Why does the net lift of an airship remain constant when the Kg of air it displaces should fall as altitude increases and pressure decreases?

Question

In the paper, Development of an Aerodynamic Model and Control Law Design for a High Altitude Airship I found a passage that makes no sense to me.
This in regards to the lift of airships.
Note: pressure altitude = max safe altitude.

It can be shown that the net lift is constant over all altitudes, up
to the pressure altitude. This is based upon the assumption that the
density of the lifting gas changes at the same rate as the atmospheric
density.

An airship's lift is based on the mass of air it displaces as per Archimedes Principle.  How can net lift remain constant when air density falls at increasing altitudes?  Although the density of the helium in the airship may also decrease the mass of helium stays constant because it is contained inside the airship.
For example:
0 meters altitude and 15 degrees Celsius.

Air pressure in Pascals is 101,325

6,000 meters altitude and -24 degrees Celsius.

Air pressure in Pascals falls to 41,413

If air pressure falls doesn't the mass of the displaced air also fall?  And if the mass of displaced air falls wouldn't the airship's lift also fall?

joseph h · Answer

Bouyant force is given by
$F_B = rho Vg$
As the airship rises, the outside air pressure does decrease as well, meaning the helium in the airship expands (which means you are correct in that its density decreases for the same mass). Therefore, the amount of displaced volume of air will also increase. This means the buoyancy force increases.
There must be a mechanism where the pilots balance the air pressure with the helium pressure inside the airship, to maintain a neutral net buoyancy.

Acccumulation · Answer

The article explains the math, so I'll try to provide some intuition.
Imagine you have some air cut off from the rest of the atmosphere with a force field. The force field has no mass, is infinitely thin, and expands to match the air perfectly. The buoyancy is zero: the air inside is exactly the same as the air outside, so there's no reason for it to rise or fall.  But suppose there's some force pushing it upwards. As it increases altitude, it expands, and it expands just the same as the air around it, so the buoyancy stays zero.
Now imagine that we have a balloon filled with helium. Since it's lighter than air, it does have positive buoyancy, so it rises. That is, the air that would be there but isn't would have weighed more, and the difference is the buoyancy. As it rises, the air around it expands, and the helium does as well. The two gasses both expand by the same factor. The buoyancy remains the weight of the air that would have been there, minus the weight of the balloon.
The key here is that the balloon matches what the force field would do: the region of "there would be air here, except there isn't because the balloon is here" matches the region that would be taken up with the air inside the force field. The buoyancy is always the weight of the air that would have been inside the force field, minus the weight of the balloon. Since the amount of air inside the force field doesn't change, the buoyancy doesn't change.

An airship's lift is based on the mass of air it displaces as per Archimedes Principle.

Technically, that link says that it's based on the weight of the air displaced.

Why does the net lift of an airship remain constant when the Kg of air it displaces should fall as altitude increases and pressure decreases?

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