Definition: Four same-model endpoint obligations

For a fixed finite code $C$, an endpoint package supplies:

1. A complete pure preparation $P_0:C\to H_{\mathrm{tot}}$, including all fixed purifiers.

2. Input-independent complete unitary evolution $U_t$, with a justified time domain.

3. Output partitions $R_t\otimes B_t$, factorwise isometries $j_t:R_t\to R_{\ast }$, $k_t:B_t\to B_{\ast }$, and an endpoint map $V_{\ast }$ such that

   $${\overline{V}}_t=(j_t\otimes k_t)V_t,{\Vert {\overline{V}}_t-V_{\ast }\Vert }_{\infty }\le {\nu }_t,{\nu }_t\to 0.$$

   The embeddings preserve access roles.

4. Either exact constancy of ${\mathcal{E}}_{\ast }={\mathrm{Tr}}_{R_{\ast }}{\mathrm{Ad}}_{V_{\ast }}$, or an explicit approximate endpoint certificate:

   • finite endpoint outputs with $d_{\diamond }({\mathcal{E}}_{\ast },{\mathcal{T}}_{{\sigma }_{\ast }})\le \zeta$; or

   • a finite endpoint proxy with complete allowance $b_{\ast }$ and residual Choi discrepancy $e_{\ast }$.

For a physical application, these data must be justified in one compatible model, with a receiver realizing the declared access.