Great, now I have prove that you have no ability to focus on the problem and provide an experimental setup...
No, but your statement is evidence of you distorting what I said, apparently because you don't want to admit that I gave you what you asked for.
Because your setup will show voltage induced by varying magnetic field as the wire is moving along the gap...
And that is Faraday's induction and not Lorentz Motional EMF...
False. The magnet is stationary, so the magnetic field is not changing. As described in more detail in dustinthewind's post, special relativity requires that whether you move the magnet or move the wire, you get the same current, since there is no special reference frame. As a result, it is not a coincidence that the prediction for the current is the same in either reference frame, they are different ways to represent the same thing, so your claim that only one of them works is self-contradictory.
If you want more room to move the wire in a uniform field, instead of a square end for the bar magnets, you can have it be rectangular with the long side along the direction you are moving the wire. The magnetic field strength will not be constant at every point on the moving wire, but that is irrelevant, since the field experienced by any given piece of the wire would be unchanging. The gradient is necessary, since you want to get the leads to the voltmeter away from the magnetic field, so their motion is not significant.
As an alternative, you can actually look at that chapter that you linked to, and see on page 10-8 where they provide a diagram where a metal rod is sliding along 2 metal bars. Use a voltmeter as the resistor, and then all of your leads can be fixed in place, so the whole setup can be within a uniform field if you have a big enough coil.
For 100 years now there is no experimental setup proving existence of Motional EMF of section of a conductor moving with constant linear velocity in uniform constant intensity magnetic field...
Untrue, you have been given multiple examples, from motion of electrons in free space, the experiment I just described and the video I linked showing a homopolar generator.
So, stop trolling nonsense!
You are relatively new here, so you may not be aware that name-calling is not acceptable behavior here. As for nonsense: I have given evidence and supported my points, while you have yet to provide a single example of an experiment where the Lorentz force does not work. In all of your examples so far, what you have described is exactly what is predicted by Maxwell's equations and the Lorentz force, you have just made incorrect assertions about what the Lorentz force predicts.
The Faraday's homopolar generator only requires physical brushing and the rotation's relativity and or direction is irrelevant...
Go look at the video I previously provided again. As I have pointed out before, the sign of the voltage changes depending on the direction the disk is spinning. If the voltage was due to friction (generating a static charge) between the brush and the plate, the sign change with direction of motion would not happen.
If magnet is in motion then its field i also in motion, but because rotation axis is aligned with axis of symmetry
all means of detection fail because at any radial point during rotation magnetic field has constant magnitude and orientation thus induction is zero thus zero EMF but Lorentz claimed otherwise...
A spinning magnet produces the exact same field as a stationary magnet when the rotation is along the magnet's axis of symmetry. The Lorentz force equation* does not predict anything different to happen with the spinning magnet than with the stationary magnet, since the fields are the same. The current due to motion of a wire relative to the magnet is very real, though you have to setup the measurement apparatus correctly. You did not even deny that the setup I provided would work, you just claimed that the field was not constant despite the fact that the magnet is not moving.
Your arguments against the Lorentz force so far have all been strawmen, where the only thing wrong is that the Lorentz equation does not predict what you claim it does. (That and the fact that you have not provided an alternative proposal that so much as has consistent units.)
*you refer to Lorentz as if you are referring to the person, rather than the equation. If you want to refer to the person, please provide a sourced quote where he is describing the situation you are talking about.
Yes, but those setups produce AC EMF of Faraday's induction:
https://www.youtube.com/watch?time_continue=36&v=-CXsXeizmZw
You need to produce DC EMF as predicted by Lorentz Motional EMF...
Ones again: no brushing, Lorentz Motional EMF must produce DC EMF without brushing...
It is AC in that video because the motion is back and forth. Again, I already provided a video where the output is DC, and the result is clearly not caused by the presence of the brush, because the sign changes with direction.
You have not provided a description of a setup that can avoid the need for a brush or similar, since something must maintain contact between the moving and non-moving pieces. The only thing you provided was a patent where the applicant failed to account for the effect of the field on 2 of the 4 sides of their coils. (Again, you base your arguments on incorrect claims about what is actually predicted.)