You're looking for logical results in a game where people summon genies to make wishes, talk to dragons, and cast magic spells? You fail at looking for logical results. Try the planet earth next time.
Stop. You officially fail logic forever. Logical =/= true, nor does it mean realistic.
The dm wouldn't be screwing you over if he rule 0'd your ring gate breaking from the punishment you're putting on it, he'd be getting the heroic fantasy game back on track, which you're trying to derail by applying external rulesets that the game shouldn't be made to accommodate. Seriously, did you try to talk your way out of losing a game of chess because your opponent's pawn just shouldn't be able to kill your rook? Cause seriously, I mean ones a serf and ones a castle, it makes no sense! Just relax and play the game everyone else is playing.
D&D operates under real-world physics unless a rule (spell, skill, feat, etc) functions otherwise. Your chess analogy is irrelevant, because the rules of chess never claim to operate under real-world physics.
As a thought experiment, I'd be much more inclined to consider what you were asking if you actually found some combination that broke the game or required resolution. Either you didn't finish reading the wall of force spell, or you're ignoring the part where it says the spell fails if the wall's surface is broken by any object. You just found a novel way to end the spell, full stop.
A ring gate is a permanent portal through which line of effect can be drawn. Or are you claiming that you can't put a wall of force through a doorway? Because that's all a ring gate is. The gate itself is not an object any more than a gate spell creates an object. So long as the ring is of sufficient dimensions to encircle the wall in some fashion, there is no interference.
Yey another tard that wants to break the rules without reading them, or by just ignoring key points, for the sake of trolling/argument.
"No my magical item isn't an object!"
You can argue that 2 sections of wall of force separated by 100 yards by an effect from an object aren't broken by that object (even while they're being joined by that object) but I don't think anyone will care what you come up with. Have fun.
Also, you fail at realizing a word can have more than one definition. Seeking a solution to contrived problems in an obsolete collaborative storytelling system is not logical. The problems will obviously remain unsolved, except in a real game, where the dm will solve them for you. Sure, its a logical conundrum, but that doesn't mean its logical or productive to try to solve it on a practical optimization board.
I guess I'll check for your answer in a couple months, mr. necromancer, just for giggles. Cya then