Curriculum·G201 Invariant Thinking·about 30 min

The invariant enforced by an assumption

By the end of this lesson you can

  • Identify the hidden assumption behind an invariant, such as an honest validator set or an uncompromised key threshold
  • State how the Harmony Horizon bridge's two-of-five multisig made its entire backing invariant depend on two keys
  • Compute how a signing threshold sets the number of independent compromises an attacker needs, and why two of five is not much of a threshold
  • Rank the assumptions behind a protocol's invariants by how many independent things must hold for the invariant to hold

Graduate · enrolled learners

This lesson opens with The Harmony Horizon bridge, 23 June 2022.

What happened
The Horizon bridge between Harmony and Ethereum was secured by a two-of-five multisig: any two of five authorized addresses could authorize a transfer of the bridge's locked funds. On 23 June 2022 an attacker who had compromised two of those five signing keys used them to sign eleven transactions that drained about 100 million dollars from the Ethereum side. Harmony's own incident report confirmed that private keys had been compromised. The FBI later attributed the theft to the North Korean groups Lazarus and APT38. No smart-contract bug was involved; the contracts did exactly what two valid signatures told them to do.
The decision point
The bridge's invariant was the same as every bridge's: withdrawals are backed by locked deposits, and only authorized withdrawals happen. That invariant was enforced by an assumption, that fewer than two of the five signing keys would ever be in an attacker's hands at once. The assumption was never written as a risk a user could weigh, and it was weak: two of five means an attacker needs to compromise only two independent things, and the five keys were not five independent parties operating under real separation. When the assumption failed, the invariant failed instantly and completely, because nothing else stood behind it. A user assessing this bridge needed to ask not is it audited but how many independent compromises would break its core promise, and the answer was two.
Recorded loss
$100,000,000

What you will be able to answer

  • What enforced Harmony's backing invariant?
  • What does a signing threshold set?
  • The question to ask about a core invariant?
  • What does a timelock do to a trusted assumption?

Orientation and Year One are open: anyone can read them without an account. From Year Two onward the lessons are for enrolled learners, because progress through the later years only means anything if it is tracked against a record.

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Sources and review

Confidence high·Volatility medium·Reviewed 2026-09-14·Owner unassigned

Contested

The exact method by which two of the five keys were compromised was not fully disclosed publicly; Harmony's report confirmed private-key compromise and the FBI attributed the attack to North Korean groups.

J301-04 owns bridge trust models as an architecture question. This lesson uses Harmony only for the assumption-behind-an-invariant framing. Keep the split.