--- Yeah, but since the input power needed to switch the relay has to be greater than zero and the output power dissipated by the load has to be less than infinite, the relay's gain can never be infinite.
You originally claimed it could, and regardless of all the hoopla and handwaving, the fact which remains is that you were wrong.
Plain and simple:
Relays, of whatever flavor, can't have infinite power or energy gain and your statement that latching relays can, is wrong.
JF