Proposition: When a basis change can be a label relabelling

Assume the $t_k$ are distinct. If

for a function $f$ on the finite spectrum, then $U_C$ permutes the one-dimensional eigenspaces of ${\hat{\mathsf{T}}}_C$, up to phases. Consequently, a general unitary basis change is not even a finite relabeling of the original pointer, much less a spacetime coordinate transformation.

Proof. The conjugated operator has simple spectrum. Hence $f({\hat{\mathsf{T}}}_C)$ must also have distinct eigenvalues. Its eigenvectors are the original $|{\tau }_k⟩$, whereas those of the conjugated operator are ${U_C}^{†}|{\tau }_k⟩$. Simplicity identifies these two collections of one-dimensional eigenspaces by a permutation.

For a concrete failure, take $D=2$, use the computational reading basis, and apply Hadamard to the input $|0⟩$. The originally deterministic reading becomes a two-outcome distribution. A deterministic relabelling of the original reading cannot do this.