Theorem: Complete spectator consistency

The guarded geometry-only event satisfies Assumption ass:blind. For an entangled extension ${\tilde{\rho }}_{MN}$ with ${\mathrm{Tr}}_N{\tilde{\rho }}_{MN}={\rho }_M$, its geometric trace, certificate, permission change, and original-name locations are unchanged, and its restricted matter output is ${\rho }_M$.

The same restriction statement holds history block by history block when prescribed local instruments are inserted.

Proof. The guard reads only $m,j,\nu$ and fixed geometric validity. The transport is the same fixed label function for each original name; it does not test how many names exist. The event's internal matter channel is the identity. For an inserted selective map $\Phi$,

by expansion in matrix units of $N$. Apply this identity along each fixed history.