Faucet-mount filter thread/size incompatibility

Question

I am trying to install a Culligan FM-25 filter onto a faucet, and it does not seem to fit. I am able to screw on the filter, but it is not secure, and water bubbles/sprays out the top. It appears that the faucet head is too long (or has too many threads)? I know that my filter can accommodate 55/64"-27 M faucet spouts. If the filter at least partly screws onto the faucet, then isn't the faucet 55/64"-27 M as well? I can't find any examples of thread spacings other than 27 threads per inch for 55/64, so I'm assuming the faucet size is also 55/64"-27 M (but I could be wrong).

I have a picture of the faucet head as well as one of the water filter and aerator that I've removed from the faucet. As you can see, the aerator appears to be much deeper than the filter mount. Not only is there an unthreaded region above the threads, but I can count 4 visible levels of threading above the washer, whereas only 3 levels are visible on the mount of my filter.

If the faucet and filter are both 55/64", and yet the two are incompatible, I have no idea how to fix this, because surely there aren't any 55/64"-27 <-> 55/64"-27 adapters out there. I suppose I could try joining two adapters together, but it would probably make the end of the faucet too low.

Edit, possibilities:

1. Threading of faucet is something like 24 per inch but same width, hence allowing me to screw on the filter, at least partially, but prevents a good seal.
2. Width is slightly different (not sure if this is possible)
3. Somehow faucet having too many threads prevents attachment of my filter

In the case of the first possibility, what thread size is the same width as 55/64"-27 M, but has <27 threads per inch?

Answer 1

It turned out that the faucet end and the filter were both 55/64-27. I was able to confirm this when 13/16 and 15/16 both did not fit. The lesson is that if it screws on, it's probably the same size. The various faucet sizes are different enough that they won't be able to thread onto each other.

But why was the filter not fitting well despite the threading compatibility? I studied my filter closely. You thread it onto a faucet by turning a metal ring. It's my belief that the ring needs some sort of flange on the faucet to brace against in order to seal correctly. As you can see in the original pictures, th current faucet head consists only of threads.

The solution I took was to use two adapters. First, a 55/64-27 F to 15/16-24 F adapter onto the faucet. Then, a 15/16-24 M to 55/64-27 M adapter onto the first adapter. Then finally the filter. In the picture below, the M/M adapter is not really visible, but it connects the filter to the first adapter. Prior to buying the 55/64-27 F to 15/16-24 F adapter, I thought that I might be able to thread the adapter in reverse (the larger side facing up) so that it could provide a flange on the faucet that I could twist the filter's ring against, but that did not work; not enough room.