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For the neocortex yes, but there are other (older & deeper) parts of the brain that are definitely task specific. If you come to me without a basal ganglia or a hippocampus. I would be very very surprised if you were "normal".

From what little I know and read it appears that the brain's neural organization is fractal in nature. The same patterns are used and reused at different levels and in all regions.

It is definitely not true. One of the many trivial examples is that Betz cells only appear in motor cortex.