The five-component output

This is the finite matching interface of VIII-A CheungVIIIA (Definition A:def:Ein), restated here. The notation ${\mathsf{Ein}}_0$ does not mean that a Hamiltonian constraint algebra or a general Einstein initial-data theorem has been proved.

For fixed face charts ${\mathcal{A}}_f$, seam map $J(\mu )$, and overlap-component exponent $n_{gf}$, the prescribed transition is

Elementary overlap components have $n_{gf}\in \{-1,0,1\}$; longer paths retain their individual component data.