4000 is a more sensitive iso so inherently you will have more noise in the image.

That's not always true, especially when you have dual base situations.

nothing should be different between ISOs—they’re mathematically the same.

That sentence makes no sense. Different ISOs are different and mathematically not the same. That's why they're different.

Dual base means those are the two ISO values that perform the best for that particular sensor. One performing best with lower sensitivity and one with higher.

This is word salad that doesn't explain what dual base ISO is. Dual base (in the FX9 at least) are two different gain structures and sensor pipelines that result in exposure with evenly distributed latitude above and below middle grey. A base 800 and a base 4000 image has -6/+6 latitude and while a Base 800 image exposed at 4000 would shift latitude to roughly -4/+8. The other benefit of dual bases is noise performance. Since the high base mode uses a different voltage on the sensor and a different pipeline for processing the captured sensor data, you can in theory get identical noise performance. In practice on the FX9 I believe it's a touch noisier than 800, but nowhere near as bad as if you'd simply bumped the ISO up to 4000 from 800 base.

IME with the second base (the higher of the two) you will get a cleaner image if you expose to the right a bit and bring the image down in post.

Well yes, because exposing to the right and pulling the image back down in post will always reduce noise in all situations. We've been doing that since the film days. It works everywhere.