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I've been trying to find the US Military's guidelines or standard for how long a soldier can go without food, for any branch of the military.
Is there any such recommendation, as to how long a man can go without food (and what activity level he can be expected to perform to)?
TL; DR - I don't think there are any actual existing guidelines for how long a soldier can go without food. There are recommended minimums for various operational capacities. As has been pointed out already, water is of much more importance, and there are numerous performance studies showing athletic impairment at levels as low as 5% dehydration.
Long answer:
I cannot find (And having been in the medical side of the military for 10 years I do not remember any minimum guidelines on food deprivation) anything current mandating that soldiers cannot go more than X days without food.
However, the minimum caloric needs of soldiers and various Special Forces (SF) units around the world has been studied quite a bit, as well as what happens with physical and cognitive performance in periods of both sleep and food deprivation. You could compare that with the extreme limits of food deprivation before death and come to an answer.
I apologize in advance, some of these I do not have access to the full study.
This study looked at both food and sleep deprivation (Admittedly for a civilian group going through survival training), and concluded that within 24 hours, subjects showed marked decline in several cognitive areas. Food deprivation was less dramatic than was sleep deprivation. (And this has been proven in real world studies, many people start hallucinating with sleep deprivation long before food deprivation would have an effect).
This is one of the better studies I found, as it compared energy requirements over different SF groups in different regions, as well as under different stresses. The section for the US Rangers showed a daily calorie deficit of 1200 calories (The data is given in MJ, which is ~ 240 kilocalories) per day. They showed a surprising amount of weight loss (16% vs an expected 10%), but not as much strength loss as they expected. This was not, however, complete deprivation but merely shortage, and presumably balanced nutrition for what they did get.
A very interesting paragraph:
Soldiers provided an additional 1.7 MJ/day supplement during the 8-week Ranger course converted approximately half of the additional food energy to body energy stores, and used the remainder to fuel higher energy expenditure [8]. This presumably reflects a lifting of the ceiling on TEE that is imposed by inadequate intakes, as in studies of undernourished Columbian school children playing soccer alongside adequately nourished children [27], when energy intake appeared to limit work performance. Earlier studies of laborers, such as sugar cane cutters and road builders, also support the idea that inadequate energy intake constrains energy expenditure 2.
This suggests that when in a period of caloric deficit, the body limits the amount you can do before exhaustion, and when calories are restored, the activity level also rebounds.
I admit, I did not read the entire chapter, but most of the methods for establishing MDRI (Military Dietary Reference Intakes, or RDA for soldiers) can be found here, albeit slightly dated (2006).
This PDF from 2001 does give a minimum of 1500 kilocalories for restricted rations, which suggests that at a minimum, soldiers must get that amount on a daily basis to be able to function at any operable level. You might be able to extrapolate from that how long without food before a soldier becomes unable to function (Again, paying attention to the water factor).
If you are interested in pursuing the research rabbit hole, there are numerous linked studies cited as reference in the studies I linked, and google scholar will have many available for perusal. I realize this doesn't quite answer the full question, but it's like trying to prove a negative, I'm not sure they even exist in the form you are seeking.
I see that nobody has weighed in on this, I think because there is an old saying that "an army marches on its stomach" (N. Bonaparte). The armies of the world would probably focus more on how to get nutrition, rather than how long it is ok to be without. Due to physiological differences (I'm not a doctor BTW) between individuals, there is sure to be a wide range of tolerance for starvation before different humans are unable to function to whatever degree is your benchmark.
I have some experience through my brother who coaches wrestlers. These young men routinely starve themselves to make lower weight classes, all while working incredibly hard and expending extraordinary amounts of energy. When I expressed my concern about long-term health effects, my brother said something that made some sense to me: "Humans are genetically built to be able to keep working through periods of fasting, how long depends on their mental and physical fortitude".
