I'm not an expert, but after reading more about it I would say between 500 - 1500 couples without trying to control the pairing.
I don't know where Dan Horvat took these assertion:
To maintain healthy genetic diversity and establish enough different
alleles to allow for sustainability of the species. I don't remember
the exact number but I think for humans it was something between 1000
or 10000, and if you get below that number the species will likely go
extinct by natural causes.
but it seems to be supported by the evidence of past bottlenecks in human population, like the Toba Cathastrofe Theory. The human population is supposed to had a maximum of 3,000–10,000 surviving individuals. I am very optimistic and take the minimum: 1500 couples. It left to match that amount with Dan's source.
The exact answer would come by the designer of the life and DNA. ;-) That info seems to have been leaked by the creator's son in a conference. Thomas has taken note from his words:
I shall choose you, one out of a thousand, and two out of ten
thousand, and they shall stand as a single one.
It seems to me that it is talking about genetic diversity, and the minimal proportions are those: 1/1000. That's why I take 500 couples as minimun.
By the way, in the previuos versicle it seems to says that that selection would be in the moment when the human race has finally ended evolving to the intended social and morphological status (very rare indeed):
When you make the two one, and when you make the inside like the
outside and the outside like the inside, and the above like the below,
and when you make the male and the female one and the same, so that
the male not be male nor the female female; and when you fashion eyes
in the place of an eye, and a hand in place of a hand, and a foot in
place of a foot, and a likeness in place of a likeness; then will you
enter the kingdom.
Going back to science, it seems that not to much effort should be made to control the pairing, because we seem to have a subconcient attraction for those individuals of the other sex that present better histocompatibility with us. In the case of humans given by odor and appeareance (mostly simetry).
Also the genetic diversity is supposed to be reached faster, if the environment is more aggresive. Normally a big mutation and adaptation happen after hostile and persistent environment over near 40 generations, something observed with the Peppered moth evolution
So I suppose that if you use well cared laboratory animals, the number can be 25 couples. But if you use a natural hostile enviroment the genetic diversity could be reached sooner.
It might be that if you build a nice human space station in another planet, maybe you need 25 couples, but if you let them nude with no technology and fig leaves you need fewer couples in order the species to survive more adapted (Although you might got many failed attempts)
Another point to consider, is how valuable are those animals for genetic studies?, how much is the study affected by the captivy of their population?, resembles their growing places the natural environment?, how hostile are these environment? resembles this environment the natural population dynamics ? how many generations ago their ancestors had a natural environment ?
Laboratory animals might have less genetic diversity and might need more couples.