Can I use Google Sync (Exchange, Push) through Mail.app in Mountain Lion?

I love using gmail through Google Sync on my iPhone because it supports PUSH through the Exchange protocol. See: https://www.google.com/mobile/sync

Essentially, they provide an Exchange compatible server through m.google.com.

I would like to setup Mail on OSX To use exchange as well, but have not been able to set it up properly.

No, you can't use Google Sync through Mail.app in Mountain Lion.

Google's exchange server m.google.com is not a fully implemented exchange server. It just implements the ActiveSync protocol which seems to be a subset of the exchange protocol for mobile devices. See also this information on Google Sync: http://support.google.com/a/bin/answer.py?hl=en&answer=135937

The problem is with Mail.app (not Google Sync), which does not suport ActiveSync. It uses Exchange Web Services instead, which must be pulled.

The options for push in Mountain Lion appear to be either use Outlook 2011, which does use ActiveSync, or configure Mail.app to use IMAP. GMail supports IMAP IDLE, which is the IMAP equivalent of ActiveSync push.