Proposition: Off-causal-plateau diagnostic

There are admissible compact data, a free scalar, and a free metric for which

The example violates (CP), not the cancellation theorem under (CP).

Proof. Work first in a dependence cylinder where the spatial cutoffs are one. Let $m=0$, $\varphi =t$, and $\gamma =t\eta$. Choose $a\ge 0$, $a\not\equiv 0$, supported in a time interval $(t_a,t_b)$ above the ramp. Let

A compact spatial cutoff will be inserted below.

Write $r={\mathcal{L}}_v(t\eta )$. Its nonzero local components are

Direct differentiation gives

Hence the linear-in-$\lambda$ companion parameter is

with zero initial data. Since $({{\partial }_t}^2+k^2)(ta)=t(a^{\prime \prime }+k^{2}a)+2a^{\prime }$,

Choose a temporal switch $g_0$ equal to one on the support of $a$, falling smoothly and nontrivially from one to zero in $(t_c,t_d)$ with $t_b<t_c$, and zero near an observation time $t_o>t_d$. Its earlier rise occurs before $\mathrm{supp}a$. Choose $k$ sufficiently small that

whenever $t\in [t_c,t_o]$ and $s\in \mathrm{supp}a$. Above $\mathrm{supp}a$,

For ${\rho }^{(1)}$, the free scalar $\varphi =t$ gives

The coordinate leg is ${\rho }^{(1)}$ by Lemma lem:no-ramp. Because $g_0=1$ on the support of $R^{\prime \prime }+k^{2}R$, the linear companion integrand is

For $t$ just after $t_d$, every nonzero source term in this integral has negative sign: $g_0^{\prime }\le 0$, $R^{\prime }>0$, and the sine kernel is positive. Thus $d(t)<0$.

Choose $f\ge 0$, nonzero, supported there and near $x^1=0$, where $\cos (k{x}^1)>0$. Its linear companion value $J$ is strictly negative. The exact finite parameter has

Linearity of the propagation and of ${\mathcal{A}}_1$ therefore gives

for a finite constant $J_2$ independent of $\lambda$. It is nonzero for all sufficiently small positive $\lambda$.

Now replace $v$ by a compact spatial realization equal to the printed expression throughout the observation dependence region. Take $g=g_0(t){\zeta }_g(\mathbf{x})$, with ${\zeta }_g=1$ on the same larger cylinder and compactly supported beyond it. Choose $U$ to contain $\mathrm{supp}{\xi }_{\lambda }$, with closure in the early plateau where $g_0=1$. The cutoffs are chosen before the calculation and large enough that their transition regions cannot influence $\mathrm{supp}f$. Finite propagation preserves (eq:R-diagnostic)–(eq:d-diagnostic) on the observation region BGP.

At the observation $g=0$, and it lies in the causal future of the parameter support. During the falling switch, $(K{\rho }_{{\xi }_{\lambda }}{)}_{00}$ has leading term $2\lambda R^{\prime }(t)\cos (k{x}^1)$, nonzero in the relevant causal past. Thus (CP) fails. The fields $\gamma =t\eta$ and $\varphi =t$ are free; one may set $b=D\gamma$ and the ghosts to zero.