  We give new proofs of general relativistic initial data gluing results on
unit-scale annuli based on explicit solution operators for the linearized
constraint equation around the flat case with prescribed support properties.
These results retrieve and optimize - in terms of positivity, regularity, size
and/or spatial decay requirements - a number of known theorems concerning
asymptotically flat initial data, including Kerr exterior gluing by
Corvino-Schoen and Chru\'sciel-Delay, interior gluing (or "fill-in") by
Bieri-Chru\'sciel, and obstruction-free gluing by Czimek-Rodnianski. In
particular, our proof of the strengthened obstruction-free gluing theorem
relies on purely spacelike techniques, rather than null gluing as in the
original approach.
