No / it depends what 'breathing' means.
First of all, the buildings' occupants need to breathe: Fresh air, either supplied through windows and doors, or by a fresh-air-ventilation system in connection with an energy recovery ventilator (ERV).
'Breathing' in the context of the building envelope, especially in connection with energy-efficient construction, is not just a waste of energy, but can actually cause problems, if breathing means air, and therefore moisture infiltration, through cracks and holes in the building envelope. Scientific testing has proven that the amount of moisture traveling through a one square-inch hole in a wall, equals 100 times the amount of moisture traveling by means of diffusion through a 4'x8' sheet of drywall. In older poorly insulated buildings with thin walls, this usually doesn't cause much of a problem, since there is enough drying potential in either direction, but in buildings with a higher degree of insulation, thicker walls and possibly vapor retardant components, this can lead to moisture being trapped, and therefore to rot, mold and decay.
That said, for a mixed-humid climate as present in Virginia and the mid-atlantic states, we recommend vapor-diffusion-open envelope assemblies, which are often described as 'breathable', meaning that the wall or roof can dry either to the outside in winter, or to the inside in summer.
So, what components function as an air-barrier, and where is it located in the Batesville house?
The concrete in slabs and basement walls is naturally air-tight. The continuity of this air-barrier to the air-barrier in the exterior walls is ensured through special tapes and sealants. In this house, the plywood sheathing functions as an air-barrier. The studs are glued and nailed to the sheathing boards, and all seams sealed with a sealant. We actually do have a belt-and-suspenders approach, because the 'black stuff' which we are mainly using as 'weather-resistive barrier' or 'bulk-water-membrane' also is an air-barrier. What is the 'black stuff'? It is a vapor-open, water-proof, weather-resistive-air-barrier-membrane. It is in place of house wrap, which usually functions as bulk-water-membranke, but isn't air-tight or water-proof. The product we used is actually new on the American market and the Batesville house is the first house where it is applied: It is called the 'Solitex Adhero' Membrane, essentially a polymer fabric, and made in Germany. The original product, the Solitex Mento or Solitex Mento Plus have been on the market on both sides of the Atlantic for a long time. The Adhero is a self-adhering membrane, using the adhesive from the tescon air-sealing tape line, which are available in different perm ratings, depending on the application. Originally conceived as a roofing membrane, we are using the Mento Plus membrane in the roof both as air-barrier and as secondary water-proofing layer between dense-pack cellulose insulation and the vent-channel below the decking. The entire Solitex line of WRBs, including ADHERO, is stocked and exclusively distributed here in North America by 475 High Performance Building Supply.