I'm not talking about any logical chain or causal issue, so it can't be backwards or not backwards, what I'm saying is independent of the "underlying geometry". Certainly in curved manifolds one can refer to the limits of a certain chart without reference to the specific underlying geometry, it is a fact that curved smooth manifolds in general can't be covered completely by a single chart.No, it isn't. Once again, you have things backwards. The EH is never "defined" in terms of its being a coordinate singularity in any chart. The fact of its being a coordinate singularity, as I said before, is *derived* from the underlying geometry plus the definition of the chart.
As I said I'm not considering your fuzzy underlying geometry concept hereI've never said it does. I've always said the EH "belongs" to the underlying geometry. Its existence and properties are independent of any chart.
It just happens that charts are needed in differential geometry, at least differentiable manifolds are defined as those equipped with an equivalence class of atlases (collections of local charts) whose transition maps are all differentiable.
r=2GM is not defined in K-S coordinates, do you dispute that?There is no coordinate singularity in K-S coordinates.
No, I'm talking about the transition map between SC and K-S, so it is a problem also with the K-S space (the whole 4-regions) since they include the outside region.So what? That's a problem with the Schwarzschild coordinates, not with the EH.
See above.Yes, there are. K-S coordinates do, so do ingoing Eddington-Finkelstein and Painleve. None of those charts are singular at the EH, so they "cover the transition" just fine.
No, I don't believe that at all.You appear to believe that, if *any* chart is singular at the EH, *all* charts are "singular" there, because the transformation from the singular chart to any other chart must be singular there. That's wrong.