"I'm 25, am 5 foot 9 tall and weigh 177 pounds."
How much of that 177# is spare body fat?
If you have, say, 15# of "healthily burnable" fat, that would provide you with ~52,500 potential calories of energy if you had adequate dietary carbs to support its metabolism plus enough protein to keep your body from cannibalizing itself. If you hiked at ~2.5 miles/hour you would probably be burning in the neighborhood of 30% carbs, 5% protein, and 65% fat. I have figured out by trial and error, some study, and post trip analysis of my body weight differentials that I burn around 4000-4200 calories/day of which ~65% come from fat. If you burn calories at even close to that rate, you would need maybe 45,000 calories for a 10 day trip, allowing some extra for your greater weight(I weigh 137#). 65% of that figure would be ~29,250 calories from fat; 30% of that figure works out to be 13,500 calories from carbs, and 2250 calories from protein. Based on this you would need to carry 13,500/4 calories per gram = 3375 grams of carbs/~28 grams per ounce = 120.5 oz/16 oz per pound = ~7.5# of carbs to burn 7.2# of excess body fat on your 10 day trip, based on parameters I have found work for me over the last several years. Even if my parameters are of by 20-25%, either for you specifically, or just for everybody but me, you have plenty of body fat to get the job done, provided you carry enough carbs to support its metabolism. I didn't do the math for 5% protein, but it's a trivial exercise. Bottom line, you should carry 7.5 to perhaps 11# of complex carbs and whatever the protein at 5% works out to be to have a chance of pulling this off without doing yourself some serious damage. Good luck if your heart is set on this.
The figure of 5% protein is meant to refer to protein for energy production. Total recommended daily intake of protein is considerabley higher. Estimates vary, but one recommendation specifically for endurance athletes in "Exercise Physiology" 6th Edition by McArdle, Katch, and Katch, p. 40, is 1.2-1.8 grams per kilogram of body mass. I carry 70-75 grams per day for a body mass of 65 kg and it seems to work fine, so far. NO detectable weakness, loss of lean muscle.