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

Theorem: One delayed experiment detects the invariant

A delayed source/readout realization › Explicit product tests · lines 1209–1258 · proof included or linked

VII-H@1#thm:experiment, VII-H@1#eq:experiment-negative

For every admissible prescription,

χ^{P} (f_{δ}, h_{δ}) = ⟨ R^{P}, F_{f_{δ}, h_{δ}} ⟩

has the same value and satisfies

χ^{P} (f_{δ}, h_{δ}) \leq - \frac{3ℏ t _{0}}{8 π ^{2}} < 0.

The parameter $δ$ is an experimental source width, not an ultraviolet regulator.

Proof. Write $y = y_{0} + ξ$ . Formula (eq:correlation-test) becomes

F_{f_{δ}, h_{δ}} (z) = \int h_{δ} (y_{0} + ξ) F_{*} (z + ξ) d^{4} ξ, ∣ ξ^{α} ∣ \leq δ .

It vanishes at zero because source and readout supports are disjoint. Every contact difference therefore pairs to zero.

Set $G = F_{f_{δ}, h_{δ}} - F_{*}$ . The mean-value theorem gives

∥ G ∥_{\infty} \leq 4 δ p_{2} (F_{*}), ∥ \partial_{r} G ∥_{\infty} \leq 43 δ p_{2} (F_{*}) .

The cone is sampled only in

q - δ \leq t \leq t_{0} + a + δ .

Its lower endpoint is at least $q /2$ , and the interval length is at most $4 a$ . Lemma lem:annular yields

∣ ⟨ R^{\circ}, G ⟩ ∣ \leq \frac{4ℏ a δ}{π ^{2}} p_{2} (F_{*}) (2/ q + 3) < \frac{ℏ t _{0}}{8 π ^{2}} .

Combine this estimate with $⟨ R^{\circ}, F_{*} ⟩ = - ℏ t_{0} / (2 π^{2})$ .

Displayed formulas in this passage

VII-H@1#eq:experiment-negative · line 1217
$χ^{P} (f_{δ}, h_{δ}) \leq - \frac{3ℏ t _{0}}{8 π ^{2}} < 0.$
unlabeled · line 1212
$χ^{P} (f_{δ}, h_{δ}) = ⟨ R^{P}, F_{f_{δ}, h_{δ}} ⟩$
unlabeled · line 1229
$F_{f_{δ}, h_{δ}} (z) = \int h_{δ} (y_{0} + ξ) F_{*} (z + ξ) d^{4} ξ, ∣ ξ^{α} ∣ \leq δ .$
unlabeled · line 1238
$∥ G ∥_{\infty} \leq 4 δ p_{2} (F_{*}), ∥ \partial_{r} G ∥_{\infty} \leq 43 δ p_{2} (F_{*}) .$
unlabeled · line 1244
$q - δ \leq t \leq t_{0} + a + δ .$
unlabeled · line 1249
$∣ ⟨ R^{\circ}, G ⟩ ∣ \leq \frac{4ℏ a δ}{π ^{2}} p_{2} (F_{*}) (2/ q + 3) < \frac{ℏ t _{0}}{8 π ^{2}} .$