This problem has come up with roommate arrangements hundreds of thousands of times.
If the person lives there, of course that person should be added to the lease and should pay rent.
The original three roommates have to agree unanimously whether it's acceptable to them to add another person, and if they don't, then the partner cannot live there, period. No wiggle room. No one roommate gets to introduce another person who lives there without the approval of the others.
How to split the rent changes, based on the number of bedroom and who has one to themselves versus who shares a room. Remember that each roommate is not just renting their bedroom but the right to use the common areas, too, so splitting in half the rent of the roommate who's partner this is may not be fair.
Example: Three bedroom apartment formerly with each of three roommates paying 1/3. Now with four roommates, the division could be 3/10, 3/10, 2/10, 2/10, with the 3/10 people having a room to themselves, the 2/10 people sharing a bedroom. Utilities, internet, etc. all divided equally four ways.