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I think of this and it just has to be whoever is most harmless.  The least threatening program to my school or to any school with a big fanbase.

In the past 50 years, who is the worst program?  First, let's remove the private scholarlies:
.336 Vanderbilt
.366 Duke
.386 Northwestern
.405 Wake Forest

Okay, with those out of the way, here are the worst public, P5 programs.  I think our "winner" will be the one who lacks a rival (Rutgers?) or who's rival doesn't return the favor.
.355 Oregon State
.383 Kansas
.386 Indiana
.412 Iowa State
.425 Illinois (wow, that's shameful)

Okay, so if their rival is really good, then it has bigger fish to fry, and the rivalry is one-sided in terms of passion.  If the rival is bad, then it's more contested and matters more to both sides.

Oregon State having the worst winning % and having a strong rival (Oregon), makes me lean towards them.  Duke's rival isn't as good (UNC), both both care more about basketball anyway, so it's sort of a big shrug.  Kansas' top rival is Missouri? or K-State, both of which are decent long-range programs, but neither is as good as Oregon, and so neither is as apathetic towards them.  

I'm going Oregon State.  They even have a harmless mascot.  

I'm going with Utah 

I'm going with Utah

That's kind of a cheat, having been G5 for so long.  I'd set Utah and TCU aside, to be fair.

What about Georgia State if we go G5?  Well, I guess they are a rival if you are a Vol fan.

I'm going with Utah

I don't know about that one, primarily because I assume they've got BYU thinking they're a bunch of godless heathens while they must think BYU is full of sanctimonious prigs. 

So they may only have one true "rival", but I think it's a real rivalry. 

What about Syracuse? They basically don't have any in-state rival in P5. Is Pitt their rival? How serious is that rivalry? 

In the Big Ten I would go with Illinois. Seriously, does anyone really care about it for football? Purdue and Indiana have each other (and ND) to compete with in-state. Iowa has Iowa State, Minnesota, and Wisconsin; Minnesota has Iowa and Wisconsin, Nebraska has too big a target on its back...Illinois is milquetoast. Rutgers and Maryland don't belong.

I think Iowa State has a serious claim to this. Sure, it has Iowa, but Iowa's big rivals are Minnesota and Wisconsin, aren't they?

Wazzu and Oregon State both have big rivalries in their home states, but outside of those in-state rivalries, no one else in their conference really cares about them (Cal is similar, with its Stanford rivalry and nothing else).

Boston College also has the Notre Dame thing. Domers do not like BC (mostly because of 1993). I think if you're in the SEC you're out--even if you are Arkansas--be nature of being in the SEC. Of the Big 12, K-State was successful enough under Snyder that there were plenty of people who didn't like it, same with Colorado back in the late 80s, early 90s. Which leads me back to Iowa State.

Kinda with you on Illinois, and conversely on Northwestern. Their "rivalry" doesn't strike me as all that real, more of a manufactured thing since they're both in the same state. And Illinois is more of a pro sports state, so it doesn't garner as much interest.  

Whereas Purdue/Indiana, as two large public institutions, on either end of the state, is much more real. Purdue/Indiana's football rivalry is of course helped by their basketball rivalry, in a basketball-crazy state, so although neither team is typically very good at football we at least care about the football rivalry. 

I kinda agree with Iowa State. At least with OU/OrSU and with UW/WSU they are in-state rivals AND conference rivals. So while maybe Oregon sometimes has bigger sights on Stanford or UW, losing to OrSU would impact their conference race. Whereas ISU/Iowa might be important in-state, but either team can lose the rivalry game with no effect on their conference race, and both teams also have conference rivals (less so for ISU though).

Where would Georgia Tech fit in this discussion?  Not consequential enough even to "not dislike"?

Heh.