The local analytic contract

The strict margin $R>r_0+T_0$ is essential. A chart containing only the observation support is not sufficient.

Such presentations exist: place a sufficiently small developed ball around a regular point and choose a sufficiently short observation interval. Larger finite operational windows can be accommodated by first fixing their size and choosing the regular point sufficiently far from the defect.

For $f\in \mathcal{D}(V)$, define its auxiliary truncated past

This is compact and contained in $K_{\ast }$. The symbol $J_{\eta }$ refers to the developed Minkowski presentation. No equality with a globally minimal causal past on the cone is claimed.