Assumption: Our weight is primarily affected by our conscious decisions of how much we eat and excercise.
Consequence: The weight of all of us (except for the most rigorous calorie counters) would fluctuate strongly (several kg per year).
The "proof" goes as follows: If our calorie balance was off by only 20-25 kcal a day, this would amount to about 9000 kcal per year, which corresponds to one kg of pure fat. 20-25 kcal is about one percent of our daily intake. It corresponds to about half a slice of bread or 300 m of jogging, respectively. This is way below the precision any of us can excercise on a day-to-day basis. In other words: if it really were up to our concious minds to maintain energy balance, we'd be in a mess. But this is a contradiction to what we actually observe: there are many people whose weight is stable over a long period of time. So, apparently, on a subconscious level there are strong feedback cycles at work, which overrule our conscious actions.
Of course, nothing is definite in medical science, but I think this is kind of an elegant way to think about such a controversial topic. Thanks to the power of maths, we do not have to rely on indirect information from epidemiological or laboraty studies.
Anyway, if we accept that it is hopeless to try adjusting our weight by quantity of food and excercise, can we do anything? Well, we can go for quality! It is well known that hormones such as insuline, human growth hormone, sex hormones, stress hormones etc. strongly affect our body composition. And we also know that we can affect the production of these hormones by our life style and the type of food we eat. Of course, at this point we do have to go into the (secondary) scientific literature to find out what we should do. But at least we know what questions to ask.
Finally, we should mention that the first law of thermodynamics certainly holds (see also this post), i.e.
calories in = calories out + calories storedBut these are things we cannot affect directly. For "calories in" you need perfect willpower. For "calories out" you need complete control over all metabolic pathways including calories leaving you in the form of heat, not completely metabolized molecules etc. And you have to be extremely precise at that ...
One last thought: the concept of calories is actually quite amazing. You can burn food in a calorimeter and measure by how much water is heated up. The number you get, quantifies the ability of this food to support life. This is a fascinating idea showing the power of abstract physical laws. But it is not the whole story.