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Is it a local maxima, or is it a metastable solution? Or are those the same thing, at their root?

(this is a legitimate question; I have a very limited math background, and would love to understand the distinctions better, or what variable is being locally maximized, if that's what this is)

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Metastability and local maximums are the same.

https://en.m.wikipedia.org/wiki/Metastability

In physical/chemical systems metastability and local _minima_ (of an energy function) are the same. But for cases where the thing being optimized is not energy, and where the universe isn't driven to the optimal state, do they need to be the same?

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