Kinda silly to do this with an idea you actually care about, especially if political (which would just increase the heat:light ratio in politics along the grain for Russian troll factories etc.). But carefully trying to make NN traps with some benign and silly misinformation -- e.g. "whales are fish" or something -- could be a great test to see if weird troll-generated examples on the internet can affect the behavior

Maybe I'll add two addenda:

1. It's easy to confuse entropy with free energy. Since energy is conserved, globally the two measure the same thing. But locally, the two decouple, and free energy is the more relevant parameter here. Living processes often need to use extra free energy to prevent the work they are interested in doing from getting converted into heat (e.g. when moving we're constantly fighting friction); in this way we're in some sense locally increasing free energy.

2. I think a reasonable (though imperfect) analogy here is with potential energy. Systems tend to reduce their potential energy, and thus you can make a story that, in order to avoid just melting into a puddle on the ground, life needs to constantly fight the tendency of gravitational potential energy to be converted to kinetic energy (and ultimately heat). And indeed, when we walk upright, fly, build skyscrapers, use hydro power, we're slowing down or modifying the tendency of potential energy to become kinetic. But this is in no sense the fundamental or defining property of life, whether we're looking globally at all matter or locally at living beings. We sometimes burrow into the earth, flatten mountains, etc. While life both (a), can use potential energy of other stuff to power its engines and (b), needs to at least somewhat fight the tendency of gravitational kinetic energy to turn it into a puddle of matter without any internal structure, this is just one of many physical stories about life and isn't "the whole story".

This is awesome!

I also wouldn't give this result (if I'm understanding which result you mean) as an example where the assumptions are technicalities / inessential for the "spirit" of the result. Assuming monotonicity or commutativity (either one is sufficient) is crucial here, otherwise you could have some random (commutative) group with the same cardinality as the reals.

Generally, I think math is the wrong comparison here. To be fair, there are other examples of results in math where the assumptions are "inessential for the core idea", which I think is what you're gesturing at. But I think math is different in this dimension from other fields, where often you don't lose much by fuzzing over technicalities (in fact the question of how much to fuss over technicalities like playing fast and loose with infinities or being careful about what kinds of functions are allowed in your fields is the main divider between math and theoretical physics).

In my experience in pure math, when you notice that the "boilerplate" assumptions on your result seem inessential, this is usually for one of the following reasons:

1. In fact, a more general result is true and the proof works with fewer/weaker assumptions, but either for historical reasons or for reasons of some results used (lemmas, etc.) being harder in more generality, it's stated in this form
2. The result is true in more generality, but proving the more general result is genuinely harder or requires a different technique, and this can sometimes lead to new and useful insights
3. The result is false (or unknown) in more technicality, and the "boilerplate" assumptions are actually essential, and understanding why will give more insight into the proof (despite things seeming inessential at first)
4. The "boilerplate" assumptions the result uses are weaker than what the theorem is stated with, but it's messy to explain the "minimal" assumptions, and it's easier to compress the result by using a more restrictive but more standard class of objects (in this way a lot of results that are true for some messy class of functions are easier to remember and use for a more restrictive class: most results that use "Schwartz spaces" are of this form; often results that are true for distributions are stated for simplicity for functions, etc.).
5. Some assumptions are needed for things to "work right," but are kind of "small": i.e., trivial to check or mostly just controlling for degenerate edge cases, and can be safely compressed away in your understanding of the proof if you know what you're doing (a standard example is checking for the identity in group laws: it's usually trivial to check if true, and the "meaty" part of the axiom is generally associativity; another example is assuming rings don't have 0 = 1, i.e., aren't the degenerate ring with one element).
6. There's some dependence on logical technicalities, or what axioms you assume (especially relevant in physics- or CS/cryptography- adjacent areas, where different additional axioms like P != NP are used, and can have different flavors which interface with proofs in different ways, but often don't change the essentials).

I think you're mostly talking about 6 here, though I'm not sure (and not sure math is the best source of examples for this). I think there's a sort of "opposite" phenomenon also, where a result is true in one context but in fact generalizes well to other contexts. Often the way to generalize is standard, and thus understanding the "essential parts" of the proof in any one context are sufficient to then be able to recreate them in other contexts, with suitably modified constructions/axioms. For example, many results about sets generalize to topoi, many results about finite-dimensional vector spaces generalize to infinite-dimensional vector spaces, etc. This might also be related to what you're talking about. But generally, I think the way you conceptualize "essential vs. boilerplate" is genuinely different in math vs. theoretical physics/CS/etc.

Nitpick, but I don't think the theorem you mention is correct unless you mean something other than what I understand. For the statement I think you want to be true, the function also needs to be a group law, which requires associativity. (In fact, if it's monotonic on the reals, you don't need to enforce commutativity, since all continuous group laws on R are isomorphic.)

This is very cool!

Right - looking at energy change of the exhaust explains the initial question in the post: why energy is preserved when a rocket accelerates, despite apparently expending the same amount of fuel for every unit of acceleration (assuming small fuel mass compared to rocket). Note that this doesn't depend on a gravity well - this question is well posed, and well answered (by looking at the rocket + exhaust system) in classical physics without gravity. The Oberth phenomenon is related but different I think