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

I think it's very cool to play with token embeddings in this way! Note that some of what you observe is, I think, a consequence of geometry in high dimensions and can be understood by just modeling token embeddings as random. I recommend generating a bunch of tokens as a Gaussian random variable in a high-dimensional space and playing around with their norms and their norms after taking a random offset.
Some things to keep in mind, that can be fun to check for some random vectors: 

- radii of distributions in high-dimensional space tend to cluster around some fixed value. For a multivariate Gaussian in n-dimensional space, it's because the square radius is a sum of squares of Gaussians (one for each coordinate). This is a random variable with mean O(n) and standard deviation $O\left(\sqrt{n}\right)$. In your case, you're also taking a square root (norm vs. square norm) and normalization is different, but the general pattern of this variable becoming narrow around a particular band (with width about $1/\sqrt{n}$ compared to the radius) will hold.
- a random offset vector will not change the overall behavior (though it will change the radius). 
- Two random vectors in high-dimensional space will be nearly orthogonal. 

On the other hand it's unexpected that the mean is so large (normally you would expect the mean of a bunch of random vectors to be much smaller than the vectors themselves). If this is not an artifact of the training, it may indicate that words learn to be biased in some direction (maybe a direction indicating something like "a concept exists here"). The behavior of tokens near the center-of-mass also seems really interesting. 

I think there is some misunderstanding of what SLT says here, and you are identifying two distinct notions of complexity as the same, when in fact they are not. In particular, you have a line  

"The generalisation bound that SLT proves is a kind of Bayesian sleight of hand, which says that the learning machine will have a good expected generalisation relative to the Bayesian prior that is implicit in the learning machine itself."

I think this is precisely what SLT is saying, and this is nontrivial! One can say that a photon will follow a locally fastest route through a medium, even if this is different from saying that it will always follow the "simplest" route. SLT arguments always works relative to a loss landscape, and interpreting their meaning should (ideally) be done relative to the loss landscape. The resulting predictions are, nevertheless, nontrivial, and are sometimes confirmed. For example we have some work on this with Nina Rimsky.

You point at a different notion of complexity, associated to considering the parameter-function map. This also seems interesting, but is distinct from complexity phenomena in SLT (at least from the more basic concepts like the RLCT), and which is not considered in the basic SLT paradigm. Saying that this is another interesting avenue of study or a potentially useful measure of complexity is valid, but is a priori independent of criticism of SLT (and of course ideally, the two points of view could be combined). 

Note that loss landscape considerations are more important than parameter-function considerations in the context of learning. For example it's not clear in your example why f(x) = 0 is likely to be learned (unless you have weight regularization). Learning bias in a NN should most fundamentally be understood relative to the weights, not higher-order concepts like Kolmogorov complexity (though as you point out, there might be a relationship between the two). 

Also I wanted to point out that in some ways, your "actual solution" is very close to the definition of RLCT from SLT.  The definition of the RLCT is how much entropy you have to pay (in your language, the change in negative log probability of a random sample) to gain an exponential improvement of loss precision; i.e., "bits of specification per bit of loss". See e.g. this article.

The thing is, the "complexity of f" (your K(f)) is not a very meaningful concept from the point of view of a neural net's learning (you can try to make sense of it by looking at something like the entropy of the weight-to-function mapping, but then it won't interact that much with learning dynamics). I think if you follow your intuitions carefully, you're likely to precisely end up arriving at something like the RLCT (or maybe a finite-order approximation of the RLCT, associated to the free energy). 

I have some criticisms of how SLT is understood and communicated, but I don't think that the ones you mention seem that important to me. In particular, my intuition is that for purposes of empirical measurement of SLT parameters, the large-sample limit of realistic networks is quite large enough to see approximate singularities in the learning landscape, and that the SGD-sampling distinction is much more important than many people realize (indeed, there is no way to explain why generalizable networks like modular addition still sometimes memorize without understanding that the two are very distinct). 

My main update in this field is that people should be more guided by empiricism and experiments, and less by competing paradigms of learning, which tend to be oversimplified and to fail to account for messy behaviors of even very simple toy networks. I've been pleasantly surprised by SLT making the same update in recent months.

Interesting - what SLT prediction do you think is relevant here?

Noticed thad I didn't answer Kaarel's question there in a satisfactory way.  Yeah - "basin" here is meant very informally as a local piece of the loss landscape with lower loss than the rest of the landscape, and surrounding a subspace of weight space corresponding to a circuit being on. Nina and I actually call this a "valley" our "low-hanging fruit" post.

By "smaller" vs. "larger" basins I roughly mean the same thing as the notion of "efficiency" that we later discuss

In particular, in most unregularized models we see that generalize (and I think also the ones in omnigrok), grokking happens early, usually before full memorization (so it's "grokking" in the redefinition I gave above). 

Oh I can see how this could be confusing. We're sampling at every step in the orthogonal complement to the gradient at that step ("initialization" here refers to the beginning of sampling, i.e., we don't update the normal vector during sampling). And the reason to do this is that we're hoping to prevent the sampler from quickly leaving the unstable point and jumping into a lower-loss basin (by restricting we are guaranteeing that the unstable point is a critical point)