Thank you Polymath257,
For that SINGLE photon that hits the mirror, we all know how the optic law works, but the task was to show how QED all-path can find the receiving detector without getting any lead from the optic law. Because, QED all-path makes that claim that it finds the path of the optic law by its own method!
Once again, if you have 100 detectors, each has a probability of detecting that photon. Each is ost likely to detect that photon from the direction given by the optic law for that detector. BUT it is not guaranteed! It is quite possible to detect a photon from a direction *other* than that given by the optic law. Because paths that are far from the optic path are canceled out when the addition of amplitudes is done, only paths close to the optical path for that detector will have significant probabilities.
In the case of 100 detectors, you can find the probability of detection at *any* of those detectors using the all-paths formulation with the paths starting from the source and going to the specific detector you want the probability for. In each case, those paths that are NOT the one given by the optic law will be canceled out by other paths.
Why do you think the all-paths formulation gets prior information about the optic law? it doesn't. It simply computes the probabilities from *all* the paths and is able through adding up those probabilities, to show that the optic law path is selected because it doesn't get canceled out.
And, again, if the 'mirror' is actually a diffraction grating, which is designed to block some of those canceling paths, then it isn't just possible to find photons that don't follow the optic law: it is very likely.