Super confused about the $7500 and leases!

How is the $7,500 tax credit used by the leasing companies such as US Bank and Ally?

From the Chevy national deal and and other posts here I understand that the savings are passed down to the leasee, but the math doesn't make sense to me.

For the example below I'm not considering any fees, taxes, etc just the basic numbers.
Approx $35,000 base price X 58% residual value = $20,300

If the car costs $35,000, and you subtract out the $7500, you are left with $27,000.

Keyes and Rydell are offering $5000 off invoice, so that comes to $22,000.

Cap $22,000 - Residual $20,3000 = $1700

$1700/36 = $47 a month.

Now I know a brand new car is not going to lease for $47 a month, but what am I misunderstanding here to get to that number?

Some other poster said it is the leasing companies money and not mine so I shouldn't have to worry about it, but then why advertise that the savings are being passed on?

For a real world example, I got a quote for a US Bank backed lease. The dealer had put the residual at 37% for a value of $12,948 and then added in the $7,500 to get to $20,448 which is 58% and some change.

From other websites the residual for the 2014 Volt is around 58% BUT that doesn't include the $7500 tax credit.

Where is the supposed $7500 savings in the US Bank lease?

Ally does something similar as well to come with a ~20K residual.

What am I getting wrong here!?