Yes, thank you very much, your explanation helped a ton. I was trying to write up the post while watching a movie, which is probably why I got the equations backwards.
I think the main part that confused me was the fact that we are summing (or integrating) over all of the differential outgoing radiance values (each of which have their infinitesimal incoming radiance that altogether compose the incoming irradiance). Do I have that right?
I noticed that in the integral we are solving for the total outgoing radiance along a given direction. It is a function that requires the BRDF to solve. It seems like somewhat of a circular definition that the BRDF includes the (diffferential) outgoing radiance as a variable, which is what we really want! I'm assuming that in practical implementations, the BRDF is a known equation of some kind that specifies the reflection ratios? I'm having a bit of a hard time mapping the theoretical into the practical implementation. I know Blinn Phong shading is a horrible example (because it lacks conservation of energy, and it's not physically accurate in any way), but is the specular component where we take the dot product between the half vector and normal technically the "BRDF" portion of the lighting equation? I hope that makes sense.
Thanks!
Zach.