Offer letter discrepancy

So basically, when I applied for my current job I was told the hourly wage upfront in my initial interview. Once I got the job offer, I inquired about more money over email but when I got a call back I was told no can do. So the only paper trail is that I was told the hourly early in the process and later in the process I emailed about negotiating but nowhere in writing is it that they couldn’t increase the salary because that happened over phone.

Ok fine whatever it was fair enough so I took the job. I got the offer letter and low and behold it was for 4k more a year, signed by the HR person I had been in contact with and myself. Great! Guess they decided to give it to me. Now I got my first paycheck and it’s for the original amount, so it was probably just a typo but STILL. I’m going to go to HR about it obviously but wondering what exactly is the legality of a signed offer letter? What would your guys response be once they tell me no (because we all know how this will go)?

Thanks!!