No doubt there is a lot of good advice in your post. Being independent is of course a key aspect of being a successful scientist and researcher, whether in indu

No doubt there is a lot of good advice in your post. Being independent is of course a key aspect of being a successful scientist and researcher, whether in industry or in the traditional pathways in academia (tenure track asst. prof, etc). It's just unrealistic to expect in new PhD students. Of course at places like Berkeley/CMU/MIT/Stanford you will find many such people, who come with "batteries included". As a result, their advisors can sometimes just let them do their thing and they will churn our fantastic results. But that is not really the norm at most of the top-25 or top-50 graduate programs, and the students there will need a lot more guidance and collaborative work to turn them from merely strong students to great independent researchers. So I absolutely agree that being able to work relatively independently will make it easier to get a PhD. I just think that that quality is something one develops during the PhD journey. Re length/rambling: It's not a personal attack on you, just a suggestion that the information you write can probably be compressed and made much more effective. Yes, this is not an English class. And yes, you're trained as a mathematician and trying to be rigorous. But often times dropping some details to convey the general ideas is sufficient, especially on a fairly technical forum like this one. Your writing would be more effective if it was less prolix.