Why does restricting size of input for NP complete problem imply a runtime of O(1)?
If the input size is limited, then you can calculate the solutions to all problems. This will take a very, very, very long time. But once these solutions have been calculated, you can just lookup the solution of any problem in your list of solutions. 

Even worse: Look at the definition of Big-O. The claim is that there is a constant $c &gt; 0$, and an integer $N_0 &gt; 0$ such that for every problem with a size $N &gt; N_0$, the problem can be solved in less than $c * 1$ steps. If you restrict SAT to $N ≤ 1000$, then this is trivially true with $c = 1$ and $N_0 = 1001$. There are no problems with a size > 1000, therefore all the (nonexisting) problems with a size ≥ 1001 can be solved in one step.