It is my understanding that Lightning is to scale with users having only few payment channels by leveraging the six degrees of separation theory to facilitate payment routing. I understand how onion routing is employed to obfuscate origin and target of the payment.

I have read claims that it is also not possible for a node in the middle to receive but not forward the payment. This appears to be implemented by having either all hops of the payment execute at once, or none at all.

How can it be assured that the whole payment chain executes the payment at once?