To explain the 4 in the PBR BRDF we need to determine how changes relative to for a specular microfacet.

The delta function enforces , so .

To make the problem simpler we can swap what's changing, that is, for some area on the unit sphere at , what is the corresponding area at , keeping as a constant vector?

We need to transforms the area onto the corresponding area following the definition of the halfway vector:

So our area at needs to be shifted from to , then rescaled onto the unit sphere to become .

Shifting to doesn't change the area, but the area no longer lies perpendicular to a sphere, so it needs to be projected onto the sphere with radius . 's normal is and the outer sphere's normal is at , so the projected area is .

We now need to rescale our area from the outer sphere onto the unit sphere. The proportion of to the outer sphere's surface area is equivalent to the proportion of to the unit sphere's surface area,

So substituting for and rearranging

We need to find , one method is to note that is parallel to so

Or equivalently, if is the angle between and , then is the angle between and so

Substituting ,