If I leave mid-year, am I still eligible for summer pay? I'm on a what appears to be a 12 month contract (Sept.- Aug.)  While we only work 9 months out of the year, the standard academic job year, we are paid over 12 months.  Well, I am now leaving my post mid-year and thus think I am entitled to half my salary because I worked half the academic year.  The administration says that I'm only entitled to one-third (because I only worked one-third of the calendar year).  As a result, I'm losing close to $7000.  Has anyone ever run into this?  Do you happen to know any sources I can cite?
I've run into this issue many times while I was serving as a department chair at a large US university. If your university's academic year salary system is similar to the standard one used at US institutions (which by your description it sounds like it is) then your analysis is correct, and your administration is wrong. 

The point is that paying your salary over 12 months is nothing more than an accounting trick to spread out your salary (paid for 9 months of actual work) over the 12 months of the year. The reasons why this is done (as far as I know) are: 1. tradition, 2. health benefits, 3. the psychological convenience to faculty of being paid each month and being continuously employed, and 4. possibly other advantages involving things like taxation and labor law. 

However, the trick is irrelevant to the fact that you performed 50% of your assigned duties for the year. It therefore stands to reason that you need to be paid 50% of your annual salary. I don't have any sources to cite, but this is just common sense.

I can also say from my experience that this discrepancy between the schedule of payments to faculty employed on a 9/12 basis and the schedule of work they are performing causes a lot of confusion, even among experienced administrators: whenever someone joined or left the department in the middle of the academic year, some calculations needed to be done to figure out how to make the salary figures work out, and people would get confused and sometimes require a bit of gentle persuasion to get to the correct answer. Thus, Hanlon's razor might apply here, and it's definitely worth going back to the administrators and trying to make them see the light. Good luck!