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!
In the US, many universities pay the nine-month (or academic-year) salary in twelve equal monthly installments so that faculty who are not teaching in the summer (or do not have research grants that pay salary in the summer) do not have to starve during the summer. (Those who teach in the summer or have research grants get additional salary for this work, over and above the 12 equal installments). The battle that faculty members are adults who are fully responsible for their finances and so should be paid their academic-year salary in nine equal monthly installments (and be responsible for putting away money for expenses during the summer when they won't be receiving a paycheck) has been lost a long time ago.
If you quit after one semester (half-year), you will likely receive a lump sum upon leaving equal to the amount that you would have been paid over the summer. Each month that you worked, you got 75% of what you "should" have been paid (i.e. one-ninth of your academic-year salary) while 25% was put away in your name for the summer. If you had worked for all 9 months, the university would be holding 9x25% of your "monthly salary" which would be doled out in three 75% chunks over the three-month summer, that is, your "monthly salary" would be the same 75% that you were getting all year long. Now that you worked for only 4.5 months, the university has put away 4.5x25% for you for the summer, and you will get that money when you leave. It will be little over one-ninth of your academic-year salary.
Read your contract carefully. If you don't find the information, you should contact your worker's union.
Just for information: In my country (in Europe), there is something called the 13th and 14th salary (twice a year, you get an additional salary). Everywhere where I worked so far, if you work only, say, 5 months, you would get an appropriate share of these extra salaries (i. e. in this case, 5/6 of a normal salary).
However, I don't understand your summer salary system enough to be sure if it is the same concept. (If you would have to work only 9 months and would get payed for 12 months, it would only be fair to get payed for 12x/9 months if you work x months in my opinion).