Manuscript En05-C@1 · En05-C.tex · proposition · aa213ea6

Proposition: Preparation-relative snapshot determinacy

Stage B: centers and restricted preparations › The exact preparation-relative criterion · lines 618–645 · proof included or linked

En05-C@1#prop:relative, En05-C@1#eq:fiberseparation

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Snapshot determinacy tested only on $S_{adm}$ is equivalent to

T (A_{out} (Y)) \subseteq D_{I}^{⊥} .

It implies literal regional containment if

D_{I}^{⊥} = A_{in} (I) .

For the full global state space, (eq:fiberseparation) holds.

Proof. Equality of the two output restrictions is exactly

(ω - ν) (T (b)) = 0 (b \in A_{out} (Y))

for the generating pairs in the definition. Taking their linear span gives the first assertion. The second is immediate. For the full state space, local elements belong to the annihilator, and Lemma lem:separation excludes every self-adjoint element outside the local algebra. Complexification gives equality.

Displayed formulas in this passage

unlabeled · line 622
$T (A_{out} (Y)) \subseteq D_{I}^{⊥} .$
En05-C@1#eq:fiberseparation · line 626
$D_{I}^{⊥} = A_{in} (I) .$
unlabeled · line 635
$(ω - ν) (T (b)) = 0 (b \in A_{out} (Y))$

Proposition: Preparation-relative snapshot determinacy

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