Well, I can try to explain it all, but for now (as I don't have my laptop with me at this current moment) I can briefly mention the fact that yes, GR does predict that beyond the event horizon it must be geodesically incomplete, and it is.. in 3+1 Dimensions (assuming you consider spacetime as a 4 dimensional concept where 'time' is the 4th dimension..) because 'geometrically' space is 3d.. however a hypersphere, caused by GR's prediction of bent spacetime due to a mass, which we then call 'gravity', bends 'into itself' causing a '4d hyperspherical structure to form, encasing the mass of the collapsed star', and so as it is a 4d shape, it would have a 3d surface, (just as a 3d shape has a 2d surface, and a 2d shape, would have a 1d surface(or edge)), and so from 3+1 dimensions (or 3d spacetime), beyond the event horizon would appear geodesically incomplete, as predicted by GR, but from 4+1 dimensions (or 4d spacetime), it would form a 'loop' around the 4d hyperspherical structure, around its '3d surface'..
As for the pre-existance of higher spatial dimensions, yes theoretically they could exists, even though we cannot 'directly see' them, as we are 3dimensional creatures, who can only truly comprehend up to 3 spatial dimensions.. but the 4d structure formed during the gravitational collapse results from the curvature of spacetime 'not just bending down into a sort of 'hole', but rather, being curved 'around' the 'center' of the mass itself, until it reaches a point when spacetime 'curves back into itself', forming a higher dimensional structure (as we and most-all objects exist in 3 spatial dimensions, this structure would then theoretically exist within 4 spatial dimensions, or 4d) which based on it 'curving around' the 'center' of mass, as it becomes ever-more 'densly compacted', it would form a sort of 'higher dimensional sphere', or a 'hypersphere'..
As for the Mathematics of the Theory, yes, I admit, I am still in the proces of refining it further and translating it into mathematics that make sense, but so far, although it has been (and still is) quite a lengthy process, I haven't run into any real 'dead-end problems as of yet', and so once I have some of the equations ready for 'peer review', I will happily provide them here.
But in the meantime, please also feel free to request for me to 'focus' upon any particular Mathematical aspect of the theory, as, once again, I really just want to share this theory with the world in the best, and most logical and reasonable sense possible
(Is it appropriate for me to keep repeating that, or is that point already been made??)