I wanted to make an air barrier around the studs to keep them in the indoor space heat but still wanted to put aircrete (aerated concrete insulation) between the studs. The attachment methods of the insulation to the studs was also a problem to eliminate thermal bridging and expensive hardware like long stainless steel screws (stainless steel conducts heat far less than other metals/alloys... but is rather expensive). One idea was to use paper or similar material as a stress-strength element to hold insulation 90 degrees relative to the wall. My latest idea is as follows: use the more ridged form of mineral wool in a 1 1/2 inch thickness to extend the 2 x 4 studs toward the inside to the 5/7-outside-2/7-inside point in the wall (in terms of R-value) where an air-barrier made of foil or reflective plastic on each side of a 1/2 inch air gap is used as both an air and vapor barrier and a radiant heat barrier. The thin material like paper to do this will extend through the stud area of the wall to also support the insulation on the outside of the studs. Because I want to do the outside later than the inside, the extra span of paper will be hidden between the aircrete and the existing outer sheathing (likely shiplap). The inside edge of this paper-lined piece of stiff mineral wool will have some material that can hold screws like wood or "fake wood" plastic composite material. After installing strength supports to prevent racking between the studs, the stud bays will be filled with aircrete. The remaining volume is filled with normal mineral wool (which has slightly higher R-value per thickness than the stiff mineral wool boards) glued to the stud extensions to hold them in place. The outside layer of reflective gap at the 5/7-2/7 point can be glued to these mineral wool batts. Horizontal furring strips can maintain the gap between the two reflective layers. New paper-wrapped stiff mineral wool "studs" can be attached vertically to these horizontal furring strips through the inside reflective layer. Finally on the inside of these with regular mineral wool between them, an inside surface can be put (vapor open but air-blocking) and a gap for drying and heating the back side of thermal-storage-PCM drywall boards (PCM is Phase Change Materials). T-shaped metal like is used for suspended ceilings can replace studs to attach two layers of this PCM drywall to the ceiling and floors of the room. When doing the outside, fire-rated ridged insulation will be used to extend the studs outward (possibly mineral wool but there are more expensive and more ridged options I might use) and like the inside, regular mineral wool glued on to those extensions. An outside surface is made with inset horizontal furring for a square-opening surface that the outside cladding can slide against. The outside cladding is then hung from the eves and attached to the ground to allow the house to move in an earthquake without damaging the cladding.