The IRS does not know that you had no tax liability from that until you tell them on your tax return. If you do not report it, then in just under a year you can expect a letter from the IRS telling you that they have information about income that you did not report. They will helpfully include a bill for penalties and interest on the entire amount of the proceeds. You can save yourself a lot of grief by filing a tax return and reporting the 1099-B.
You can't avoid paying taxes by not filing. The IRS gets copies of your 1099-B and they do check to see if you claim it. If you don't file and claim it, then when they catch you in about a year you will pay a lot more, and there will be no way around it.
I don't think we're on the same page. I don't want to avoid paying, I've just heard about a Zero Percent Tax that is used specifically for people who are under the 15% tax bracket that do not have to pay on additional capital gains income. I just want to know what to expect.
All that means is that if you qualify you don't owe any tax. It does not mean that you don't have to file and tell the IRS that you have no tax. As I said earlier, the IRS will not know that you don't owe tax until you file a tax return and tell them.