Going back to the 35mm equivalent discussion, consider this:
On APS-C (like the 7D) compared to full frame (like the 5D Mark III)
The sensor is 1.6 x 1.6 times smaller.
35mm equivalent aperture - Multiply F-Number by (1.6 ) . (an f stop is a base 2 log, so even though we have 1.6x1.6 times as much light we take the square root, which is 1.6 to multiply the F number by. (example 2.8 x 1.6 = 4.48, 4.0 x 1.6 = 6.4, 1.8 x 1.6 = 2.88))
You'll have to clarify what you mean by "35mm equivalent aperture." The fact is, f/1.8 is f/1.8 whether mounted on FF or crop.
35mm equivalent focal length - Multiply by 1.6
35mm equivalent ISO or light sensitivity - Multiply by (1.6 x 1.6) (bet you haven't heard of that, but if you do the math the an APS-C sensor amplifies the signal 1.6x1.6 times more at a given PIXEL than the a full frame camera, so even if both say ISO 800, ISO 800 on the an APS-C it is multiplying the light from each individual pixel the same as ISO 2000 full frame, assuming they had identical resolution, if resolution differs)
You're right, I haven't heard of that and the math eludes me. Please could you show it to us.
Edit: I reread your statement and saw that your calculation is "assuming they had identical resolution." So, in other words, the signal amplification of a sensor is related to pixel size, not sensor size. On that principle, we are agreed.
And on that same principle, the relationship of a lens's aperture to sensor size that has been made throughout this debate is spurious. It boils down to the same thing: small pixels are noisier than big pixels. If that is the only contention that this thread has been about, then we're all "in violent agreement."