THE FOLD / CO-OP / SPLIT-SCREEN / THE STABLE ROOMMATES
THE STABLE ROOMMATES
the co-op with no settlement
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Gale and Shapley proved in 1962 that the marriage problem — two groups, ranked preferences — ALWAYS has a stable matching. Then they flipped one structural bit: what if everyone lives in ONE group, pairing off as roommates? The guarantee dies. There are preference profiles where every possible pairing has a blocking pair — two people who would both rather dump their partners for each other. The classic witness needs only four people: three who cyclically prefer each other and a fourth nobody wants. No algorithm can find what doesn’t exist; Irving’s 1985 algorithm decides existence, but existence itself is no longer promised.
LIT verified live: the 4-agent cyclic witness — all 3 perfect matchings checked, each has a blocking pair, no stable matching exists (exact); random 4-agent instances go unstable-free about 3.6% of the time (4,000 sampled); and the marriage CONTROL: 2,000 random 3+3 bipartite instances — every single one has a stable matching, exhaustively confirmed (window.__roommates). FIG Gale–Shapley 1962 and Irving 1985 cited; the ~3.6% is a measured rate for n=4 uniform preferences, not a universal constant.
LIT verified live: the 4-agent cyclic witness — all 3 perfect matchings checked, each has a blocking pair, no stable matching exists (exact); random 4-agent instances go unstable-free about 3.6% of the time (4,000 sampled); and the marriage CONTROL: 2,000 random 3+3 bipartite instances — every single one has a stable matching, exhaustively confirmed (window.__roommates). FIG Gale–Shapley 1962 and Irving 1985 cited; the ~3.6% is a measured rate for n=4 uniform preferences, not a universal constant.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at split-screen — the co-op: two players on one couch works because the screen SPLITS — a bipartite structure. Merge everyone onto one side and some parties can never settle: the co-op mode that cannot always be played. AVAN (AI) built the instrument: the exhaustive blocking-pair auditor and the bipartite control.
Credit as content: David Gale & Lloyd Shapley (1962); Robert Irving (1985); Tan (the characterization). The weave: David names the couch that won’t split; I check every pairing and the quarrel never ends.
Credit as content: David Gale & Lloyd Shapley (1962); Robert Irving (1985); Tan (the characterization). The weave: David names the couch that won’t split; I check every pairing and the quarrel never ends.
3 ONE DIMENSION
Four people, three pairings, three blocking pairs — no rest.
4 TWO DIMENSIONS · INTERACTIVE
Cycle the three pairings; each one's blocking pair lights up.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the cyclic triangle spinning, the fourth outside.
AVAN’s addition (the inverse-companion): don’t blame the preferences — blame the topology. The inverse of ‘stability is about what people want’ is ‘stability is about how the wanting is WIRED’: identical desires, bipartite wiring, always settles; one-sided wiring, sometimes never. Magenta is the endless cycle of defections; green is the split screen that would have saved them. Some conflicts are unsolvable only because of the shape of the room.
LIT Verified live: the 4-agent cyclic witness — all 3 perfect matchings have a blocking pair, no stable matching exists (exact); ~3.6% of 4,000 random 4-agent instances are unstable-free; the marriage control: 2,000 random 3+3 bipartite instances ALL have a stable matching, exhaustively (window.__roommates.ok).
FIG Gale–Shapley 1962, Irving 1985 cited; the 3.6% is a measured n=4 rate, not a constant. The AVAN inverse — blame the topology, not the preferences: identical desires settle under bipartite wiring and cycle forever without it. Magenta is the endless defection cycle; green is the split screen that would have saved them. Some conflicts are unsolvable because of the shape of the room.
FIG Gale–Shapley 1962, Irving 1985 cited; the 3.6% is a measured n=4 rate, not a constant. The AVAN inverse — blame the topology, not the preferences: identical desires settle under bipartite wiring and cycle forever without it. Magenta is the endless defection cycle; green is the split screen that would have saved them. Some conflicts are unsolvable because of the shape of the room.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SPLIT-SCREEN · David Lee Wise (ROOT0), with AVAN