How does domestic law in the State of Georgia define whether a man
or a woman has "residency" in another's home?  This is assuming
that one party is in a dating relationship with the actual owner and
sleeps over several nights a week.  Not listed on the deed or
mortgage, pays no household bills - just "camps out" several
nights/week.  This person thinks that if asked by the owner to leave and
he/she refuses, there is legal argument on his/her side that he/she
has established residency.

Just a free comment.  I expect that if the person still has an own
residence, rented directly or shared without their name on the lease,
they wouldn't have much of an argument, but this is no legal advice,
of course.

You have a much better chance of getting a researched answer if you
increase the price of your question.

Myoarin is 100% correct, what you have described is not residency,
it's called being a "GUEST."

The person does not pay rent or utilities, in no way contributes to
the household living expenses....the persons name is not on a lease or
rental agreement, --the utilities, phone, and cable are not in his/her
name, and he/she gets no mail at there.

She/he maintains a residence elsewhere, that address is documented as
being connected to him/her, utilities, pays rent, gets mail from the
bank, their work has that address, etc.

In some states in the case of "roommates" you have to evict a roommate
as you would a tenant. In this case however, if a person refuses to
leave, you simply call the police and show them a couple utility
bills, no clothes of his/hers in the closet, no dresser that has their
clothes, and out the person goes.

It's called Trespassing if the person returns, but you have to be
willing to enforce it.