Light falloff and the f-stop

On the way into work this morning I was thinking about dark background flash photography. Essentially I was wondering how you would work out how far away your background would have to be from your flash and subject in order for the background to be recorded as completely black.
I’m sure you’re all well aware of the rule that light falloff follows an inverse square law. It also occured to me on this drive that f-stops on a lens also follow a square law. This led me to wonder the following:
If you know the dynamic range of your sensor in terms of stops – for example 9 – and you know the distance from you flash to your subject – 1m – is the black level distance really very easy to calculate? 4.5m beyond subject.
Let me walk you through what I’ve just done:
I’m assuming (incorrectly but this is just a quick example) that your correctly exposed subject falls dead in the middle of your dynamic range, i.e. 4.5 stops from peak white and peak black. I’m also assuming that if you double the distance between your subject and your flash (to 2m) that you’d have to open your aperture up one stop to compensate. This would give evidence to my idea that the square law fall of light is directly proportional to the square law nature of ‘stops’ in camera lingo. My final assumption is that if you took a meter reading and the underexposed by 4.5 stops (in this case with a dynamic range of 9) that you would get a pure black output.
All of this is just conjecture at the moment, but I have an idea of how to test it… more info when I get the time to experiment