You MIGHT get there with the .223, and maybe some 30s or 35s, but I doubt you'd find the accuracy needed with those shorter, light bullets.
If you desperately need 4000fps, you're in need of a bigger case, like the .22/250. Then you can use the useful 40 grain weight, and not crowd high pressures, and hit your goal of 4000fps. The .22/250 can do 4000 easily, and can crowd 4300 with a caseful of Varget. I just don't think you can do 4000 safely within SAAMI specs, with a standard .223.
