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Openai/68ec50da-cf00-8005-b5f6-b683506e5853
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=== User: Here is a sketch of a complete proof. === Here is a sketch of a complete proof. I am using various variants of your argument for Lemma 1 in the first message you sent me throughout. Please turn this into a completely structured latex document with all the lemmas spelled out. The first lemma should be the critical one (essentially a variant of lemma 1 which can win the 1-6/pi^2 or 1- \delta as in the document). There are several cases please incorporate this casework carefully. Please structure the proof so that is is easy to edit. Also place all numerical facts used at the top of the document and double check each of then using python.
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