UPDATE ON ERROR 3:
I got a call from "Darren", one of the Stanford CA Apple Store Managers late Tuesdau. He told me that based on my "feedback" that my "case" had been escalated to Tier 2 which apparently brings some of the Apple Care engineers that look for "strange new things" into the mix.
Darren said they would be happy to replace my iPad 3 (16GB w/Verizon LTE) for free (with a factory certified refurb). We chatted a bit and he appeared to indicate that the Apple Care engineers wanted my machine to evaluate it.
For those of you not aware, Apple reportedly has a "bounty" of sorts for Apple Genius's, mostly in play when new devices come out, to grab and return Apple devices that have unique failure modes. This is smart on Apple's part as it give the manufacturing and software engineers a chance to catch manufacturing "corner cases" so they can tweak the manufacturing line or software, minimizing future customer returns. Looks like "Error 3" might be one of those "corner cases" they might be wanting to jump on (hopefully).
I don't know what this means for everyone else having Error 3 issues, but if the problem is a Flash erase/write verify issue as my "Freezer Physics" strongly suggests, there is a possibility that Apple can modify the eraser/write verify alogorithm to solve the issue, or even add in a test for "weak Flash bits" and "map out" bad bits (puts "weak" bits into an unusable address location), to solve the problem. The Apple software guys are smart. I believe they can totally fix this with a software tweak. I suggest waiting a few days and visiting and Apple Genius bar again. I don't know what they might do, but "IF" they issue some sort of a quiet replacement for devices with Error 3, you might luck out. I'm not hopeful they will do this, but they already pleasantly surprised me once.
Keep in mind "BE NICE" when you meet with your Apple Genius (I was both times). "Sundown" (my Genius tonight, no kidding on his name, a cool guy!) was fast and efficient. He grabbed my replacement device from Darren, moved my Verizon sim card over to it, had me finger sign on his iPad the swap, and I was on my way. These guys and gals deal with angry people all day who want a new device for free after dropping it into a toilet, or some such, so being NICE can go a long way.
I'll report if I hear more. For those who don't have luck with their Apple Genius, and my "Freezer Physics" didn't solve your "Error 3", hang in there. I am now hopeful that Apple engineering is going to dig into this and come up with a software fix. Don't expect it to hit with 7.1.0, which is already in beta with us Developers, but maybe the release after that, "IF" they find a software solution. From another report on this thread, they're also looking at log files, so there is action going on.
When you get your Apple Genius post meeting email survey, be sure to mention Error 3. The more they see that, the better. Plus, maybe your Apple store manager will see it and ask you to bring in your device for a free replacement if the Apple Care engineers continut to gather up some different Error 3 samples for evaluation.
Let us all know what happens with your device no matter how long it takes. Persistance sometimes can pay off for all.