Just to add to this. Another way could be to find a specific construction. If you could for example find an algorithm that given any even integer returns two primes that add up to it and you showed this algorithm always works. Then that would be a proof of the Goldbach conjecture.
Yeah. “Calories in calories out” is correct in theory but it is an oversimplification. Most people burn a vastly majority of their calories “at rest” by just maintaining their body temperature.