Nope, I'm not brain damaged.
So you have no excuse and you are just lying to everyone?
No, there is no proof.
Yes, there is.
The simplest (other than things falling) is the Cavendish experiment.
You remember in third grade or so where they had you learn greater than or less than? That's all the math you need.
No, it isn't.
Saying one is greater than the other provides no reason at all for why it should move at all.
It provides no reason for why it should move in a particular direction.
It provides no reason for why it should accelerate at a particular rate.
It provides no reason for why this rate should vary with location.
It provides no reason for why this results in a force on an object (e.g. scales) needing to support the object above, this includes when it is a higher density (e.g. a steel table supporting a piece of aluminium or a fish tank full of water).
It provides no reason for the pressure gradient.
It provides no reason for why this pressure gradient doesn't push everything up.
In short, it explains NOTHING!
You say so, that things must fall because of a force.
Because all the evidence shows that if you want to change the motion of an object, it requires a force.
If such a force existed, buoyant objects would constantly be in a struggle to stay afloat.
Considering that same force is responsible for buoyancy, no they wouldn't.
Without such a force, everything should stay afloat.
Picture this.
Picture this. A kid finds a see saw, and places a small weight on it, and sees that side go down.
They then place a heavier weight on the other side, and see the small weight go up.
The realise that it goes up because the heavier weight has a greater force pulling it down.
They can transfer this to the garbage you describe, where they recognise the wood rises to the surface, because the water goes down because the force trying to moving the water down is greater than the force trying to move the wood down.
All your garbage does is show buoyancy is caused by gravity.
But then they can also look at other things, which behave similar to buoyancy.
If you hang a weight from an accelerating vehicle, we see the weight appear to fall backwards.
But if we do this with a balloon full of helium, we see it appear to fall forwards.
This is quite comparable to gravity and buoyancy.
The fluid creates a pressure gradient which pushes things forwards.
Just like with gravity, a pressure gradient is created with pushes things upwards.
Again, if you want to claim your delusional BS works, you need to address the points above.
See if you can explain the pressure gradient, and why this doesn't push things up.