Why does accessing an element by index in an array take constant time?

Contiguity turns a lookup into arithmetic. Every element occupies the same number of bytes, so the address of index i is the base address plus i times the element size. One multiply, one add, one fetch. The cost is identical whether the array holds ten items or ten million, which is what constant time means. The C++ standard states the layout guarantee plainly: an array is a contiguous container.

This is also why an array beats a linked list on lookup and loses on insertion. Reaching the thousandth node of a list means following a thousand pointers; reaching index 999 of an array is a single calculation. NIST defines the random access model as one in which arithmetic operations are allowed to compute the address of a memory register, and that is the mechanism.

D describes the linked-list traversal that arrays exist to avoid. B and C are not descriptions of arrays at all. NIST calls an array an assemblage of items randomly accessible by integers, the index; random there means any index costs the same, not that the ordering is scrambled between accesses. Layout is fixed at allocation.