Finite evaluation is not an endpoint

If the constant endpoint residual is pure, the endpoint radiation channel is an isometric encoding. If it is mixed, the syndrome normal form applies. An infinite endpoint syndrome need not have a finite decoder program.

The access condition is essential. For $V\psi =|0{⟩}_R{\psi }_B$, swapping the two outputs changes radiation recovery from impossible to exact. Such a swap is not an access-preserving identification.

The theorem refines the same finite-code endpoint structure in CheungHawk04A (thm:endpoint). The scattering endpoint of CheungHawk02D belongs to its quadratic model. In the supplied convention its outgoing one-particle map is ${\Omega }_{+}^{†}{J}_S$, with the adjoint required by the domains of the wave operator. No interacting asymptotic-completeness theorem for (eq:H) follows from that comparison.