Do you work in a right-to-work state?
You shouldn't have signed the performance plan. You should have written a rebuttal, request that it be placed in your file, and requested a meeting with HR.
Also, request to see your employee file. Contest anything that is inaccurate.
Contact the EEOC and file a grievance.
Find out if what you did/didn't do warrants a verbal warning, writeup w/action plan, or (up to) termination. Don't just assume that they are following the chain of command with their actions as it relates to your infraction.
IMO, once you start getting disciplined you need to transfer or resign once you have secured gainful employment elsewhere. Don't ever quit a job as you will not be eligible for unemployment if you should need it.
There is a lot of bs politics in Corp: You need to always fiercely protect your livelihood / reputation while simultaneously displaying professionalism, being a team player, and doing your work to the best of your ability. Document everything and always remember that you do not have any friends at work.