Manuscript VII-H@1 · VII-H.tex · proposition · d20d2a9f

Proposition: Propagated ambiguity

Propagation and reconstruction before observation · lines 391–423 · proof included or linked

VII-H@1#prop:propagation

Let $P : U \to E^{'}$ be a defined continuous linear map and let $u^{P} \in U$ . If

u^{P} - u^{Q} = c_{PQ} r,

then

P u^{P} - P u^{Q} = c_{PQ} P r .

The observation condition is $⟨ P r, F ⟩ = 0$ , not $⟨ r, F ⟩ = 0$ unless the two objects have actually been identified.

More generally, if

R^{P} = R_{*} + ν \sum P_{ν} c_{ν}^{P},

the ambiguity is contained in the span of the images of the allowed joint changes $(c_{ν}^{P} - c_{ν}^{Q})_{ν}$ . It equals a proposed image span only when those changes exhaust its generators.

Proof. The first assertions are linearity. For the last assertion, subtract the two response formulas. Independent variation of every catalogue entry is not implied by the formula; correlations among allowed changes must be retained.

Displayed formulas in this passage

unlabeled · line 396
$u^{P} - u^{Q} = c_{PQ} r,$
unlabeled · line 400
$P u^{P} - P u^{Q} = c_{PQ} P r .$
unlabeled · line 408
$R^{P} = R_{*} + ν \sum P_{ν} c_{ν}^{P},$

Proposition: Propagated ambiguity

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