solution
well, i had a look at the solution to my problem with one of my professors, which i posted on the 14th. it fails toward the end of the proof. any mathematics graduate student beyond his or her first year will see it immediately. it turns out that proving my claim requires more machinery than i had thought. i am frustrated that my solution doesn’t work, but i learned a lot in the process. as one progresses in mathematics, one learns that past a certain point, learning takes place entirely on one’s own. more and more of the effort of learning is pushed off onto the student from the teacher, until this idea reaches its natural fulfillment in the ph.d. thesis.

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