This paper gives a mathematical model of when Goodharting will occur. To summarize: if

(1) a human has some collection $s_1,\dots ,s_n$ of things which she values,

(2) a robot has access to a proxy utility function which takes into account some strict subset of those things, and

(3) the robot can freely vary how much of $s_1,\dots ,s_n$ there are in the world, subject only to resource constraints that make the $s_i$ trade off against each other,

then when the robot optimizes for its proxy utility, it will minimize all $s_i$'s which its proxy utility function doesn't take into account. If you impose a further condition which ensures that you can't get too much utility by only maximizing some strict subset of the $s_i$'s (e.g. assuming diminishing marginal returns), then the optimum found by the robot will be suboptimal for the human's true utility function.

That said, I wasn't super-impressed by this paper -- the above is pretty obvious and the mathematical model doesn't elucidate anything, IMO.

Moreover, I think this model doesn't interact much with the skeptical take about whether Goodhart's Law implies doom in practice. Namely, here are some things I believe about the world which this model doesn't take into account:

(1) Lots of the things we value are correlated with each other over "realistically attainable" distributions of world states. Or in other words, for many pairs $s_i,s_j$ of things we care about, it is hard (concretely, requires a very capable AI) to increase the amount of $s_i$ without also increasing the amount of $s_j$.

(2) The utility functions of future AIs will be learned from humans in such a way that as the capabilities of AI systems increase, so will their ability to model human preferences.

If (1) is true, then for each given capabilities level, there is some room for error for our proxy utility functions (within which an agent at that capabilities level won't be able to decouple our proxy utility function from our true utility function); this permissible error margin shrinks with increasing capabilities. If you buy (2), then you might additionally think that the actual error margin between learned proxy utility functions and our true utility function will shrink more rapidly than the permissible error margin as AI capabilities grow. (Whether or not you actually do believe that value learning will beat capabilities in this race probably depends on a whole lot of other empirical beliefs, or so it seems to me.)

This thread (which you might have already seen) has some good discussion about whether Goodharting will be a big problem in practice.