Any fire that consumes fuel behaves exponentially
that's likely to be roughly true when the fire is starting - there is abundant new fuel and oxygen; heat is the limiting factor for the rate at which new fuel is ignited, so it is plausible that at least under some circumstances rate of growth could be proportional to current heat output, leading to exponential growth. Nevertheless, anyone who does not have a brain resembling that of a hydrophobic monkey must see that this is not true in the long run - people heat their houses with fires which spend the majority of their time burning at roughly constant intensity. They do not grow until someone takes a fire-extinguisher to them. Some process (I would guess the difficulty of rapidly getting enough oxygen to the fire) limits how quickly it burns.
Since I'm more of a harmless forum troll than an actual scientist, I don't know what process this is in the sun. Based on what you've posted here, you don't either. (hint - this involves neither vibrating ether, nor vibration frequencies gravitating discrete quanta together into energy bonds). A guess would be that if it burns too fast, that increases the gas pressure, and thus lowers the density in the core of the sun to the point that the reaction slows down.
Let's do a simple calculation (these are allowed here, or do we primarily wax poetic using ill-defined terminology and colorful metaphors?). I don't understand your mechanism for powering the sun via electromagnetic interactions with the earth's dipole field, but surely you must agree that the energy must ultimately come from the earth's orbit or rotation? The earth is traveling roughly 30 km/x. It weighs about 6*10^24 kg. Therefore its kinetic energy (mv^2/2) is 3*10^33 Joules. The rotational energy is a small add-on to this. This is enough to power the sun for about 3 months. If the earth is providing any significant fraction of its orbital energy to the sun, then its orbit would decay on the timescale of human lives. The same is true for any other planet.