Theorem: Relative non-entailment of an unconstrained assignment

Suppose $T_{\mathrm{phys}}$ has a model and contains no premise constraining an added predicate $\mathcal{C}$. Then the physical premises do not determine that predicate. In particular, there are two expansions with identical physical objects and predictions but different assignments.

Proof. Take one model of $T_{\mathrm{phys}}$. Expand it once with ${\mathcal{C}}_1=\varnothing$ and once with ${\mathcal{C}}_2=E$. The physical reduct is unchanged, so every physical premise and prediction has the same interpretation. The assignments differ because $E\ne \varnothing$.