I have been struggling with this issue a bit and I think that it is a difference of language. You define politics as the unspoken way things get done at work, and I think that is how I would define organizational culture, with politics being one component of culture. I believe that it is a very common component, but I also believe that it is an optional component.
It may not be easy to do away with politics, because we have bought into the idea that it is just part of business, but I do believe it can be done and I think that there are organizations doing some really good work in this direction, with relentless focus on transparency, authenticity, and other characteristics that make less room for politics.
The truth is that politics, as I understand politics, makes already complex dynamics increasingly and unnecessarily complicated. Politics as I understand them are a waste of valuable resources (such as time, attention, trust, etc.).
I think that your five rules can help people be successful, but I do not necessarily think that means or requires that they playing politics. We might be viewing / defining politics differently… could you share a few comments on how you would define politics vs. organizational culture?