As per DOJ and DOL posted by you, you have worked for 5Y 10M 4days………..app.
So you are eligible.
Payment of Gratuity Act 1972 covers educational Inst.
However you may confirm if your dept has its Gratuity Rules and attach the same.
Assuming that you have erred in writing your DOJ, DOL and your correct period of service is 4Y10M4days even then if your have worked for 240 days in a year you shall be eligible for payment of gratuity…………………
Even if the employee has not submitted any application of Gratuity the employer should supply the notice of determination of Gratuity and payment within 30 days or the employee shall be eligible for interest @10%p.a for the delayed period.
You may submit Form under proper acknowledgment, preferably by redg. post.
You can download the Act from :
https://labour.up.nic.in/labourcommissioner/payofgraact72.htm