Manuscript Hawk02-E@1 · Hawk02-E.tex · proposition · 4841592b

Proposition: Omitted-output counterexample

Disorder, omitted residuals, and cutoff discipline › Source relaxation is not complete residual relaxation · lines 1673–1689 · proof included or linked

Hawk02-E@1#prop:omitted

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A fixed source marginal does not imply complete residual decoupling.

Proof. The isometry

V ∣ ψ ⟩ = ∣ 0 ⟩_{S} ∣ 0 ⟩_{R} ∣ ψ ⟩_{E}

has a constant source and a constant accessible output, but the inaccessible system $E$ retains the code. The complete residual distinguishes orthogonal inputs perfectly, so its distance from every replacement channel is at least $1/2$ for a nontrivial code.

Displayed formulas in this passage

unlabeled · line 1681
$V ∣ ψ ⟩ = ∣ 0 ⟩_{S} ∣ 0 ⟩_{R} ∣ ψ ⟩_{E}$

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section: Source relaxation is not complete residual relaxation