Manuscript IV-B@1 · IV-B-v1.tex · theorem · dfb66408

Theorem: Ramsey–Enter identity

The same-model dictionary and Ramsey–Enter composition › One complete composed channel · lines 1659–1689 · proof included or linked

IV-B@1#thm:composition, IV-B@1#eq:composed

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For each of the three generators and every trace-class input $Z$ on $X_{i}$ , the complete output is

E_{i} (Z) = s = \pm 1 \sum ∣ k_{0} ⟩ ⟨ k_{0} ∣_{R_{T}} \otimes ∣ s ⟩ ⟨ s ∣_{Y} \otimes (I_{L} \otimes K_{s} (A_{i})) Z (I_{L} \otimes K_{s} (A_{i}))^{*} .

This identity holds with every untouched reference. The time code is deterministic, the Ramsey outcome need not be, and all $L_{1}, L_{2}, L_{3}$ factors survive.

All contrast and convergence statements in Theorems thm:model1, thm:spectral-channel, thm:model2, and thm:model3 survive this attachment in their stated topologies.

Proof. Apply Theorem thm:instrument to the two-factor clock and use Lemma lem:ramsey. The $C_{T}$ bra annihilates every branch except $k_{0}$ ; the $C_{B}$ bra gives $K_{s} (A_{i})$ . The rod identity and the reference identity tensor through the calculation. Attachment of a fixed classical code is an isometry on the comparison interface and changes no record probabilities. Compatible CPTP composition preserves the stated error bounds.

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IV-B@1#eq:composed · line 1663
$E_{i} (Z) = s = \pm 1 \sum ∣ k_{0} ⟩ ⟨ k_{0} ∣_{R_{T}} \otimes ∣ s ⟩ ⟨ s ∣_{Y} \otimes (I_{L} \otimes K_{s} (A_{i})) Z (I_{L} \otimes K_{s} (A_{i}))^{*} .$

Theorem: Ramsey–Enter identity

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