Theorem: Conditional one-model certification

Under Assumption ass:physical,

The channel bounds preserve arbitrary external references. If complete history comparisons are used, the corresponding history-record bound holds with their sequential budgets. For the exact benchmark of Section sec:model, $r_0={\delta }_0=0$.

These conclusions certify only the declared recovery and infalling experiments. Physical horizon regularity additionally depends on the independently proved content of (H5)–(H6).

Proof. Trace the full-output comparison over $K$ and apply the same comparison decoder. Contractivity and the triangle inequality give (eq:physicalrec), taking an arbitrarily near-optimal comparison decoder if necessary.

Trace instead over $R$ and compare to an arbitrarily near-optimal replacement channel for the comparison complement. This gives (eq:physicalleak).

For the instrument outputs, insert ${{\mathcal{I}}_t}^{\mathrm{phys}}{{\mathcal{V}}_t}^0$ between ${{\mathcal{I}}_t}^{\mathrm{phys}}{{\mathcal{V}}_t}^{\mathrm{phys}}$ and ${{\mathcal{I}}_t}^0{{\mathcal{V}}_t}^0$. Channel contractivity and the instrument comparison bound the first two differences by $e_{\mathrm{em}}$ and $e_{\mathrm{inst}}$. Reading only the classical record cannot enlarge the distance. Add the stated regular-reference allowance $e_{\mathrm{reg}}$.

All comparisons were in diamond norm, so the proof holds with ${\mathrm{id}}_Q$ tensored throughout. The history assertion follows from Lemma lem:telescoping. The exact benchmark values follow from Theorem thm:modelchannels. Nothing in this argument constructs the independent physical hypotheses (H5)–(H6).