Unless one has connections, there is no meaningful appeal process for polygraph false positives with any federal agency. The best you can do to mitigate the damage caused by your polygraph operator's false accusation that you lied is to write a reply to the "better qualified applicant" letter you received explaining that you told the truth (and without offering any explanations or rationalizations for why you might have "failed"). Your letter should be added to your file and will document the fact that you didn't tacitly accept the accusation that you lied.

I hope you might consider contributing a statement about your polygraph experience for inclusion on our Personal Statements page (the 3rd one is mine):

https://antipolygraph.org/statements.shtml

Hello George,  I recently applied for position with the USSS and the process was all professional and great until the polygraph. I was told I failed one of the questions and for the next 2 hours the examiner tried every possible way to make me confess to something. I had done nothing to confess to. It ended with the person saying it's up the powers that be to decide. Couple of days later I went from being very qualified to receiving the BQA letter. Everybody is very secretive about this unfair test and I can not get any info. I was wondering if you could help on how I can contest the results. Thank you.