Hash tables are bounded below by the speed of arithmetic. A trillion-bit number takes a while to multiply. You may have never touched a trillion-bit number, but it's called "asymptotic" for a reason.
You only get O(1) if you use a model of computation in which arithmetic is constant time. It's simple to work with, even though it's not realistic, and opens up scenarios where you can encode a really complex operation into adding/multiplying some huge numbers, and then claim the overall computation is still fast.
Some theorists use a model of computation in which, if your input size n, then multiplying a log-n bit number is O(1), but multiplying an n-bit number is O(n). I have no idea how this is well-defined.
Hash tables are bounded below by the speed of arithmetic. A trillion-bit number takes a while to multiply. You may have never touched a trillion-bit number, but it's called "asymptotic" for a reason.
You only get O(1) if you use a model of computation in which arithmetic is constant time. It's simple to work with, even though it's not realistic, and opens up scenarios where you can encode a really complex operation into adding/multiplying some huge numbers, and then claim the overall computation is still fast.
Some theorists use a model of computation in which, if your input size n, then multiplying a log-n bit number is O(1), but multiplying an n-bit number is O(n). I have no idea how this is well-defined.