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So, the description for the Beta badge says that one must "Actively participated in the private beta".
Is Programmers not a "private" beta or something? Because I'm wondering what all would actually be needed to achieve this badge otherwise.
The private beta for Programmers lasted for 7 days immediately following the conclusion of the commitment period. It is unattainable now. We're currently in the public beta period.
But you can still fulfill your Area 51 commitment to the site -- assuming you committed to the site proposal -- for a badge on http://area51.stackexchange.com