Why under red light the additive synthesis of primary colors gives white where one primary color is absent? Under White Light

Under Red Light

For instance, red & green under red light gives white.
 A: Under red light, your picture never shows white. It only shows bright red, which you are interpreting as meaning white. 
If you take a piece of paper and put a small hole in it, then hold the hole over any part of the screen image, you will see that it is red, not white.
When doing this in real life (which I suspect is what you're talking about), three things are happening. First, your illumination is probably not monochromatic, so a small amount of other color information is being reflected by your chart. Second, the color receptors in our eye are not actually all that selective, 
https://en.wikipedia.org/wiki/Color_vision#/media/File:Cone-fundamentals-with-srgb-spectrum.svg
so even pure red is also perceived as slightly green (at the ocular sensor level).
Finally, the brain does a quite astonishing amount of processing in order to determine what a color "really" is despite wide variations in illumination color. This is called "color constancy", and is described by the Land Color Theory.
https://en.wikipedia.org/wiki/Color_constancy
As a matter of fact, there is a Lord Peter Wimsey story (Dorothy L. Sayers) which claims that the process can be subverted. At a masquerade, a guest dressed as the Red Queen from Alice Through the Looking Glass is briefly illuminated by a spotlight and becomes a suspect, but is later cleared of suspicion for other reasons. Lord Peter realizes that the spotlight had been changed to a red beam, and the guest was actually a different person, who was dressed as the White Queen, but appeared red in the red beam.
So basically, your brain works very hard to create the illusion that red light reflected from a white surface means that the surface is actually white, and it has just enough clues about other colors to do the job. Seeing is not believing. I suspect that if you were to use really high-quality monochromatic red light, such as a collimated laser beam, and you made sure that there was no other source of light available, your "white" would go away.
A: To add to WhatRoughBeast's answer, this is because you are not really viewing the image under pure red light. The "red light" you used to create the image has small amounts of green and blue wavelengths, causing the white areas to appear more "white" than the red areas, even though they are both just slightly different shades of red. To quote WhatRoughBeast, I suspect that if you were to use really high-quality monochromatic red light, such as a collimated laser beam, and you made sure that there was no other source of light available, your "white" would go away.
To simulate this, I took the original image and removed the green and blue channels. (Just a note, I made some small modifications to the original image, such as white-balancing and small saturation changes to make this affect more apparent.) The only difference between the first image and the second image is that the Blue and Green Channels were removed - no additional retouching. As you can see, the red and white areas become identical shades of red.


(I noticed these images have slightly different appearances on different monitors. On one of my monitors, you can still see a slight difference between the "red" and the "white" portions, but on my other monitor, they are identical. In an ideal world, they would be identical)
