Theorem: Exact monitored cost law

For fixed sampled controls and every supported complete measurement outcome, the conditional pure state's local entanglement costs are

After summing all outcomes in the complete attempted-run law, these costs are independent with common law

For $p>0$, this is not an exponential or a positive-shape Gamma law.

Proof. A number outcome $r$ on one endpoint of a measured geometric pair leaves the normalized pair state $|r⟩|r⟩$, which has zero entanglement. An unmeasured pair retains cost $w_e$. Products of the selective maps preserve the product-over-edges structure.

The masks and controls were sampled independently. Summing the complete quantum outcomes gives total probability one for every mask and control realization, so it does not alter their joint marginal law. The stated independent mixture follows.

This does not assert that selecting one prescribed outcome word leaves the control distribution unchanged. Such postselection would require a separate posterior calculation. The atom at zero excludes both specified positive continuous laws.