What happens, isn't a 'false' positive (unless she's on fertility drugs, or has certain kinds of cancer (that's where one WILL get false positives)....
... What happens is a 'chemical pregnancy'. Women having unprotected sex, on average, are 'pregnant' 4-6 times a year. What happens? The ovum AND the uterus (and every other organ), CONSTANTLY check for screwed up DNA. When eggs fertilize, MOST of the time, the DNA dosen't zipper correctly. These wouldn't be 'birth defects', but instead are fatal zippers that code for 'no bones' or 'acid for blood' or 50 arms and nothing else, or, or, or, or.
But the body constantly self monitors. No matter what, a fertilized ovum starts the chemical process for pregnancy. BUT if the ovum notices bad zippering, it self destructs (so does every other cell in our body, when the DON'T AND the backup fails... Neighbor cells also check adjacent cells... When the self check AND the backup fails? That's known as cancer. And the cancerous cells keep replicating, and replicating the bad DNA.). If it DOESN'T self destruct, then when it implants, the uterus does the normal 'backup check' on neighboring cells, and calls a destruct sequence on both the fertilized egg, and itself (the lining).
The body is VERY good at these checks and backup checks (we typically have dozens of cancerous cells in any given minute, but either the cells self destruct, or when the bad coding disables the check the neighboring cells catch it, phone for help, and then several hindered neighboring cells all self destruct.
SO.... In the first 2-4 weeks MOST 'pregnancies' self correct. But the chemical process is already started, so even though there is no more embryo, a woman's blood and urine dtill register as 'pregnant' because the chemicals we test for are still there.
Hence: Chemical Pregnancy.
No embryo, period on time or a week or two late, and ready to try again the following month (unlike miscarriage, which has weeks of bleeding and up to several months of wacky hormones, and is a HUGE stress on the body... Chemical pregnancies are super common and don't even blip our fertility cycle).
We have the tech to test for pregnancy super early, but that will never make the market. There's not only nothing to be done, not only do we not want to do anything -no live baby possible-, but even though they don't mess with our bodies like miscarriage does, they'd be devastating to most peoples minds and hearts each and every single time fertilization results in bad zippering of the DNA. As it is it's usually only those TTC and testing early and often end up with a positive one day and negative (or period) the next.