But there is proof for the impossibility of both, at least insofar as proof is possible in science. For the first kind, the proof is basically Noether's theorem, which forbids creating energy based on the assumption that physics is time translation invariant. The theorem is mathematical truth and time translation invariance is experimentally verified except on a cosmological scale.

The second kind is impossible due to the 2nd law of thermodynamics, which is a statistical law. It boils down to the fact that if we randomly disturb a system with a lot of possible configurations, it is more likely to end up in a state that has more corresponding configurations than in a state that has few corresponding configurations. And calculating the probabilities with anything that is even remotely macroscopic, it is all but a certainty because of the large number of configurations.

And by "all but a certainty" I mean: "In the entire lifetime of the universe, the probability that it has never happened is overwhelming."

@RootGroves if the Lagrangian has time translation symmetry then the resulting physics conserves energy at all time scales, both short and big.

Note that reality trumps theory, always. If someone did build a working perpetual motion machine, it would disprove conservation of energy. We know it's impossible to create energy because everything we've ever tried has always failed, and there are good theoretical reasons too, and it makes a lot of math work nice, but there's an extremely tiny chance we're wrong. Conservation of energy is an observed effect, not a cause.

@Vercassivelaunos "The theorem is mathematical truth" Mathematics is not physics. If mathematics were physical truth, anything would be possible.

@user20574 "We know it's impossible to create energy because everything we've ever tried has always failed" This is a strange argument given that, not that long ago, we knew that mass and energy were completely different things and that each was conserved separately.

I am not sure to understand the comments. I stated that machines of the first kind, which violate conservation of energy cannot exist if the laws of physics we know are valid. But this was well explained in another answer. Here I focused on the second kind, in which there is no definitive mathematical proof that the "basic" laws of physics forbid them, basically, no definitive proof that the second law cannot be violated by the basic laws of physics.

@Dale no the weak interaction literally breaks energy conservation. In fact if it didn't the baryon and lepton number would be significantly less.

@JimmyJames They are both still conserved separately, btw.

@user20574 How does a nuclear power plant work, then?

@RootGroves The weak interaction breaks CP symmetry. This is not equivalent to breaking conservation of energy. Note that it is in some ways equivalent to breaking time-reversal symmetry, but the symmetry associated with conservation of energy is not time-reversal but time-translation symmetry (essentially the symmetry that states that when an interaction occurs has no bearing on the outcome of the interaction), which has never been observed to be violated.

@JimmyJames It captures the excess energy that is given off as atoms rearrange themselves into different states with lower (more negative) binding energy than before.

@user20574 And that energy is a contribution to the mass of the atoms in question. The decay products, taken together and cooled to the same temperature as the original atom, will weigh (very slightly) less, as they have less total energy.

@Hearth Yes, because you the reactor took some of the mass away. That is not surprising. What is surprising is that energy has mass. But this doesn't mean that energy and mass weren't both conserved. If I take the water out of a water balloon it weighs less because the water has mass, but the mass and the water were both conserved.

@Hearth the neutron decay would be impossible if the weak interaction doesn't violate energy conservation. The W boson is so much heavier than the difference in mass between Proton and neutron so if energy was always conserved the neutron would be stable which in practice it is not.

@RootGroves The W⁻ involved in neutron decay is a virtual particle, which does not necessarily have all the same properties as a "true" W⁻ boson; in particular, its mass does not have to be the traditional 90-ish GeV. In modern quantum field theory I believe this would be considered as simply a fluctuation in the W field, not properly formed to become a particle. Though don't quote me on that, it's probably more nuanced than that.

@user20574 I believe this is a nuance of interpretation between two (or perhaps more) equivalent models of what's going on, then, down to how exactly one defines the words "mass" and "energy".

@Hearth did you really linked in Wikipedia?Anyway how do you explain neutrino mass oscillations then?

@RootGroves I mean irrelevant to this physics forum , I really like what I did.For regular verbs im gonna use from now on both "did" and the "-ed" ending when forming a question.

@RootGroves Yes? It's a readily accessible source with no requirement that you have access to academic journals or obscure textbooks. You can start there, and then get more information from other sources depending on which ones you have access to. As for neutrino oscillation, that's a bit out of my realm of knowledge; I know it happens, but I don't know the mechanism behind it. I would expect, at a guess, that the mass-energy difference corresponds to lost or gained kinetic energy, but that is only a guess.

@Hearth in Wikipedia anyone can post up though so I wouldnt say its a reliable source.

@RootGroves It's more reliable than your typical news article about some scientific breakthrough, and it has a convenient list at the bottom of much more reliable sources on the topic.

@Hearth you think I read that on a news article?No no sir.I actually watched a video from Fermilab's youtube channel.That is a reliable source.

@RootGroves I'm not saying you did, I'm merely remarking that wiikipedia, while not the most reliable source, is hardly the least reliable either. || On an unrelated topic, I noticed your self-reply above saying "For regular verbs im gonna use from now on both "did" and the "-ed" ending when forming a question."; I feel it should be pointed out that in English, this is generally considered incorrect grammar and/or an error.

@Hearth well yah but im a native speaker so I can talk however I want.In my region for verbs which end with -nk or any double consonant which dont sound well , we add a -ed to the end regardless of if we have "did" already.

@RootGroves I mean, you can talk however you want even if you're not a native speaker; I'm just saying how it might be perceived.

I mean yah its a very regional thing , I should write standard english when online.

@RootGroves you are incorrect. The weak force conserves energy. The virtual particles you mention are not on shell.