I haven't had time to absorb what you've written yet, but what I'm after is the same thing the OP asked about. Why does f/8 give the same brightness for both a 100mm lens and a 50mm lens when they have quite different aperture diameters? The explanation I offered, and which I've seen in several books and articles, explains this by applying the inverse square law to the focal length; that is, the distance light travels from the lens to the sensor or film (which you described in that other thread you gave a link to as being from H to F'). I've been told in the past that this explanation was in error, but I still don't see why or how. Maybe I will after I read this post of yours a few more times.
It has nothing to do with the inverse square law. That's a coincidence.
It has to do with the angle of view.
So the angle of view is what I will get at with this formula you gave?
f# is f# * (1 + m/p)
At first blush angle of view wouldn't seem to matter much; after all, in the example of the 2" lens and the 6" lens on different film sizes, the angle of view was the same and the brightness was equal, and brightness is still equal when using a 100 mm lens at f/8 and a 50 mm lens at f/8 on the same camera despite having different angles of view. Well, I'll keep working at it, but probably not until tomorrow--right now, my head hurts.
No, that's "apples and oranges".
Angle of view doesn't relate to effective f#
The question wasn't about effective f-stop, but I think that has to do with the idea that f-stops assume infinity focus, and that illumination is decreased for close-up photography because the lens has to be extended further from the film plane to maintain focus.
"The lens, projecting the image, acts as the source of the image-forming light. And with increasing distance to the image plane, the area over which its 'output' is spread increases too (and, by the way, the size of the image itself does as well: magnification increases). Consequently there is less light per unit of area. The projected image is less bright.
The relation between image distance and illumination is a quite simple one too: illumination decreases with the square of the distance. Double the lens to film distance, and the illumination in the projected image decreases by a factor 4 (2²)."
http://www.hasselbladhistorical.eu/HT/HTComp.aspx
Angle of view has to do with how much light is gathered by the entrance pupil.
All light gathered by the entrance pupil is delivered to the exit pupil.
The exit pupil then projects that toward the image plane.
(Which is where effective f# comes in.)
Sorry, but I'm left not knowing what parts of what you've written relate to why a 100mm lens at f/8 gives the same brightness as a 50mm lens at f/8.
I did re-read what you wrote above to Leonard:
"The amount of light collected is the product of the area of the entrance pupil and the solid angle of the light collected. The solid angle is in turn a function of the angle of view.
So the area and angle are what cancel out.
eg. 100mm has a larger area and an offsetting smaller angle of view."
Doing a search for solid angle, I found:
Light collection
The light collection efficiency is the solid angle that an optic makes with an object. The f-number describes this angle:
f-number: f/# = l/d
where l is distance and d is diameter of lens.
The solid angle that a lens collects is approximately:
OMEGA = pi d² / 4 l²
The fraction of light that an optic collects is this solid angle divided by the total 4 pi steradians:
fraction collected = OMEGA /4pi = pi d² / 16 pi l² = 1 / 16 (f/#)².
http://elchem.kaist.ac.kr/vt/chem-ed/optics/lenses.htm
If I'm understanding the formula given above for the solid angle, and it's entirely possible I'm not, then when set to the same f-stop, all lenses regardless of focal length will have the same solid angle.