My partner and i just added a third. Our third is afraid of splitting my partner and I up. Is there anyway we can prove to our third that we both want it to be the three of us for the long haul?
Short answer, no: there is very little you can do to change how someone else thinks or feels, and there is nothing you can do in the present to provide 'proof' of something in the future.
However, you can try and talk things out and help everyone understand where everyone else is coming from. Ask your new partner: where are these feelings coming from? Is there something we're saying or doing that's sparking this anxiety? What could we do to help you feel more secure?
You can be reassuring, and let them know that you're both happy with the way things are now, and that you will let them know if concerns come up or something starts to change. Stick to your word on that - be open, honest, and vulnerable. People often find it easier to trust you after you've demonstrated that you're willing to say awkward, uncomfortable truths and share difficult feelings, even if it's a smaller-stakes issue.
Let them know that this isn't their problem to worry about, that you two are committed to making the triad work, and that if something comes up in the future, you'll handle it then. Talk about what you like about this new triad and your best-case-scenarios for the future.
Sometimes, things like this fade with time. Newness and change are scary, and our brains sometimes funnel that nebulous anxiety into specific fears, whether or not they're grounded. Stay in the present, knowing that the three of you can cross future bridges when you come to them, trusting your future selves to handle what comes up, and doing your best not to 'borrow trouble' if things are working out right now.