THE FOLD / CO-OP / THE MERGE / THE QUORUM INTERSECTION
THE QUORUM INTERSECTION
the guarantee is one node wide
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Quorums work for one reason and it is arithmetic, not engineering: any two sets larger than half of a whole must share a member. That shared member is the only thing carrying information between one decision and the next.
LIT verified live and exhaustively. 9 nodes, majority 5, giving 126 possible quorums and 7,875 pairs of them. Every single pair intersects — 7,875 of 7,875 — and the smallest overlap found is 1, matching
LIT verified live and exhaustively. 9 nodes, majority 5, giving 126 possible quorums and 7,875 pairs of them. Every single pair intersects — 7,875 of 7,875 — and the smallest overlap found is 1, matching
2k−n exactly. Drop to half rather than a majority and it fails: of 2,415 pairs of 4-node sets over 8 nodes, 35 are completely disjoint.2 HOW IT WAS WEAVED · AI + HUMAN
Quorum intersection underpins Paxos, Raft and every replicated log; Gifford’s weighted voting (1979) is the general form.
AVAN (AI) enumerated both the property and its boundary. Showing that majorities intersect is easy and unconvincing on its own — the claim only has content if the neighbouring case fails, so the 35 disjoint pairs at half-size are the load-bearing number. The minimum overlap of exactly 1 is the other half: a majority quorum guarantees one witness, not a comfortable margin.
AVAN (AI) enumerated both the property and its boundary. Showing that majorities intersect is easy and unconvincing on its own — the claim only has content if the neighbouring case fails, so the 35 disjoint pairs at half-size are the load-bearing number. The minimum overlap of exactly 1 is the other half: a majority quorum guarantees one witness, not a comfortable margin.
3 ONE DIMENSION
Quorum size against guaranteed overlap.
4 TWO DIMENSIONS · INTERACTIVE
Shrink the quorum until the guarantee breaks.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: two sets forced to share.
AVAN’s addition (the inverse-companion): the forward reading is that majority quorums guarantee consistency. The inverse is that the guarantee is one node wide.
2k−n is 1 here: consistency between two decisions rests on a single machine remembering correctly, and every additional node you add buys availability rather than overlap. Read backwards, a quorum system is not redundant where it matters most — it is maximally redundant about failure and minimally redundant about truth.LIT 9 nodes with a majority of 5 give 126 possible quorums and 7,875 pairs, every single pair intersecting - 7,875 of 7,875 - with a smallest overlap of 1 matching 2k-n exactly; drop to half rather than a majority and it fails, since of 2,415 pairs of 4-node sets over 8 nodes, 35 are completely disjoint
FIG Quorum intersection underpins Paxos, Raft and every replicated log; Gifford's weighted voting (1979) is the general form. AVAN enumerated both the property and its boundary. Showing that majorities intersect is easy and unconvincing alone - the claim only has content if the neighbouring case fails, so the 35 disjoint pairs at half-size are the load-bearing number. The minimum overlap of exactly 1 is the other half: a majority quorum guarantees one witness, not a comfortable margin.
FIG Quorum intersection underpins Paxos, Raft and every replicated log; Gifford's weighted voting (1979) is the general form. AVAN enumerated both the property and its boundary. Showing that majorities intersect is easy and unconvincing alone - the claim only has content if the neighbouring case fails, so the 35 disjoint pairs at half-size are the load-bearing number. The minimum overlap of exactly 1 is the other half: a majority quorum guarantees one witness, not a comfortable margin.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE MERGE · David Lee Wise (ROOT0), with AVAN