Manuscript IX-C-ST@1 · IX-C-ST.tex · theorem · 892f50c4

Theorem: Fixed-interface evaluation certificate

Shared finite clock, certificates, and regular transport › Finite evaluation records and a normed comparison · lines 802–840 · proof included or linked

IX-C-ST@1#thm:table, IX-C-ST@1#eq:tabledistance

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The map (eq:tablechannel) is CPTP, reference preserving on its surviving factors, and has conditional sign mean $v_{k i}$ . For the same labels, pinching, report words, and payload interface,

d_{⋄} (I_{v}, I_{v}) = \frac{1}{2} k, i max ∣ v_{k i} - v_{k i} ∣, d_{⋄} (Φ, Ψ) = \frac{1}{2} ∥Φ - Ψ ∥_{⋄} .

Proof. Kraus operators are

p_{y} (k, i) ∣ k, i, y, r (k, i)⟩ (⟨ τ_{k} ∣ ⟨ i ∣) \otimes I_{P} .

Completeness follows by summing first over $y$ . This also proves the complete surviving-block statement.

For the distance, include a reference and use orthogonality of all recorded flags. The trace norm of the output difference is at most

k i max ∣ v_{k i} - v_{k i} ∣ k i \sum ∥ X_{k i} ∥_{1} \leq k i max ∣ v_{k i} - v_{k i} ∣∥ X ∥_{1} .

The last inequality is trace-norm contractivity of pinching, which is an average of unitary conjugations, on arbitrary operators. A basis input at a maximizing index attains the bound.

Displayed formulas in this passage

IX-C-ST@1#eq:tabledistance · line 809
$d_{⋄} (I_{v}, I_{v}) = \frac{1}{2} k, i max ∣ v_{k i} - v_{k i} ∣, d_{⋄} (Φ, Ψ) = \frac{1}{2} ∥Φ - Ψ ∥_{⋄} .$
unlabeled · line 820
$p_{y} (k, i) ∣ k, i, y, r (k, i)⟩ (⟨ τ_{k} ∣ ⟨ i ∣) \otimes I_{P} .$
unlabeled · line 831
$k i max ∣ v_{k i} - v_{k i} ∣ k i \sum ∥ X_{k i} ∥_{1} \leq k i max ∣ v_{k i} - v_{k i} ∣∥ X ∥_{1} .$

Theorem: Fixed-interface evaluation certificate

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