Iron from the core is then processed to create massive shoring I-beams which keep these ever expanding caverns from collapsing as the mine-bots home in on one another to create a single chamber within an iron geodesic sphere.

Meanwhile on the lunar surface Plasma engines, shaped charges and solar sails begin to slowly increase the rotational rate of moon.

Several tens of decades go by...

The moon is now roughly twice the diameter it was when construction began and no longer has a side considered dark as it rotates twice a day.
Since no substantial mass has been added or removed from the moon, there has been no effect on the Earth's tides, though several Terran species are observed becoming spontaneously nocturnal as now even a crescent moon casts the amount of light that the full moon once reflected.

and several more decades...

It will be many years yet before enough spin has been imparted for colonists within Luna to experience one Earth gravity or 1G as centrifugal force holds them to the inner walls along the equator, but the plant life, which helps to filter and scrub the trapped atmosphere, has grown remarkable quickly. Unfettered by gravity's restraint the plants compete their way toward the central light source at a manic pace.

The main complaint of the colonists is a slight feeling of vertigo and nausea when looking across the chamber to see people miles away walking upside down above them.
These complaints have decreased substantially as the canopy has begun to fill in.

The epic Global effort to construct drive engines capable of moving an object the mass of Luna are well underway and are estimated to be ready well in advance of the last of Earth's surface vegetation and animal life dying off from our expanding Sun's increased radiation output.

The Clism of 2012 Coupled with the strict procreation laws enacted in 2126 have reduced humanity’s numbers to the point where there will be space enough for all when the Lunark embarks.