Lemma: Faithful algebraic comparison and physical representatives

The maps ${\iota }_n$ and $j_n$ are mutually inverse unital $\ast$-isomorphisms between $\mathcal{B}({\mathcal{H}}_0)$ and ${{\mathfrak{B}}_{\mathfrak{n}}}^{\log }$. In particular,

For every point-labelled continuum observable, its actual live-register representative at scale $n$, pulled back through the physical unitary circuit, is ${\iota }_n(A)$. Its physical localization approaches the declared continuum point or retained path location.

Proof. Equation (eq:iota) is tensoring by the identity followed by unitary conjugation. It is a faithful unital $\ast$-homomorphism and is isometric, proving (eq:productcomparison).

For the second assertion, track all finite logical dilation factors, including the dormant apparatus factors. At any circuit cut, routing swaps implement a tensor permutation of those factors and the other reserved blank factors. Each payload unitary acts on the current logical factors as its prescribed logical unitary and acts trivially on the rest.

Inductively, the unitary evolution to that cut is, under the initial and current tensor identifications, the logical prefix unitary tensored with the identity on the other factors. Pulling a current live-factor observable back therefore gives exactly ${\iota }_n(A)$. Lemma lem:finegeometry controls its current physical location.

This argument uses unitary factor tracking on a selected subalgebra. It does not assert that a general encoding compression or a general channel adjoint preserves products.