This is not as suggestive as one might imagine given the title or the discussion here so far. It's not "Turing machine but the tape is limited to 1024 cells" -

This is not as suggestive as one might imagine given the title or the discussion here so far. It's not "Turing machine but the tape is limited to 1024 cells" - i.e. you don't get 1024 units of memory and a conditional branch instruction. You can't branch at all. In order to simulate the Turing machine, you need to run the Rule 110 automation, which necessarily implies expanding linearly into that 1024 cell space - i.e. your simulated program really isn't going to get very far. Edit: see comments by dTal and Dylan16807