  We prove a strong localized gluing result for the general relativistic
constraint equations (with or without cosmological constant) in $n\geq 3$
spatial dimensions. We glue an $\epsilon$-rescaling of an asymptotically flat
data set $(\hat\gamma,\hat k)$ into the neighborhood of a point
$\mathfrak{p}\in X$ inside of another initial data set $(X,\gamma,k)$, under a
local genericity condition (non-existence of KIDs) near $\mathfrak{p}$. As the
scaling parameter $\epsilon$ tends to $0$, the rescalings $\frac{x}{\epsilon}$
of normal coordinates $x$ on $X$ around $\mathfrak{p}$ become asymptotically
flat coordinates on the asymptotically flat data set; outside of any
neighborhood of $\mathfrak{p}$ on the other hand, the glued initial data
converge back to $(\gamma,k)$. The initial data we construct enjoy
polyhomogeneous regularity jointly in $\epsilon$ and the (rescaled) spatial
coordinates.
  Applying our construction to unit mass black hole data sets $(X,\gamma,k)$
and appropriate boosted Kerr initial data sets $(\hat\gamma,\hat k)$ produces
initial data which conjecturally evolve into the extreme mass ratio inspiral of
a unit mass and a mass $\epsilon$ black hole.
  The proof combines a variant of the gluing method introduced by Corvino and
Schoen with geometric singular analysis techniques originating in Melrose's
work. On a technical level, we present a fully geometric microlocal treatment
of the solvability theory for the linearized constraints map.
