The Soviets researched their experience form 43-45 and quantified a lot. Their conclusions pointed rather at a 6:1 ratio, and I don't remember older literature with focus on force ratios than 50's stuff right now.

The necessary ratio (usually a ceteris paribus thing) depends a lot depending on the mission and level anyway. A 1:1 ratio can suffice on the offence as demonstrated in 1940, whereas a 3:1 ratio can be insufficient.


I personally think that the culminating point distance is a more useful metric - and a badly neglected one.