does all of buddhism depend on dependant co arising? If co-dependant arising is shown to be false or contradictorary, would all of Buddhism fall ?

anatta, shunyata, nirvana as extinction-cessation, momentariness and and so forth seem to be based on this law-paticasamutpada/pratityasamutpada ?

is this Buddhism's crutch?
It's not like "buddhism" "depends" on a "law" - it's just that D.O. is the most fundamental observation about how things work.

It is an observation about how the world of representation develops, from complete nondifferentiation to having appearance of distinct sentient beings who are born, die, suffer, argue and fight with each other.

And it's basically like, "hmmm, this appearance of distinct sentient beings with all their troubles only exists in the world of representation, and outside of representation none of this is applicable!" and "therefore, if we learn to deconstruct our own representation, we can transcend its limits and griefs" and "so, it makes sense to study how representation develops, this might give us clues into how to deconstruct it".

But if you want to prove it false, you'd need to prove false the very idea of representation (aka subjective experience aka consciousness). As well as the idea that subjective experience is something that does not just pop out of nowhere but grows over time.

Both of these are obvious facts known first-hand. We all have subjective experience. And we all remember how it developed when we were children. Do you want to try and prove this false?