Residue geometry and the diagonal limitation

Each observation equality diagonal ${\Delta }_{ij}\subset X$ has dimension eight and codimension four; the total diagonal has dimension four. The inherited embedding $\Psi$ parametrizes an eight-dimensional relation $\mathrm{Rel}$. On it,

Consequently $\mathrm{Rel}\cap {\Delta }_{13}=\Psi \{t=t^{\prime }\}$ has dimension seven and codimension five in $X$. Moreover,

These are the VII-D observation relations VIID (eq:chartrelations; Section “Observation coverage'). The initial Cauchy surface is an auxiliary surface, not an observation face inside $X$.

local density. For the local transverse-degree calculation below, recall the VII-D chart

Its inherited density and residue representation are

There is no $\delta (t-t^{\prime })$ VIID (eq:density, eq:rlocal). We do not re-prove this coordinate formula. Positivity and the separated witness use the parameter measure in (eq:r), not this chart.