It is not a “mathematical principle”. Not everything in physics that you don’t understand is some kind of complex math.
If fluid was to flow out of the bottom of the bottle submerged in the margarita it would leave a void in the top of the bottle, a vacuum without any air. Outside air pressure needs to be balanced with air pressure inside the bottle, and the weight of the fluid in the bottle is not enough to overcome ambient air pressure. That means it stays in the bottle.
For water the height of water that can be supported by air pressure is around 32 feet.
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