Cumulatively building a mask

I know there's a clever way to do this, but I'm stumped at the moment.

I want to use the "future data stream" graphic from the content/gauges menu to build a transition. The graphic is a nice, overlapping stream of white rectangular boxes. I'd love to have this graphic running, but on the left half of the screen, let the graphic start remaining on screen so that after 30-40 frames, the left half of the frame is solid white, and the graphic is still running on the right side of the frame with a soft feathered transition between them.

Once that's built, then I can use the whole thing as a matte to key in picture in another comp.

My first thought was to put a solid white rectangle with a feathered edge underneath the 'future data stream' graphic and fade it in slowly, but the effect is not as sharp. I want it to feel like the boxes are revealing the image on the left side of the screen.

Second thought was to build up layers of the 'future data stream' at offsets, until they cumulatively fill the left half of the frame. That gets close, but it's supremely inelegant, and still feels funky.

Third thought is that the secret is in the emitter. The 'future data stream' is just a white rectangle manipulated by the emitter. So I could create a second layer of the 'future data stream' and futz with the emitter settings until I get a solid white, then use a mask to limit it to half the screen. This seems like the best solution, but my emitter-manipulating skills so far don't seem up to the task.

Advice, anyone?