I read your response 4 times and I THINK? I understand your explanation. Going out on a limb, what I think your saying is: the Radius control does not work directly as a control for the output of the sharpening edge adjustment. But, more like a sampling control, that controls the edge sampling.
Yes, a sampling control in the sense that it determines which surrounding pixels are included in the modification of the central pixel that's being processed at that moment, and how much weight is attached to those pixels (less with distance).
That in turn is used to calculate the applied (output) edge adjustment.
Not only edge, could also be any other detail or noise or smooth area, although the latter will then undergo little change. Actually, the underlying algorithms are not very intelligent, they don't see edges or other structures. They just see a pixel surrounded by other pixels within a circle radius, and calculate a new value for the central pixel.
I am going to postulate that the width of the applied sharpening, is controlled most directly by the Amount slider?
No, 'Amount' is more of an amplifier to boost the amount of contrast enhancement. If the calculations with the surrounding pixels result in a brightness increase of e.g. 2 of the central pixel, then an Amount of 200 will result in an increase of 4 instead of 2, and an Amount of 50 will result in an increase of 1 instead of 2. The actual boost amount may be a bit different but you can view the Amount as a percentage.
The Detail slider is closest to an "edge finder" control?
Not as you think it does. It is a control that mixes between two sharpening algorithms used. At its minimum position it only uses something like a traditional USM sharpening algorithm, and at its maximum position it only uses a sort of 'deconvolution' sharpening algorithm. The in between positions of the Detail control produces a weighted mix between the two methods.
At its minimum position the sharpening algorithm basically boosts edge and point contrast by subtracting a blurred version of the original image and adds that result back on top of the original (while multiplying the 'amount' it will add back). It boosts local contrast and that tricks the human eye into a suggestion of sharpness.
At its max position the deconvolution algorithm attempts a real restoration of sharpness, by subtracting some of the blurred content from surrounding pixels, and adding it back to the central pixel, for each pixel in turn. It really restores original detail that was spread/blurred around. Unfortunately it cannot distinguish noise from detail, so it might increase noise as much as detail if one's unlucky.
Since its clear you have a really deep understanding of the tool, can you explain (My last request, I promise) why the radius is in values of 1/10s of a pixel? Is going below 1.0 making some kind of pivotal change in the sharpening as compared to values over 1.0?
No problem, just ask if needed. Blur is not something that abruptly stops at a pixel boundary, and it is also not constant within the radius. There is usually an underlying model that describes how much influence/weight the surrounding pixels have on the central pixel that is being processed. that model usually resembles a Gaussian curve. Most of the contribution comes from the pixels that are close by (near the 'peak' of the bell shaped curve), and progressively less contribution comes from more distant pixels.
Therefore, a radius of 1.0 doesn't mean that only one pixel will have influence, because that wouldn't change anything. It just means that say, 75% of the result is dominated by the influence of the 9 central pixels (inside the radius of 1), and 25% by all other
pixels outside the radius (although very distant pixels will have negligible weight/effect or even none). When the radius becomes smaller than one, then that contribution of surrounding pixels is further reduced, but is not yet zero (after rounding). When the radius gets smaller than approx. 0.3 then hardly any noticeable effect can be expected from surrounding pixels anymore.