The 2x factor reason is pretty intuitive if you understand that video compression is still fundamentally representing superpositioned signals from 2D Fourier Analysis and that multiplying by 2x the number of pixels in each direction is no different for perception than if we doubled the DPI. Double the DPI yields up to 2x the possible frequencies needing to be represented up to the Nyquist frequency that would cover each possible interpolated pixel. This is part of why noise in film or audio makes representation much tougher - noise is typically higher frequency and rather random (although distribution depends upon brown, white, pink, etc. noise).
4x number of points in real space -> 4x points in Fourier space, the Fourier space is still 2D. I don't get your reasoning.
OTH if you increase the DPI you bring in higher frequency components that are not that important for perception, so you can compress them more heavily.
I was confused with a different concept, disregard that part. FFT is by definition reversible for a discrete signal like a quantized image so each pixel must be reversible, so it has to be 4x total space used with no further operation, correct.
Quantization and filtering are the more important parts of the encoder than the FFT / DCTs since the transform is 1:1 reversible. Compression isn't just the mathematical accuracy of a signal when it comes to lossy algorithms as you know. A 720p video upscaled to 1440p should theoretically be exactly the same size for the sake of effective quality but encoders don't care about just the math and apply perceptual filters because simply doubling pixels looks really bad perceptually it turns out.
