That structure is a great starting point. All 4 walls decoupled, preferably also disconnected from the original joists. Ceiling drywall decoupled. All mass is damped. Absorption in the cavity. All great. Install a mat and float a plywood floor on it.
Hopefully you can avoid having the theater share a wall with the stairwell. Shield any support beams and columns. Shield ductwork in the ceiling.
Now you have this great bunker, and now need to cut a bunch of holes in it. Door(s), ventilation ducts, outlets, recessed lighting, access panels all will detract from your soundproof bunker. That's where the details count.
The Soundproofing Company