The rules that pick the head of the chain and lock blocks in for good (finality).
The idea: Validators vote on blocks with BLS signatures. Once two-thirds of all stake has voted for a checkpoint, and then for the next checkpoint building on it, the first one is finalized. Reverting it would require at least a third of all stake to sign contradictory votes, which is provable onchain and gets slashed.
Why it matters: The heart of Ethereum's security.
On the map: Gasper: LMD-GHOST + Casper FFG
Gasper combines LMD-GHOST[1], the rule for picking the head of the chain, with Casper FFG[2], which finalizes checkpoints so they can't be reverted without slashing at least 1/3 of all stake.
It depends on validators' BLS signatures, SHA-256 Merkle roots of the beacon state, RANDAO for picking proposers and committees, the honest-stake assumption, and, for nodes joining or returning after a long time offline, a recent trusted checkpoint (weak subjectivity)[3].
If any of these breaks, this breaks too.
If this breaks, these break with it.