Handwriting analysis matched Patsy's handwriting, not John's, but I think they determined the phrasing was really unlikely to be Patsy's because parts of it seemed to be lifted from a "dude" movie that she probably wasn't familiar with it (can't remember which one). They also found a "practice" note on a notepad from the Ramsey's kitchen. The ransom note is one of the weirdest things about the case. It was three pages long, which is apparently way longer than "legit" ransom notes. It addressed John by name multiple times, and according to at least one linguistic forensics expert, that suggested that the author/killer had a lot of personal rage towards him--which, in my non-expert opinion, sounds unlike Patsy, because they stood by each other and remained fiercely close, and that's unusual for parents who lose a child: upwards of 80% get divorced. Additionally, the amount of money requested as ransom was exactly the amount John had just received as an annual bonus--something like $118,000, which was not a lot at all for the Ramsey family--can't remember their net worth, but they were definitely multi-millionaires: they flew privately and the house was something like 10,000 ft2.
(I feel bad writing this comment without taking the time to link sources, so don't just trust me on the details, but I'm pretty sure I'm remembering it all correctly. I've been obsessed with the case for years, to the point where one of my favorite boyfriends came out of us bonding over hours of debate on it.)
If you look at the case as a study of the politics of a wealth and influence in a small town, of how alien the mourning process of the ultra-rich can be (circling the wagons to protect privacy and assets, prioritizing being composed over being emotionally open in the media), and how impossible the truth becomes when you have experts and witnesses and investigators that all claim proof for opposite theories mixed with media obsession and political pressure, it's insanely interesting.