Even though I've never partially filled a corny (I'm way more likely to fill above the gas dip tube!) for some reason I was curious about this.
Intuitively it seemed like the net CO2 usage to properly flush, carb and dispense a keg starting with 5 gallons would be very close to equal to that used for 3 gallons in the same keg. But I couldn't make up my mind which would in theory use more CO2. So I thought I'd give the math a try right here, right now.
So assume 5 gallons in a corny leaves roughly two quarts of headspace, after filling and lidding the keg you fill that volume with CO2 and release it three times, you're carbing to 2.5 volumes, and with no change in CO2 pressure (ie: purging pressure = carb pressure = dispensing pressure).
So it takes 2 gallons to flush and then refill the headspace, 2.5 x 5 gallons to carb, and 5 gallons to dispense for a total of 19.5 gallons or 2.60 cubic feet of CO2 at pressure.
Starting with 3 gallons you'd have 2.5 gallons of headspace, use 7.5 gallons to carb, and use 3 gallons to dispense. Four shots of CO2 to flush and fill the headspace = 10 gallons, plus 7.5 gallons to carb to 2.5 volumes, plus 3 gallons to dispense, for a total of 20.5 gallons or 2.75 cubic feet.
Unless I missed something (quite possible - going from sealed to unsealed model during the purging could mess that part up) that's pretty darned close. That 3 gallons would use more gas reflects the power of multiplication
Cheers!