Basically: upon reaching the boss area wipe out all minor enemies, then immediately destroy the middle core's barricades and get up close to the core itself. This'll cause the left and right cores to fire their spheres straight across at you, trapping them at the top of the screen (the middle core won't fire any at all). Keep it up, and eventually most of the spheres that'd normally be bouncing around the screen will be harmlessly, permanently stuck up top. With four options, laser and missiles all firing during this, you can then make short work of all three cores. Even if a couple of spheres get loose in the baiting phase, it's a cinch to avoid them and the cores won't last long. I like to bait four or five shots apiece, then make a break and speed-kill all three cores.
I've tested it dozens of times in MAME and the Deluxe Pack (PS1) - worked every time (Loop 1, default difficulty). The margin for error while baiting the spheres is pretty generous - you have to really try to collide with the middle core, and even if the spheres aren't baited dead-straight across, there's an invisible line they'll very likely bounce off of.
There's also the lower-right safespot timeout method, which Spy Yamato goes straight for in his Loop 10 no-miss (available at super-play.co.uk). Probably worth practicing that in case later loops have any nasty surprises. I can do it consistently, but I don't like it much since it's pretty easy to bump the wall and die. If that happens I just rush the leftmost core, destroy it and hide in its recess until timeout (nothing can hit you there, either, unless you're slow to reach the top of the screen - then a sphere might get you from the side). Here's a screencap of a location that works for me (the background is helpful).
Holy shit. Congratulations! I'm getting closer and closer to our first encounter. Once my converter for Saturn sticks to PC gets here, I should be set to get to the second loop. Any words of advice for aspiring EVAC DOOM annihilators?
