I wasn't able to find anything via search, and don't know enough about the Circle to be able to ask using accepted terminology. As context, I'm a guitar player working out the theory/geometry behind the keys, using the circle of 5ths.

Please see my Circle of 5ths "number line" below. There are three rows. From top to bottom row, each row denotes number of sharps/flats, key name, and note added, respectively. In the middle column, we have 0, key of C, and no added note. To its right, we have 1 sharp, key of G, adding F#. The other way, we have key of F adding Bb, and so on and so forth in both directions, adding one flat or sharp at a time.

I noticed that each flatted note added becomes the name of the next flatted key. I'd known about this for a while, but I was surprised to discover the same pattern continues past C, and into the "sharp keys"! It is clear that 'BEADGCF' is seen in the order of key names, the order of flats added (Bb/Eb/Ab/.../Fb), and the reverse order of the sharps added (F#/C#/G#/.../B#).

1. My biggest question is, what do repeated interval jumps of perfect fifths have to do with the business of naming keys and the order in which notes are sharped/flatted?

Some other questions I had:

2. Why do we need to "un-flat" the sharp note when it becomes the name of the next key? So what I mean is, Why is it key of F, but not key of F#? Why key of G, not G#? I would expect it to be symmetric to the flats, but I am interested in knowing whether there is a deeper geometric pattern to this.
3. Why does this pattern span two keys in the sharp side, but only 1 in the flat side? For ex: D# is the 4th sharp note added, but key of D has 2 sharps. The difference is 4-2=2 sharps. In contrast, take Ab -- it is the third flat note added, but it is the name of the key of 4 flats. 4-3=1.
4. What happens to this pattern in enharmonic equivalent keys past 4 #/b? Key of Db/C#, Gb/F#, Cb/B.

I'm very stuck and would appreciate at least a tip. If it is easier to explain just by giving its name rather than the concept, just let me know it as a hint and I can Google search it.

I wasn't able to find anything via search, and don't know enough about the Circle to be able to ask using accepted terminology. As context, I'm a guitar player working out the theory/geometry behind the keys, using the circle of 5ths.

Please see my Circle of 5ths "number line" below. There are three rows. From top to bottom row, each row denotes number of sharps/flats, key name, and note added, respectively. In the middle column, we have 0, key of C, and no added note. To its right, we have 1 sharp, key of G, adding F#. The other way, we have key of F adding Bb, and so on and so forth in both directions, adding one flat or sharp at a time.

I noticed that each flatted note added becomes the name of the next flatted key. I'd known about this for a while, but I was surprised to discover the same pattern continues past C, and into the "sharp keys"! It is clear that 'BEADGCF' is seen in the order of key names, the order of flats added (Bb/Eb/Ab/.../Fb), and the reverse order of the sharps added (F#/C#/G#/.../B#).

My question is, what do repeated interval jumps of perfect fifths have to do with the business of naming keys and the order in which notes are sharped/flatted?