I see no line, neither as the image just appears in the post, nor when clicking it; it becomes a smaller image up and to the right. When I close the latter, and it fades downwards to the left, there seems to be a slanting gap (downwards to the right) underway, presumably just a side effect of the fading.
I believe it may be screen dependent. Does it show in a PDF and/or in print? Sometimes, non printing artifacts are reported, annoying but harmless.
I can see your line; you're not imagining it. I can replicate it by cutting a black object with the knife tool and moving one resulting piece .0001 inch. It's certainly smaller than the resolution of my low budget printer, so whether it's a problem or not will depend on where it goes from here. Someone more knowledgable about printer resolution will have to tell you whether that size gap will show up anywhere.
Dear Jacob and Peter,
Thank you both very much for your help.
Please see these photos showing my problem more clearly.
I am using Illustrator to draw the regions, Photoshop to consolidate the layers, and Apple Motion 4 to build my animation.
 : Imperial Germany in 1914.
 : Germany after Versailles.
In Illustrator CS4, the images  and  have the colored layers in "Multiply" transparency mode. It's how I want the layers to look when I import them into Apple Motion 4.
My idea was to color all the layers black in "Normal" mode and then apply "colorize" filters in Motion 4 and set the transparency to "multiply" in Motion.
Now with Imperial Germany (all layers colored in black with transparency set to "Normal"), when I bring the image into Photoshop, I suddenly get very fine white lines between the layers and these lines show up in Motion which is the big problem. Like I said, when you click on this last photo, and you see the image full size, you can clearly see the problematic lines which appear exactly where the layers touch each other. In the photo below, the territories ceded to Poland (West Prussia) and Lithuania (Memel) are not yet red. They are still black but you can see the lines.
I don't understand because in Illustrator there were no gaps whatsoever between the layers. In Illustrator CS4, I simply cut the "territorial losses" from imperial Germany using the "Trim" function in the Pathfinder Menu. I created the red and yellow layers from the original single "imperial Germany" layer. In Illustrator I can colorize all the layers black and still never see any gaps. The fine white lines, appear in Photoshop and in Motion.
(NOTE: The documentary is being in edited in Final Cut Studio which is why I use Motion for animating.)
Sorry to ramble on. Thank you again for any further help you can give me.
Could it be because the background canvas of Illustrator is always white?
There is no background canvas in Illy. But the whiteness of the empty screen shows (through) wherever there is emptiness before its emptiness.
How do you carve out the territories given to neighboring states?
I realize that it may not be harmless even if it shows bara på några bildskärmar.
One option might be to cheat by extending the underlying unchanging path to cover the gap from below.