United Empire generates Influence through Industry but what do you do when you've run out of system improvements to build, and you already have a ton of ships? Does the infinite Dust and Science generation still generate Influence via Industry? From my testing it doesn't seem to but I'm hoping I'm wrong and someone can clarify, or point to a relevant thread.

This is kind of a weird question, but it's a "sort of" yes. Infinite Improvements have a secret property called "Consumes Industry", and you can't tell without testing which infinite improvements have this property and which don't. What it does is make it so that if an infinite improvement with that property is currently at the top of the queue at the end of the turn, all excess industry will be consumed (which counts for the industry -> influence). An easy way to test this is to put the improvement in your queue for a turn, and then afterwards drag something with multiple turns to the top (2 turns, in my opinion, is the easiest to test). If the infinite improvement doesn't consume your industry, then you'll instantly complete 2 turns worth of industry towards that improvement (because of the industry you generate that turn, and also the stored industry from the previous turn, since you only save up to one turn's worth of industry). If you only get one turn worth of industry, then you know it consumed last turn's industry. Another thing that doesn't count for industry -> influence conversion though are improvements that list a number of turns as their cost rather than an amount of industry.

Thank you, I'll definitely have to test this with the infinite dust and science improvements.