Definition: Four same-model endpoint obligations

A finite-code endpoint package consists of:

1. Complete preparation. A justified finite code $C$, its embedding, and a fixed complete preparation are supplied in one microscopic model. All purifiers are retained in the complete description.

2. Complete evolution. The model supplies unitary evolution on its complete physical system and therefore the complete interfaces $V_t$, not merely radiation marginals.

3. Outgoing access and endpoint. The model supplies the meaning of $R_t,B_t$, an access-preserving comparison, and a map $V_{\infty }:C\to R_{\infty }\otimes B_{\infty }$ such that

   $$\Vert {\hat{V}}_t-V_{\infty }\Vert \le {\kappa }_t,{\kappa }_t⟶0.$$

   The spaces in this finite theorem are finite dimensional. A continuum or infinite-output application needs its corresponding normal-channel and convergence theorem, rather than an unproved truncation.

4. Complete endpoint complement. For one endpoint state ${\sigma }_{\infty }$, either

   $${\mathcal{E}}_{\infty }={\mathcal{T}}_{{\sigma }_{\infty }},$$

   or the declared approximate allowance

   $$d_{\diamond }({\mathcal{E}}_{\infty },{\mathcal{T}}_{{\sigma }_{\infty }})\le \zeta$$

   is proved, where ${\mathcal{E}}_{\infty }={\mathrm{Tr}}_{R_{\infty }}{\mathrm{Ad}}_{V_{\infty }}$.