Definition: Endpoint certificate inputs

A conditional endpoint recovery claim supplies:

1. Identified complete interface. A named code, physical preparation class, accessible output, complete complement, and endpoint or outgoing representation. Any gravitational dictionary is separately justified.

2. Model complementary certificate. A complete finite model isometry $V_0$, with complement ${\mathcal{C}}_0$, and a state $\tau$ satisfying

   $$d_{\diamond }({\mathcal{C}}_0,{\mathcal{T}}_{\tau })\le \delta .$$

   Selected thermal modes or retained source factors do not replace the complete complement.

3. Controlled physical transfer. On common identified finite interfaces,

   $$d_{\diamond }({\mathcal{N}}_{\mathrm{physical}},{\mathcal{N}}_0)\le a,$$

   with regulator, truncation, collection, and validity errors accounted for once.

4. Deterministic executable return. A model decoder $D_0$ with a proved recovery bound and an actual implementation $D_{\mathrm{imp}}$, including every failure output, satisfy

   $$d_{\diamond }(D_{\mathrm{imp}},D_0)\le k.$$

   Its quantum access, schedule, workspace, and output type are declared. Information-theoretic decoder existence alone does not prove this implementation obligation.