# Proof that blockchain converges to the longest chain [duplicate]

Is there a mathematical proof that shows that blockchain converges to the longest chain? It seems not obvious to me how a scattered group of nodes and transaction signals will eventually converge to a single longest chain under the Proof of Work/Stake/etc.

Thank you!!!

Without further qualifications or assumptions, no such proof exists because the statement isn't true. Proof of work, at best, makes it expensive for miners not to converge, but if miners had no regard for their own financials, and simply wanted to construct multiple chain branches that the rest of the nodes keep disagreeing about, there is absolutely nothing stopping them. Of course, anyone can become a miner, allowing them to stop such an attack - but that again becomes an economic argument, not an unconditional proof.

Next, demanding "a single longest chain", depending on how you interpret that, is not even the goal of proof-of-work. PoW aims to make the uncertainty a node has about a produced block go to zero in function of the number of blocks on top, but that does not imply much about the latest block. It could well be the case that there are always multiple competing blocks at the tip, but nodes only rarely disagree about what the Nth (say, 6th) ancestor of that tip is. Think of it as a tree that keeps growing, with old branches being cut off, while there are increasingly more branches at you get close to the tip. Note that this is rare in Bitcoin's parameterization in practice (forks more than 1 block deep only happen every few months), but things could very well be different for other parameters.

Yet, even with the above two qualifications, I don't believe a general proof exists. The difficulty is expressing conditions to capture things like economic assumptions. If you go even further away from reality, there does exist a proof that if a majority of the hashrate is "honest" (in that they always work on top of the best block chain tip they know of), and every party sees every block produced without delay, then the probability of needing to switch branches does approach 0 as more blocks have been built on top; this proof is in the Bitcoin whitepaper even. There may be proofs in later work under somewhat general assumptions, including ones that permit a (bounded) delay for communication before miners hear about each other's blocks, though I don't have any references right now.