Free to me (theoretical)

[webby1]

Remember in the first one air was used inside and outside so for the water use and to be the same- water would need to be used inside and outside.

1) In the "first one" the internal medium was being compressed and the external was not.
2) With air inside the cylinder and water outside, the internal medium was being compressed and the external was not.
3) With water inside and outside the cylinder, both media are not compressed.

Thus the behavior of system 3 is more different from system 1 than system 2.

I think there is a miss.

In the first one the air inside the cylinder is at the same pressure as the outside, after all it comes from the same source, open the valve move the piston to approximately 1/3 the distance of travel, close the valve.  Do this in water and water is in the cylinder.

You have done the electrolysis work up already, I read it.

Simple thing is that air under the influence of gravity can be change in a fashion that allows the air to store some gravity work/potential by compressing into a smaller volume via pressure created by gravity,, you could probably state that much better than me.  
Water can not do this so you can not shift one observation and storage point to another point of observation for release for very little to no cost because there is no storage in water, except for the dissolved gasses.

The air at 5000ft has a pressure to it ,so it is compressed, but that value of pressure, aka force, has a gradient that increases the lower in altitude that you move.  1 cubic meter of air at 5000ft has a smaller quantity of air at a lower pressure than 1 cubic meter of air at sea level.  For the cylinder to have the same internal pressure up at 5000ft the piston must move out increasing the volume occupied by the quantity of air contained within the cylinder.

[Verpies]

I think there is a miss.

In the first one the air inside the cylinder is at the same pressure as the outside, after all it comes from the same source, open the valve move the piston to approximately 1/3 the distance of travel, close the valve.  Do this in water and water is in the cylinder.

Well of course! You are getting bogged down in practical implementation minutia instead of focusing on the proposed gain mechanism.
When I wrote that the external medium might as well be water, I did not propose to let the water inside the cylinder during the initial setup (the "prep").
System #2 has air inside the cylinder and water on the outside during the working cycle (during which the valve is always closed so the mass of the internal medium is constant).

Simple thing is that air under the influence of gravity can be change in a fashion that allows the air to store some gravity work/potential by compressing into a smaller volume via pressure created by gravity...
Water can not do this so you can not shift one observation and storage point to another point of observation for release for very little to no cost because there is no storage in water, except for the dissolved gasses.

There is still air in System #2 so your point is moot.

The air at 5000ft has a pressure to it ,so it is compressed, but that value of pressure, aka force, has a gradient that increases the lower in altitude that you move.

The external water exhibits a depth-dependent pressure gradient which is caused by gravity, too. This pressure is not qualitatively different than air.

The comprehensibility of the external medium is not used in your 5000ft tower system so it does not matter whether the external medium is compressible or not - all that matters is that the external medium exerts a pressure on the outside of the piston ...and that piston does not care whether water or air is pressing on it from outside.

The behavior of the internal medium is quite different, but in system #1 and #2 the internal medium is the same (air).

1 cubic meter of air at 5000ft has a smaller quantity of air at a lower pressure than 1 cubic meter of air at sea level.  For the cylinder to have the same internal pressure up at 5000ft the piston must move out increasing the volume occupied by the quantity of air contained within the cylinder.

I agree but the air inside of the piston does not care whether it is compressed by the external pressure of air or by the external pressure of water.

You have done the electrolysis work up already, I read it.

Nobody has done it here.

[webby1]

I think there is a miss.

In the first one the air inside the cylinder is at the same pressure as the outside, after all it comes from the same source, open the valve move the piston to approximately 1/3 the distance of travel, close the valve.  Do this in water and water is in the cylinder.

Well of course!  You are getting bogged down in practical implementation minutia instead of focusing on the proposed gain mechanism.
When I wrote that the external medium might as well be water,  I did not propose to let the water inside the cylinder during the initial setup (the "prep").
System #2 has air inside the cylinder and water on the outside during the working cycle (during which the valve is always closed so the mass of the internal medium is constant).

I am actually looking for my error since I take the view that I am usually wrong.

There is still air in System #2 so your point is moot.

It is not actually, and it is in the behavior of the mediums involved that I am looking.

The thing is that the mass density of a gas under the influence of gravity will create a gradient as such that the mass density of the gas goes down as the altitude goes up, translation less and less buoyant lift the higher you go.
A non-compressible medium such as water does not exhibit this behavior so 1 cubic meter of displaced water will have the same lift value anywhere within the water column.

[Verpies]

The thing is that the mass density of a gas under the influence of gravity will create a gradient as such that the mass density of the gas goes down as the altitude goes up, translation less and less buoyant lift the higher you go.

That's correct. Buoyancy in atmospheric air of constant temperature decreases faster with altitude than in water.
To obtain the net work due to that buoyancy force, it needs to be integrated over the vertical distance of travel (in both directions).

[webby1]

The thing is that the mass density of a gas under the influence of gravity will create a gradient as such that the mass density of the gas goes down as the altitude goes up, translation less and less buoyant lift the higher you go.

That's correct. Buoyancy in atmospheric air of constant temperature decreases faster with altitude than in water.
To obtain the net work due to that buoyancy force, it needs to be integrated over the vertical distance of travel (in both directions).

This chat mini bot has done that but I am still not "feeling it" if you know what I mean.
So here is the wrap up from the AI

Practical constraints and when net work ≠ 0 Net zero work requires the ideal assumptions listed above. Nonzero net work is required if any of these apply:

• Pistons/pulleys have mass or friction.
  
• Moves are not simultaneous or are not equal and opposite.
  
• Piston areas differ.
  
• Processes are irreversible (heat losses, turbulent flow).
  
• You extract energy during steps and do not return the same energy later.
  

Final concise conclusion

• In the idealized closed two‑piston cycle you described, a full cycle (simultaneous locked swap + release + harvest) yields zero net work input and zero net harvested energy.
  
• Extracted energy during expansion or compression comes from the atmosphere/other part of the system; capturing it without returning it requires an external source to restore the cycle.
  
• Sustainable net energy extraction requires coupling to an external, ongoing non‑equilibrium source (e.g., solar‑driven atmospheric flows).
  

So it says that if I run many many cycles I will deplete the atmosphere of energy

---

This the full text file of the write up from mini bot

[webby1]

I ran this setup through 3 AI bots and none of them correctly identified the buoyant lift input required until I focused them on the lift by displaced volume, the expanded cylinder is 0.2 cubic meters larger in volume.

All 3 bots determined initially that I could not extract any energy from the compression of the cylinder until I had them view that condition by itself.

In the end there is approximately 18.5 kJ of energy I can extract from the expansion and another 18.5 kJ from the compression for each change in position.  For each change in position I need to supply approximately 3.7 kJ of energy to make the change, so approximately 33 kJ of "free to me" energy that comes from the environment.

If I had no expectations then I would be misled by the answers the bots initially provided.

Now considering Verpies assertion that it should work using water. I at first pushed back on that consideration but, if I change the systems observation of what I have and what I need then it can work using water and gas as well.

If I used Helium gas and then extracted the heat out of the environment and used the gas change in pressure caused by heat the system could function.  Down low in the water column I heat up the gas and it expands displacing more water, or at least makes it much easier for me to expand the cylinder, then, near the surface, I take the heat out of the gas and that makes it easier for me to compress the cylinder into a smaller volume making it easier to sink.

This then is primarily a heat exchange with the atmosphere.  Using that heat to cause a pressure change in the gas allowing for an easier change in displaced volumes.  With enough exchange of heat my costs could be next to nothing.