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Your Universe of Digital Possibilities
Seven generals must attack together or not at all, and some of them are traitors. Relay what you heard, vote what you were told, and the loyal agree — right up to a third. The tables count the general who breaks it.
With oral messages, agreement among the loyal survives f traitors exactly when they are fewer than a third of the camp. The margin is a count: The Generals’ worst-case search breaks OM at n = 3, 4, 5, 6 for f ≤ 2 and finds no break at n ≥ 3f + 1. With unforgeable seals the third disappears: SM(m) handles any number of traitors up to m.
Round r of OM(m) relays every value along every path of r distinct generals, so the courier count is a falling product per round: 3 + 6 = 9 for OM(1) at n = 4, 6 + 30 + 120 = 156 for OM(2) at n = 7, 3 609 for OM(3) at n = 10 — all counted by the engine. Signed SM(m) relays each order once per lieutenant instead.
The Clock gave the camp an order of events; The Generals ask whether it can share a decision when some of its members lie. The answer is a counted margin — the loyal win while they are more than two thirds of the camp, by relaying everything — and seals buy the rest. The same margin sits under every Byzantine-tolerant replica set, from Castro & Liskov’s PBFT to the validators of The Chain.