Is written evidence of a contract always required to establish validation? How?

In California, with a few minor exceptions, oral contracts are just as valid as written ones. I am guessing, since you do not give us enough information, that the problem is the HOA feels you have no proof the work was done in the amount claimed. An invoice is a form of contract. ?Find out what the HOA wants and provide it; if they still will not pay, and the HOA is responsible for the work that was done, you will need to sue in Small Claims Court, but you will need to prove the work was done, the charge was reasonable, you paid for the work, and the HOA is responsible for it.

No oral evidence may be presented. It pretty usually to call example for a plumber to provide a service and that the service provider send an invoice. No written contract is need only oral evidence that a service was requested and he was given permission to enter the building and that service was provided and that service fixed the problem.