Manuscript Hawk04-D@1 · Hawk04-D-v1.tex · section · 39b16f67

Information-free is not physically absent

Stage 5: endpoint obligations and order of limits › Information-free is not physically absent · lines 1941–1962

A constant complement contains no logical input dependence in the declared full-code task. It need not be one dimensional, pure, empty, or zero energy. Conversely, a zero-energy source can have a degenerate memory retaining the entire code:

V ∣ ψ ⟩ = ∣ 0 ⟩_{R} ∣ vac ⟩_{S} ∣ ψ ⟩_{M}, H_{S} ∣ vac ⟩ = 0.

Radiation is constant, while the complete residual recovers the input.

The explicit endpoint maps in the supplied Hawk01-B excerpts transfer all pending code and records to radiation and leave a fixed residual CheungHawk01B. That rule is an additional construction. Calling a source “evaporated” does not supply it.

A complete classical-output channel on $d \geq 2$ cannot have a one-dimensional pure-dilation environment: otherwise it would itself be an isometry and have an exact decoder, contradicting Lemma lem:benchmarks. Thus an empty list of live program names does not prove an empty complete physical complement.

Displayed formulas in this passage

unlabeled · line 1947
$V ∣ ψ ⟩ = ∣ 0 ⟩_{R} ∣ vac ⟩_{S} ∣ ψ ⟩_{M}, H_{S} ∣ vac ⟩ = 0.$