Theorem: Sourced semantic packet

On ${\mathcal{D}}_{\mathrm{src}}$, the selected packet and complete geometric family at ${\mu }_{\theta }(\rho )$ are well defined. The rule formally returns the same matter state. For fixed frame and named embedding, it is invariant under spectator extensions preserving the source restriction.

Moreover,

Zero source, zero coupling, or $\theta =0$ selects the reference family, not necessarily the previously stored catalogue entry.

Proof. The interval $I$ is convex and contains ${\mu }_0$. Thus the selected deficit belongs to $I$. Sections sec:family–sec:matching were proved for every scalar member independently of the source, so substitution is legitimate. Trace duality proves (eq:source-stability); partial trace proves spectator invariance. The source-off statements follow by direct substitution.