Theorem: Spectator invariance

Suppose $\tilde{\varrho }$ extends $\varrho$ by spectator names $N$ and ${\mathrm{Tr}}_N\tilde{\varrho }=\varrho$. Then:

1. the named source functional, including $I_f$, is unchanged;

2. for fixed declared geometry and frame, the constructed geometry and certificate are unchanged;

3. under Assumption ass:control, equal complete declared inputs and accessible source functionals give equal applicability, output certificates, original-name transport, and successor controller data.

Entangled spectator extensions are allowed. The sourced step preserves the original matter reduction.

Proof. For every $a\in {\mathcal{A}}_{\mathrm{src}}$,

The geometry depends on matter only through the selected scalar deficit, proving the first two assertions. The additional controller conclusion is exactly conditional on Assumption ass:control. Identity action on matter preserves its original reduction.