Witness, calibration, and record

Choose a nonzero real TT packet with compact Fourier support away from zero and initial velocity zero, as in the construction of Section sec:model1. Its Model 2 label $z_{\ast }$ belongs to ${\mathfrak{d}}_g$. The Hessian argument of Lemma lem:injective, applied to this wave, gives a compact-data free scalar ${\varphi }_{\ast }$ such that ${\psi }_{\ast }=S_{t_0}(k_{z_{\ast }}){\varphi }_{\ast }$ is nonzero at a future point. Choose a nonnegative bump $b$ there and put

The scalar label $v_{\ast }$ is Schwartz by its compact smooth data.

The coherent contraction of the four kernels is

It follows first by inserting the one-point functions in (eq:Bclassical); boundedness and the Hilbert tensor norm then extend the pairing continuously in the labels.

Let

If $z_{\ast }$ is supported in $r_{\ast }\le |p|\le R_{\ast }$, then for $R\ge \max \{2,r_{\ast }^{-1},R_{\ast }\}$, ${\alpha }_R{z}_{\ast }=z_{\ast }$. For every integer $j\ge 0$,

This uses $\Vert (1-{\beta }_R)v_{\ast }\Vert \le R^{-j}\Vert |q{|}^j{v}_{\ast }\Vert$. Exact eventual equality is not asserted, because the scalar label need not have compact momentum support.

Fix $a\ne 0$, and use

Put

The cutoff masks are contractions, so $\Vert B_R\Psi \Vert \le C_B\Vert (\mathsf{N}+2)\Psi \Vert$, uniformly in $R$, including $R=\infty$. Finite number bandwidth gives

Therefore define

The mean contrast obeys

This error cannot be suppressed: particle-number projections do not commute with coherent displacement.