Manuscript IX-D-ST@1 · IX-D-ST.tex · proposition · b1dc631a

Proposition: A nonzero observation-spacelike control

The settled witness class and the unfinished bridge · lines 1615–1652 · proof included or linked

IX-D-ST@1#prop:control

There are nonnegative nonzero compact tests $f, g$ supported in the laboratory with

∣ t - s ∣ < R < t + s on F \times G,

for which $C_{f, g} \neq = 0$ . Observation-spacelike separation does not certify both actual functional-support orderings.

Proof. Choose $λ > 0$ small enough that neighbourhoods of

x_{*} = (λ, a), y_{*} = (λ, a + λ e_{1})

and their retained shadows fit in the chart. At the central pair, $0 < R = λ < 2 λ$ . The strict inequalities persist on sufficiently small product neighbourhoods.

Take nonnegative bumps there, set $\partial_{t} ϕ = 1$ near $F$ , and $h_{00} = 1$ near $G$ . Then

C_{f, g} = - \frac{i}{8 π} \int \frac{f ( x ) g ( y )}{∣ x - y ∣} d^{4} x d^{4} y \neq = 0.

This uses observation supports only. The functional $B_{f}$ can also depend on metric configurations on its retained past and initial surface. No two-way certificate has been supplied for the control.

Displayed formulas in this passage

unlabeled · line 1619
$∣ t - s ∣ < R < t + s on F \times G,$
unlabeled · line 1630
$x_{*} = (λ, a), y_{*} = (λ, a + λ e_{1})$
unlabeled · line 1642
$C_{f, g} = - \frac{i}{8 π} \int \frac{f ( x ) g ( y )}{∣ x - y ∣} d^{4} x d^{4} y \neq = 0.$

Proposition: A nonzero observation-spacelike control

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