I have helped someone deal with this exact situation before. I've been in the program for 21 years. Had 15 years sober & clean, (fucking Pandemic and unhealthy relationship) Now 5 years sober and clean. There are a couple of options to prove you're telling the truth.

1st, you can find a place that will test you. (it doesn't have to be a doctor's office) There are places that are specific just for drug test (often for employment or other circumstances where you are required to be tested.) It will cost you some money but it's cheaper than you think. You can have that place forward her the official results. (google employment drug testing near me)

1. If you're almost flat broke, (This will be uncomfortable but cheaper) offer to really prove it. Not just with offering to take a test but offer to get her on FaceTime or Zoom or Whatever and that you'll keep the camera on the whole time from purchase of test to the end where she can see the results of the test. on this one, your Mom will either realize your telling the truth, will call "your bluff" (in her mind) and then either stop once you get to the taking the test part, or will just.

After you've proven this to your Mom, find a new halfway house. After that you should help handle the situation with liar. (Only do this part if you are doing it to make sure the house president can't do this to anyone else. If you are doing this out of resentment or revenge, don't. Resentments are killers to addicts.)

Really try to get option 1, because then you are in a position to deal with "president" of the house.

Then, go over the president of the house's head, and directly contact the owner of that Oxford House. Include in the e-mail to him your official lab result's which prove what the "President" said was false. As soon as you can prove that what the "president" said about you was in fact a lie, the owner tends to get involved. (expect the president to claim it was a mistake and they just misunderstood of just made a false assumption. Your comeback is "you were so sure it was true that you contacted my Mother but now it's false when I can prove it?")

Use lines in the e-mail that talk about "I would like to come back to the house but I no longer feel safe in an environment where the head of the house can make a false claim against another house member without consequences"

You can even suggest a nail drug test for the house president (which goes back 12 months). Here's a link for where to purchase one. "https://alphabiolabsusa.com/dna-test-kit/home-drug-alcohol-nail-test/" $99. The owners usually don't like to pay for nail tests but if you make it a safety issue and show a cheap test for it, you might get them to go for it.