Lemma: Exact physical refinement

The physical Hilbert space ${\mathcal{H}}_{\mathrm{phys},n}$ has dimension four. The map

that replaces each arc's $X$-eigenvalue by the same value on all its sublinks is unitary between the physical spaces. Define

Then $J_n:M_4\to \mathcal{B}({\mathcal{H}}_{\mathrm{phys},n})$ is a unital $\ast$-isomorphism and a complete isometry.

Proof. The constraints give exactly the four configurations (eq:theta-basis), now constant along each arc. They form an orthonormal basis. The stated map sends one physical basis bijectively to the other. The conclusions follow from unitary conjugation.