Hi, non-mathematician here, looking for any proof regarding gimbal lock being unavoidable for 3-value rotation representations

Question

So while there are many CG-oriented guides explaining gimbal-lock, they always jump into quaternions without first stating the necessity for doing so in that it's a dead end for deriving rotations using three separate angular values with any specific order. Has it already been shown mathematically (and programically) that there's no way to make unavoidable singularities, flipping, and/or gimbal lock for 3d rotation constructed from any arbitrarily selected triplet of numbers?