You stating a thought in your own words is not an explicit finding

True. However, me repeating the explicit findings of MANY others in my own words doesn't somehow unmake the explicit findings themselves. As much as you desperately (and irrationally, i should urge you to recognize) hope it would :(

As i've said many times now, if you feel the explicit findings (and/or my redescribing of them in intelligible and plain english) are incorrect simply cite or record the measurements demonstrating they are incorrect! The fact that you can't do that (and what that inescapably means) should eventually sink in...

I'm happy to be proven wrong if you can find even a single book stating this law as you have

I can, but i won't (because it will only make you a lesser student). This isn't a "contest" to me, and i have no interest in "proving you wrong". It's just a discussion. Many modern hydrostatics textbooks contain the law (albeit in different words) - but if you don't want to read them, so be it. Just don't delude yourself into believing you have read them, or know what they do or do not contain in the meantime!

If so, why is the air so much thinner at higher altitudes? Shouldn't the gas be able to expand to be a constant pressure/density throughout our livable space, since it's not at absolute zero?

Good question! It certainly tries to become completely equal in pressure, to the best of its ability (largely dictated by the available thermal/kinetic energy).

The reason it ultimately fails, and the cause of the density gradient in all things - solids/liquids/gasses etc) is because it is being pushed upon by the weight of the gas above it (which ultimately itself is touching, and deriving pressure from the container ceiling).

Consider a sealed gas cylinder at constant temperature, for simplicity. The gas inside seeks equilibrium (rest). The gas expands to fill the container and then the gas settles. The gas above the gas below is settling on top of it! As a result the gas below is at a higher pressure/density - AT equilibrium (at rest).

An analogy that might help is considering a piece of compressible foam. With nothing sitting atop it (aside from air) it has one volume/density. When it has something with weight sitting above it - compressing it, it has a decreased volume and increased density. It works exactly the same way with the gas layers below the gas layers above.

The above gasses weight is greater than the force of expansion of the lower gas AND it is also (ultimately) pushing back down upon it by that same expansion force derived from the container ceiling.

You stating a thought in your own words is not an explicit finding

True. However, me repeating the explicit findings of MANY others in my own words doesn't somehow unmake the explicit findings themselves. As much as you desperately (and irrationally, i should urge you to recognize) hope it would :(

As i've said many times now, if you feel the explicit findings (and/or my redescrbiing of them in intelligible and plain english) are incorrect simply cite or record the measurements demonstrating they are incorrect! The fact that you can't do that (and what that inescapably means) should eventually sink in...

I'm happy to be proven wrong if you can find even a single book stating this law as you have

I can, but i won't (because it will only make you a lesser student). This isn't a "contest" to me, and i have no interest in "proving you wrong". It's just a discussion. Many modern hydrostatics textbooks contain the law (albeit in different words) - but if you don't want to read them, so be it. Just don't delude yourself into believing you have read them, or know what they do or do not contain in the meantime!

If so, why is the air so much thinner at higher altitudes? Shouldn't the gas be able to expand to be a constant pressure/density throughout our livable space, since it's not at absolute zero?

Good question! It certainly tries to become completely equal in pressure, to the best of its ability (largely dictated by the available thermal/kinetic energy).

The reason it ultimately fails, and the cause of the density gradient in all things - solids/liquids/gasses etc) is because it is being pushed upon by the weight of the gas above it (which ultimately itself is touching, and deriving pressure from the container ceiling).

Consider a sealed gas cylinder at constant temperature, for simplicity. The gas inside seeks equilibrium (rest). The gas expands to fill the container and then the gas settles. The gas above the gas below is settling on top of it! As a result the gas below is at a higher pressure/density - AT equilibrium (at rest).

An analogy that might help is considering a piece of compressible foam. With nothing sitting atop it (aside from air) it has one volume/density. When it has something with weight sitting above it - compressing it, it has a decreased volume and increased density. It works exactly the same way with the gas layers below the gas layers above.

The above gasses weight is greater than the force of expansion of the lower gas AND it is also (ultimately) pushing back down upon it by that same expansion force derived from the container ceiling.