Which only goes to show that you can't apply real world reasoning to explain an academic concept. In this purely theoretical concept, World (which cannot be inspected) does contain the entirety of the world, including all other actions running in parallel, the current time and the state of every atom in the universe. So given the exact same state of the universe and assuming purely deterministic laws governing it (or the state of the world containing the future of the world too), yes, the action will produce the exact same output, because all other actions running in parallel or the things that will lead to the future actions are also part of that input World state.

Thats my best attempt, and it still sucks! I bet someone will start wondering about the uncertainty principle there... So lets just use a free monad or thunks in a strict language

  function pureAction() {
    return function() { return impureAction(); }
  }

  function chain(pureAction, fnReturningOtherAction) {
    return function() {
      var impureResult = pureAction();
      var newAction = fnReturningOtherAction(impureResult);
      return newAction();
    }
  }


  function interpreter(mainAction) { return mainAction(); }

Everything is pure, since all the functions don't do anything but return thunks for the interpreter to run. Except the interpreter which actually runs those. This isn't even a theoretical explanation - its exactly how PureScript's Eff works.