  Ricci-flat solutions to Einstein's equations in four dimensions are obtained
as the flat limit of Einstein spacetimes with negative cosmological constant.
In the limiting process, the anti-de Sitter energy--momentum tensor is expanded
in Laurent series in powers of the cosmological constant, endowing the system
with the infinite number of boundary data, characteristic of the asymptotically
flat solution space. The governing flat Einstein dynamics is recovered as the
limit of the original energy--momentum conservation law and from the additional
requirement of the line-element finiteness, providing at each order the
necessary set of flux-balance equations for the boundary data. This analysis is
conducted using a covariant version of the Newman--Unti gauge designed for
taking advantage of the boundary Carrollian structure emerging at vanishing
cosmological constant and its Carrollian attributes such as the Cotton tensor.
