Proposition: Source, observation, and coherence

The branches are complete and their probabilities are

The diagonal algebra ${\mathcal{A}}_Z={\mathrm{span}}_{\mathbb{C}}\{P_0,P_1\}$ is branchwise invariant under the adjoints. All geometric history laws therefore depend only on the initial populations.

Before feedback, the summed source channel preserves populations and multiplies off-diagonal entries by

For $x=\delta q$, one active increment has

Also the unconditional observable identity is

Proof. The effects are positive and sum to the identity, proving completeness and the probabilities. All branch matrices are diagonal, proving adjoint closure. Their two eigenvalue cross products each equal $r_{\eta }/2$; their squared eigenvalues sum to one on each sector. This gives the coherence factor.

The increment is $\delta a$; its second moment is ${\delta }^2$, proving the moment formulas. Finally, $Q$ commutes with every branch matrix and ${\sum }_a{M}_a^2=I$, giving the operator identity.