I got to thinking about the problem and that jarred my memory. I do recall (vaguely) a discussion about the bright bands you observed. I believe it involved an LPL enlarger. The conclusion—which seemed to make sense—was that the negative carrier is reflecting light from the 4 edges of the film aperture of the carrier into the bright bands that you see projected onto the baseboard.
But why haven’t I ever observed this on my enlargers? The film apertures on my carriers are similar.
I looked at the images you sent.
What they show contradicts the above idea. When you projected a 6 x 4.5cm image you got the same bright bands arranged in the SAME SIZE SQUARE as was the case with the 6 x 6cm projection. Clearly, the edges of the film aperture in the negative carrier are not the cause because they are of obviously different aspect ratio (3:2 vs 1:1).
I notice that the LPL uses a thick black piece with a handle below the film holding plates (photos 9 and 10). I think that this plays a role in the problem on this LPL enlarger because it provides greater that usual distance from the negative to the opening in the negative stage. This makes it easier for the light to illuminate the edges.
Photos 2 and 4 show the light colored aluminum opening in the negative stage. Notice that the opening is SQUARE and that the vertical edges are relatively reflective. The size relation between the film aperture in the carriers and the opening in the film stage is about the same as the relation as between the projected image though the carriers and the bright bands. (Lightbulb comes on).
Here is what I think is happening. The light that passes through the back-illuminated negative radiates in every direction from the bottom surface. Some of it hits the 4 vertical sides of the square opening in the negative stage.
The edges are “lit up” and are slightly within the image pickup circle of the lens at the magnification that you’re using. Recall that the bands disappear at the corners. That’s because the image pickup circle is too small to include the corners.
The lens sees these bright bands and does what all good lenses do. It projects them faithfully onto the baseboard. I think this is the most likely explanation of your observation.
That makes sense. The 50mm lens has a smaller diameter pickup circle, so it doesn’t see the bright reflections from the edges of the opening in the negative stage.But I did switch it to a Nikon 50mm lens. It causes vignetting on the 6x4.5 and 6x6 negative but I didn't see the lines of light leak.
You can cover the edges of the opening in the negative stage with black tape to test the idea. If you cover them with black tape and the bright bands disappear, then you have positively found the problem and can easily devise a permanent fix.
The important things to realize are: there is nothing wrong with your enlarger or lens.
The projected bands, while annoying, are of modest intensity and are well outside of the image area. They won’t affect the enlargement quality and I don’ think they will affect the contrast of the main image.