws agree for the two processes, then every compatible finite causal port-only context gives the same retained law. This does not assert a strong actual-state quotient.

Proof

For a fixed realized action/record path, multiply the process's word probability by the context's conditional probabilities of choosing each action from the preceding retained history. Those conditional factors are the same in both descriptions. Equality of word probabilities therefore gives equality of every path weight. Summing over internal policy randomization or stopped leaves preserves equality. If a reference is initially correlated with the process and can affect the context, the matched law is the joint word/reference law; equality of its unconditioned marginal alone is insufficient.

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The proposition explains why the alternative four-state generator in Example 6.4 can be behaviorally adequate without furnishing the failed actual quotient. It also explains why the fixed-input counterexample in Proposition 6.6 must concern approximate agreement: exact equality permits the pathwise multiplication, whereas separately small errors can be selected and amplified by a feedback policy.