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This answer mentions several effects that tend to move geostationary satellites out of their "box" over time, but one really surprised me.

According to the plot below (from there, but uncredited) the semi-major axis of a satellite in GEO will climb by 21 kilometers in six months.

Is this really true? If so, could someone explain how this rapid orbit-raising is possible? I don't think there are tidal forces at work here, the spacecraft would not be massive enough to cause shape changes in the Earth big enough to do this.

If it is possible please include a sourced mathematical expression that reproduces approximately this rate of orbit-raising as well.

From here:

Graph 1 : Shows the drift of semi-major axis for a satellite placed at nominal R of 42164.2 km propagated for about 6 months shows an increase of about 21kms.

enter image description here

Source: [Book] Li, Hengnian. Geostationary satellites collocation. New York: Springer, 2014.

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The cause of this increase is irregularities in the gravity field of the Earth. The rate shown in your graph is entirely dependent on the location of the satellite. If there's a 'lump' in the Earth's gravity field ahead of the satellite, this will make gravity point somewhat in the direction of the velocity of the satellite, raising the orbit. The opposite is also possible, lowering the orbit. Since the satellite is in GEO, the position with respect to the 'lump' will remain roughly the same. Of course over time, this perturbation will move the satellite out of GEO and the dynamics will change considerably.

I don't have a mathematical derivation for you but I propagated two different orbits to demonstrate this effect for you. I used a gravity field of degree and order 15, and turned off all other perturbations (SRP, 3rd body perturbations, tides,...).

The two orbits differ only in their position over the Earth, which is shifted by 90 degrees along the orbit.

Orbit 1: SMA raised

Orbit 2: SMA lowered

I haven't checked which 'lump' in the gravity field it is exactly that causes this but if you check a map of the gravity field you can visually find multiple examples of this kind of situation

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