A transformation of spatial coordinates is a transformation of space-time coordinates, just a special case. A rotation is a special kind of Lorentz transform, even though it doesn't act on time coordinates.I still see that you are still hung up on your brilliant discovery that Christoffel symbols are nonzero for the case of polar transforms.
I see that you still can't tell spacetime transforms from space transforms.
It depends what you mean by 'the wave equation'. The wave equation typically used by anyone doing basic mechanics is the flat space form, ie $$\partial^{2}_{t}\phi = \Delta \phi$$. Would someone doing GR call this 'the wave equation'? No, they'd use the general form which Guest has given, the one where partials become covariant, ie $$\nabla_{a}\nabla^{a}\phi$$ where a goes from 0 to N, where N is the number of spatial directions. That is covariant and it reduces to the standard one when you have flat Cartesian systems.Congratulations! After much proding you did it. Now try using it in showing that the wave equation is covariant.
I can see why you're saying what you're saying but I seriously think you're not understanding the points Guest is trying to make. If someone knows the standard special relativity formulation of something then to upgrade to full GR typically you change all partial derivatives to covariant ones, $$\partial \to \nabla$$. This means that if you view the special relativity formulation as the general relativity one for a specific coordinate system in a specific space-time then it is covariant but only because you know you've got connection terms which just happen to be zero. If you don't know you should include connection terms then the special relativity formulation isn't covariant. It's a subtle different but an important one and I think you're labouring under the misconception that Guest doesn't know the difference.
Your comments about him and Eugene being made for one another (or whatever it was you specifically said) are just ad homs since at no point has Guest been saying Eugene's paranoid nonsense about physicist is right but instead he's been making a point about the general concept of coordinate transformations and the mathematical formalism behind them. Implicit definition of coordinate transformations doesn't make them invalid, as the consistency of a (supposed) coordinate transformation is entirely independent of how they are defined, it rests in the interdependency between old and new parameters throughout the manifold.
I'm sure Guest would have no problem rolling his sleeves up and going to town on the subject of tangent bundles, gauge connections, coordinate atlases etc but I wonder whether you'd be up to it. Every time Guest or I have pushed any kind of detail you've not replied in kind, you just reply with "You don't know what you're on about". Since clearly there's wires being crossed surely it would be to everyones benefit (ie those of us who claim to understand the details) if the discussion became more formal and we were all very precise with our definitions and notion and provided algebra where needed? Even Eugene, for all his delusions, at least realises attempts at mathematical formalism are needed to have any hope of getting his claims listened to. Out of the people in this thread who've pitched in I've seen the least mathematics from you[. Hopefully its just because I'm not entirely paying attention and you've not shown your best side so if we all kick up the details and formalism a few gears maybe that'd go a long way to sorting out whatever the problems seem to be?