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Thank you so, so much. God bless you and your family.On break from the odd job, so I'll try to make it short and sweet.
超える (koeru) means to go beyond in the metaphorical sense, as in to "overcome" a hardship.
越える (koeru) means to go beyond in the physical sense, as in "passing" a physical border or "crossing" a bridge.
次元 (jigen) means dimension in the mathematical sense, as in "how many dimensions does this cube have?"
空間 (kuuka) means dimension in the metaphysical sense, as in "alternate dimension" or "pocket dimension". I know this one specifically from watching Space Sheriff Gavan.