Your social security number is broken into parts which you noted, the first basically what state (variations in the number designate which county) and the middle is group (basically if you were to slice the country into strips vertically, which one you are in) and finally the last 4 which do appear to be random but are actually based on unique algorithms used from each state that generate the first 3 numbers based on your DOB, last name, DOI and the last digit is what number of people have that number and you increment.
So for example, if your info yields 123x and there are 4 people with that as well, you become 1235 and if you are say the 46th person with that, you become 1276 and etc. Now the combination of state, group yield which algorithm to use and applies it to the last 4 to see if its valid, if it doesn't pass validation, it will be rejected immediately and they usually send you a letter asking you to provide copies of stuff.
If it passes the validation, then it allows them to request a soft inquiry of your credit with whichever bureau they happen to use, I believe TU for most, EX for far west and EQ for far east, and go from there. Just keep in mind, its not JUST your DOB and Last 4, they also have your name. If there is a vague response from the system (meaning matches multiple people hits) then OSCAR will use some heuristics to find the best match with your peripheral information but if not, then it will kick it to "further review" where a human needs to take over. Hope that helps.