Peace be with you my son.
Again, I'm a regular guy, not a biblical historian. I know I don't have all the answers. Though I could research them as I often do, there are times I ask myself, "To what end?"
This is often the case when arguing Conservative points with Libertarians. Regardless of what I come up with, a quibbler will respond, "Well then, what about the Blah blah blah...", ad nauseam.
I wasn't born yesterday, and I think I've come to recognize when someone is merely trying to lead me into a quagmire of words. Though, I was a little slow to pick up on it in this case.
You wouldn't happen to be a Libertarian would you?
Contrary to what someone of your persuasion might believe; though there may have been a four-year period of time that there was not a pope, the Church continued. It had merely been driven underground by the Diocletian persecution, which was the most terrible of the ten persecutions the early Church endured.
You can read about it from the links found on this page.
http://www.newadvent.org/cathen/09640b.htm
So, it was under extenuating circumstances that the hiatus resulted. I imagine the election of the next pope can seem like a lower priority matter when tomorrow you can be thrown to hungry lion to the delight of a decadent crowd, or hung on a cross and set on fire along with a hundred of your fellow believers to provide a dazzling night-time spectacle.
Thanks for asking me about this. Though I know it will only draw another, Well, what about the blah, blah, blah?, response from you, I learned some new things, and was reminded of the sacrifices the early Christians endured to ensure the survival of the faith that we are blessed with 1,700 years later. Suddenly, I see that enduring your antics is a small price to pay.