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Re: mesosaurus

BTW, how does PAUP differentiate massive convergence versus homology? Or
does it take human intervention (weighin characters, etc>) to note the key

Erm... huh? PAUP* does cladistics. This means that it will give you those distributions of character states across trees ( = it will give you those trees) that can be explained with the fewest assumptions of convergence ( = the most parsimonious ones). It tries to distribute all characters so that each state change happens once, and balances them against one another so that the fewest additional changes must be assumed. If... I hope this sentence isn't too long... :-/ if there are enough characters the most parsimonious distribution of which contradicts the assumption that the character change in question happened only once, the character change in question will be shown as having happened several times convergently.
If you don't trust the result because the character is related to (say) burrowing, then add more characters (see "enough" above) and/or more taxa to test your distrust. More taxa, especially more basal taxa, help because if the basal taxa of two clades retain plesiomorphies that the derived ones lack, the corresponding apomorphies can no longer come out as synapomorphies of the two clades.
Something tells me I should use PowerPoint to explain this...

The answer will not only affect the mesosaur question, but another one
that recently came up about "worm lizaards" and "legless lizards", which
are all definitely lizards, but are they all related? Or does burrowing
impose convergence?

Which ones do you mean, amphisbaenians and anguids? They seem to be rather distantly related... yes, burrowing does impose a lot of convergence.