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The pressure of the tires pushes horizontally outward on the sidewalls and helps them hold their shape, so they can absolutely be in compression and not fold up.

I am not convinced by this. It would seem that, if it works the way you say, a bubble's surface could be under compression.

Consider a section of tire wall, with the same pressure on either side. It is curved outwards, and if you put compression on it, it will buckle outwards. If there is higher pressure on the inside of the curve, that would help with the buckling, not prevent it.

In an inflated tire, as in a bubble, it is the tension along the curved surface that resists the internal pressure. What keeps the sidewall of an inflated tire from buckling is that the buckling would decrease the radius of the curve locally, increasing the net inwards force of the tension over the area that is buckling.

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