How does one earn the Yearling badge on meta if there is no reputation? If it is just judged using the reputation on MSE, how can the member be judged 'active' on the meta site?

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Despite its description, the Yearling badge does not judge your activity:

You'll get the yearling badge as long as you have at least (Years * 200) rep by the target date. We don't actually test that you got the reputation within the specified year.

And yes, it uses the main site reputation but the meta site account creation date. You can visit meta once, get 1000 points on the main site, and in due course, both meta and main will issue you 5 yearling badges even if you forget about both of them entirely. Just for illustration: Robin Chapman has two Yearling badges on both main and meta despite not visiting either since January 2011. Many users with Yearling on meta never posted anything on meta, and Yearling is their only meta badge. In fact, it's the most common badge on meta.

Arturo Magidin will keep receiving Yearling badges at least until 2761, but actually for much longer because his answers still earn reputation. For how long will they earn at least 200 rep per year? The answer may depend on whether the Universe has enough matter to undergo a Big Crunch.

Also, it turns out that the premise "there is no reputation [on meta]" is false. Meta sites have a reputation count that is never displayed to users, and is used to award Mortarboard, Epic, and Legendary badges. Some quotes from SE developers:

Is there a reason why we gain ghost reputation on the meta of an SE website?:

In the SE system, votes tie to rep. A side effect of it being the same code-base, but I don't see it as a big enough deal that it would warrant changing for the meta sites.

SE 2.0 Meta doesn't award rep, but it refers to "rep":

Because it doesn't provide any utility to the site functionality, it isn't important for people to see what others have for reputation. It's primarily useful for measure of participation by the Team. Who, as mentioned earlier, are the ones who have clear access to it.

Should I really have been awarded mortarboard on a Per-site meta?:

Just redefine it as "you received a lot of upvotes in a single day". So I think it is correct in spirit.

It seems that the meta site code is much the same as the main site code with hacks to hide or disable certain things like bounties. Meta.StackOverlow even has a visible reputation count, independent of StackOverflow. Which is reasonable because (i) it also serves as Meta.StackExchange, for questions about SE in general; (ii) it existed with visible reputation long enough for it to acquire "historical value".

So, there are two reputation counts on which meta badges could be based. Someone decided to use main site count for Yearling but meta site count for Mortarboard... I guess I agree in both cases.

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    $\begingroup$ +1 for the fact about Arturo! $\endgroup$ – Parth Kohli Jan 6 '13 at 8:17

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