No one seems to want to directly address the design but instead dance around the subject.
I completely explained the operation of the machine in post #23.
Where’s @Lucien at when you need him
You missed my answer as well, so I'll say it again:
The 'SeaPower' machine is a pneumatic motor, a.k.a air-motor, in which air expands from one volume to another while doing mechanical work on a linkage. The SeaPower differs in constructional details from conventional air-motors by having a balloon (or inverted bucket) and liquid medium to confine the air, rather than a piston sliding within a cylinder or a rotor with sliding vanes mounted eccentrically in a cylindrical housing. The liquid serves as a mechanical linkage to transfer to the output shaft the force exerted by the boundary surface of the air, just as the piston and connecting rod, or vanes and rotor do, in the normal type of air-motor.
Intuitively, we can see that driving the shaft (using an external prime mover) so that the balloons descend, will cause the machine to function as a compressor. Absent any losses, it will compress the same volume of air to the same pressure as would be required to make an identical machine operate as a motor and deliver the same shaft torque and speed as output. Intuitively again, we can see that coupling the two machine shafts and air pipes together would result in equilibrium and all would remain stationary.
If, however, you want to ignore the obvious symmetry and prove the result numerically, you will first need to learn the basics of integral calculus (and I do mean just the basics.)
1. Learn calculus.
2. Find an expression in terms of depth for the force exerted on the belt by a balloon containing a unit volume of air.
3. Integrate this expression with respect to depth to obtain the work done on the belt by each rising balloon.
4. Compute the work done by a unit volume of air injected at arbitrary depth, and divide by the work done compressing that air against the head of water at that depth.
5. If the result is over unity, you probably made a mistake!
Anyone who is still reading at this point might have sufficient interest in water-pistons as to be familiar with the Humphrey pump. And if you aren't, you should be. I don't know where in Texas
@justcurioustwo is located but one of the few Humphrey pump installations in the world was in Del Rio TX. The Chingford installation is just ten miles from here.
Humphrey Pumps Wikipedia article