Manuscript Oai282-A@1 · Oai282-A.tex · proposition · 5b6134ec

Proposition: Fixed modular and Ward data do not determine recovery

Stage 3: modular null cuts and recovery are different interfaces › An explicit record-only separation · lines 1320–1341 · proof included or linked

Oai282-A@1#prop:recovery-separation

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For any fixed record and $d \geq 2$ , there are independent completions with identical $d$ -dimensional accessible and complementary spaces, one exactly recoverable and one not recoverable. Their selected accessible entropy is zero for every pure logical input.

Proof. Set

W_{good} ψ = ψ_{R} \otimes ∣ 0 ⟩_{B}, W_{bad} ψ = ∣ 0 ⟩_{R} \otimes ψ_{B} .

The first accessible channel is identity and its complement is constant. The second accessible channel is constant and its complement is identity. Theorem thm:recovery separates their recovery verdicts.

For each pure input, both accessible outputs are pure. All original Ward and modular record entries remain fixed.

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$W_{good} ψ = ψ_{R} \otimes ∣ 0 ⟩_{B}, W_{bad} ψ = ∣ 0 ⟩_{R} \otimes ψ_{B} .$

Proposition: Fixed modular and Ward data do not determine recovery

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