More Than Half the Computing Power
The edge of the one-ledger basin has been approached once on Bitcoin and crossed many times on smaller currencies built the same way.
The second basin is one ledger: when two versions of the newest block appear, every copy settles back on the version that took the most guessing to build. Its edge is anyone who controls more than half the computing power, because that party can always build the winning version and rewrite recent history. The Spending a Coin Twice page explains how such an attack works.
Bitcoin, 2014: GHash.io
Miners usually work in pools, sharing the guessing and splitting the rewards so each member gets a steadier income. In June and July 2014 one pool, GHash.io, grew so popular that its share of Bitcoin's computing power approached half, and by some counts briefly reached it. Nothing was attacked, but the community was alarmed. Miners moved to other pools, and GHash.io publicly promised to keep its share below 40 percent.
What pulled Bitcoin back from this edge was not a rule in the code. It was miners and pool operators deciding that a currency one party could rewrite would be worth less to all of them, including that party.
Smaller Currencies: the Edge Crossed
Several smaller currencies use the same design with far less computing power behind them, which makes half of it cheap to rent. In May 2018 attackers took majority control of Bitcoin Gold and spent the same coins twice on exchanges, a theft widely reported at about $18 million; it happened again in January 2020, for about $70,000. Ethereum Classic suffered the same attack in January 2019 and three more times in August 2020.
Those currencies did not disappear, but on each occasion their ledgers were rewritten and real money was stolen. That is what the far side of this edge looks like.
What It Shows
The edge is real, and Bitcoin's protection is size. Taking half of Bitcoin's computing power would mean buying and running more mining machines than everyone else on Earth combined, and the attempt could still fail. The same design with a small amount of computing power behind it has been crossed repeatedly.
Sources
TechCrunch (16 July 2014). Popular Bitcoin mining pool promises to restrict its compute power to prevent feared "51%" fiasco.
Bitcoin Gold attacks: reports of May 2018 and January 2020; Cointelegraph (January 2020), "Bitcoin Gold blockchain hit by 51% attack leading to $70K double spend."
Ethereum Classic attacks: January 2019 and August 2020, widely reported by exchanges and the Ethereum Classic developers.