The quantity the author describes is commonly called the working set. His explicit thesis is that the time to process the working set is related to the size of the working set regardless of computational complexity. He spends the entire first part of his discussion belaboring this point with a linked list example. He goes on to do a systems analysis of the results and concludes that time of execution is system dependent not complexity dependent. Again using the same algorithm engaging with a larger and larger memory system.

Generally engineers would experience this important principal in a more binary way, once their program started swapping to disk, its performance would fall through the floor.

If you hold the size of the datum constant (and that is important), then processing fewer of them will take less time than processing more of them.

The argument you make, as I understand it is as follows; Consider two algorithms that process N and P datums respectively to achieve the same computational result, and where N > P. There exists an algorithm such that P datums takes less per unit time to process than N. Resulting in a violation of my Lemma because Oi(P) > Oi(N) but Ot(P) < Ot(N). And you cite cache obliviousness as the principle that enables that violation.

Would you agree that I have accurately summed up your argument?

And in this case of O(sqrt i) memory access times, a binary tree and b-tree stay within a constant factor even as you vary b. (The reason is, the binary tree accesses that a single b-sized access replaces get exponentially more "local.")