Let me guess: even if you "prove" yourself, your sites will never rank well again.
Oh goodness, no. What would be the point in that?
Frankly, I tell anyone who starts out new to expect AT LEAST 18 months before the search engines even pay attention to you (of course, I am sure there are all sorts of people who could counter me on this and likely their sites only have a lifespan of about 18 months). The search engines were done with the churn and burn sites a long, long time ago. They wanted to make sure that you had some skin in the game, so to speak.
All pages go through a kind of sandbox really. When I publish a new page, I don't expect it to do anything for at least 12 months, and statistically, they don't really hit full stride until after 3 years.
There are millions of sites out there who have been working for years to get traffic. So why should yours be suddenly blessed just because you got a domain name and a hosting account?
You are truly a little cub in the internet world. The sandbox is just G's way of telling you that your sites have a little growing up to do.