If you link a logic block to a weapon computer, the weapon computer loses its link with the ship core.
Yet now, if you look at your power consumption, you'll see that as soon as you link a logic block to a weapon computer, power consumption on the ship drops. The "at rest" charge is completely gone, which makes me conclude that it simply not part anymore of the new power system.
There seems to be 2 cases:
# Linking a logic block to a computer that's already a **support** of something else, links it properly to the logic block and it gets linked to the ship core too. Power consumption is fine and logic firing works 24/7
# Linking a logic block to a standalone computer, the computer loses its ship core link and power consumption of that weapon is gone. You can only fire the weapon once, assuming it was already fully reloaded.