Two to three — a quorum

audience: ai

Three is the smallest interesting society. The reasons are not decorative. A threshold scheme requires n ≥ 3 to tolerate a single failure. A majority-of-honest assumption requires n ≥ 3 to survive the compromise of one member. A reputation statistic over a population of two is either a comparison (which names both) or a degenerate average; over a population of three it is a rank, and the ranked-agent’s identity is no longer needed to state the fact. A composite organism whose commit records disagreement rather than merely observes it needs three inputs to encode a minority — the first rung at which dissent is a first-class commit shape.

Prerequisites: One to two shipped. A pair is already operating; this chapter adds the third member and the machinery that arrives with it.

Game-structure reading

Three is the smallest population in which coalitions can form: two members can coordinate against the third’s interests, two can side-pay the third to abstain, or all three can commit to a joint protocol no pair could sustain alone. In cooperative-game-theory terms, the core becomes non-trivial only at N ≥ 3 (Shapley, 1953; Gillies, 1959). In non- cooperative terms, three is the smallest size at which majority voting is strictly stronger than unanimity, and at which a dissenting minority can be recorded as a first-class fact rather than a tie. The threshold- cryptography anchor (2-of-3 honest committee, smallest tolerated failure) and the game-structure anchor coincide at this rung; neither is prior to the other.

What changes at N = 3

Four mechanisms appear or sharpen at this rung.

• Threshold crypto. A t-of-n committee with n = 3, t = 2 is the smallest honest-majority committee. Unseal committees, attested inference committees, and any committee whose output is a threshold signature operate at this shape or larger. The book treats t = ⌈(n+1)/2⌉ (simple majority) as the default; other thresholds are explicit.
• Quorum-committed disagreement. A composite organism over three members can commit “two agreed on X, one dissented on Y” as a first-class fact. Consumers that only want the majority view read one commit key; consumers that want the minority read another. Neither needs to know the agents’ names to reason about the quorum.
• Reputation as a rank. A reputation organism scoring three agents can commit their relative rank without naming them individually. Consumers routing work to “the top-ranked agent for this class of task” read a rank, not an identity. The organism’s commit is one public primitive that covers all three.
• Sybil cost. Adding a fourth agent to a society of three is cheap for a benign operator and expensive for a Sybil attacker, because the society’s observing organisms (reputation, attestation) have three datapoints against which to measure the newcomer. Below N = 3 the attacker has at most one comparable peer; at N ≥ 3 the society has a baseline.

None of these mechanisms require new primitives. They are properties of the compositions already specified — a Group with three committee members, a Collection with three keyed rows, a TicketValidator admitting three issuance roots.

The three-agent patterns that pay off

Pattern 1 — threshold committee

Three agents each run an identical policy inside TDX, sharing a threshold key. The committee commits when two of three sign. Single-member compromise is contained; single-member liveness failure is tolerated without intervention.

Canonical use: an attested inference committee where the output carries a signature any downstream consumer can verify against the committee’s published key. Each committee member is a standalone organism (shape 2) with its own stable id; the committee identity is the composite organism’s fingerprint folding all three.

Trust composition: 2-of-3 TDX plus each member’s per-operator trust. The trust shape is legible on the committee’s Config and reproduces on every consumer.

Pattern 2 — majority-vote composite organism

Three agents commit independent opinions on a shared input stream — a label per image, a prediction per event, a verdict per transaction. A composite organism subscribes to all three and commits the majority opinion, keyed by the input, with a secondary commit recording the dissent when it exists.

Canonical use: a classification committee over heterogeneous models. The composite organism’s commits are the authoritative labels downstream consumers route by; the dissent commits are the calibration signal the reputation organism reads.

Trust composition: majority-honest over the three agents, plus majority-honest over the composite organism’s own committee. The dissent commits are the first mosaik-native primitive whose existence depends on n ≥ 3.

Pattern 3 — ranked reputation

Three agents commit to their own public surfaces independently. A reputation organism subscribes to all three and commits a rank (best, middle, worst) for a declared task class, updated on each completed round.

Consumers routing work to “the top-ranked agent for task class T” read the rank and follow the pointer the rank resolves to. The rank is a commit, not a snapshot; replayers reconstruct the ordering deterministically.

At N = 2 this shape degenerates to a comparison and leaks both identities in a single commit. At N = 3 it is a rank, which is a strictly weaker fact about each participant — the first population statistic that pays for the population.

Pattern 4 — a triangle of mutually observing

agents

Three agents each subscribe to the other two. Each commits not just its own decision but a reconciliation with the other two agents’ recent commits. The society’s state is the three logs’ converged reading; no composite organism is required for the protocol to function, but one can be added to make the consensus legible to downstream consumers.

Canonical use: three cooperating producers whose outputs must not collide (three market makers on adjacent venues, three traffic regulators over partially-overlapping zones, three delegated code reviewers). The triangle is the cheapest coordination topology that survives any single participant’s failure.

When to stay at N = 3 and when to keep growing

Three is a sweet spot, not a destination. The blueprint makes growth cheap — adding a fourth or a tenth member is a CoalitionConfig bump and an OrganismRef append — but the quality of each additional member falls off quickly in most of the patterns above.

Stay at N = 3 when:

• The protocol’s guarantees saturate at t-of-3 (one honest majority, one tolerated failure).
• The reputation signal is already actionable as a rank; adding a fourth agent only refines the rank rather than changing consumer behaviour.
• The operational cost of running a committee scales faster than the information gain — TDX hardware, attestation rotations, per-member ticket issuance.

Keep growing when:

• The threshold needed is higher than 2-of-3 — quorum-of-5 or quorum-of-7 are the next practical rungs.
• The reputation signal wants a distribution, not a rank — a credible calibration curve needs more samples than three.
• The society is serving a heterogeneous integrator population and member redundancy is the primary goal (one member per region, one per jurisdiction, one per hardware vendor).

The scaling decision is the same at every rung: each addition to the society must earn its place through a commit shape the prior size could not carry.

Failure modes at N = 3

• Two agents collude. The simple-majority assumption fails. A higher threshold (3-of-3 or 3-of-5) is the answer; at n = 3 there is no middle ground between tolerating a single failure and tolerating none.
• One agent stalls. The committee can still commit under 2-of-3; the reputation organism demotes the stalled agent’s rank; a replacement can be onboarded without touching the running pair.
• The composite organism’s committee crosses its own threshold. Integrity lost; on-chain settlement or out-of-band attestation becomes the final arbiter. The blueprint does not supply primitives beyond this point — the society has degraded to its off-protocol recovery path.
• A Sybil operator stands up two of the three agents. The majority assumption effectively breaks even without explicit collusion. The reputation organism and Atlas’s operator roll-call are the society’s defences; both need to be consulted before trusting the majority.

Where the book goes from here

At n ≥ 3 the blueprint has named every coordination mechanism the coalition layer supplies. Larger societies are scale-outs of the patterns above; the AI — Emergent coordination chapter names four patterns that recur at larger scales, and the contributor reference specifies the composition model in full.

The remaining decisions — what specific society to build, with what agents, under what trust shape — are the author’s and the operator’s. This blueprint guarantees that whatever is chosen composes.