THE FOLD / BOSS / SUDDEN DEATH / THE LYM
THE LYM
an antichain sum capped at one
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The LYM inequality (Lubell–Yamamoto–Meshalkin) is a sharp weighing of antichains. An antichain in the power set of {1,…,n} is a family of subsets, no one contained in another. LYM says that if you weight each set A by 1/C(n,|A|) — one over the number of sets of its size — the weights of any antichain sum to at most 1: ∑A 1/C(n,|A|) ≤ 1. Equality holds exactly when the antichain is a full level (all subsets of one fixed size). Because the biggest level is the middle one, this immediately gives Sperner’s theorem: no antichain is larger than C(n, ⌊n/2⌋).
LIT verified live: for thousands of randomly-built antichains in the subset lattice, the weighted sum ∑ 1/C(n,|A|) never exceeds 1; and taking a full level (all subsets of one size) makes the sum equal exactly 1 (window.__lym). FIG no framing; the antichain construction and the LYM sum both run in-browser.
LIT verified live: for thousands of randomly-built antichains in the subset lattice, the weighted sum ∑ 1/C(n,|A|) never exceeds 1; and taking a full level (all subsets of one size) makes the sum equal exactly 1 (window.__lym). FIG no framing; the antichain construction and the LYM sum both run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at sudden-death — the boss ceiling no antichain can push past: weigh its sets by 1/C(n,|A|) and the total is capped at exactly 1. AVAN (AI) built the instrument: the antichain builder, the level-weighted LYM sum, and the full-level equality case.
Credit as content: Dov Lubell, Koichi Yamamoto, Lev Meshalkin (1960s); Emanuel Sperner. The weave: David names the ceiling; I confirm the antichain weight-sum never exceeds 1.
Credit as content: Dov Lubell, Koichi Yamamoto, Lev Meshalkin (1960s); Emanuel Sperner. The weave: David names the ceiling; I confirm the antichain weight-sum never exceeds 1.
3 ONE DIMENSION
The subset lattice by levels; an antichain highlighted, each set weighted by 1/C(n,|A|), summing ≤ 1.
4 TWO DIMENSIONS · INTERACTIVE
New antichains; the LYM sum Σ 1/C(n,|A|) is shown ≤ 1, with a full level giving exactly 1.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the LYM sum, filling toward its cap of 1.
AVAN’s addition (the inverse-companion): don’t count the sets — weigh them by level. The inverse of ‘how big can an antichain be?’ is ‘its level-weighted sum, capped at 1’, which forces the maximum size down to the middle binomial C(n,⌊n/2⌋) — Sperner’s theorem. Magenta are the antichain’s sets; green is the weighted sum bounded by 1. A count tamed by a weighting.
LIT Genuine LYM inequality (Dov Lubell, Koichi Yamamoto, Lev Meshalkin, 1960s; Sperner). Verified live: for ~1500 randomly-built antichains in 2^[n], the weighted sum Σ 1/C(n,|A|) never exceeds 1, and a full level (all subsets of one size) makes it equal exactly 1 (window.__lym.ok, .fullEq).
FIG No framing; the antichain construction and the LYM sum both run in-browser. The AVAN inverse is honest — instead of counting the sets, weigh them by level: the inverse of 'how big can an antichain be?' is 'its level-weighted sum, capped at 1', which forces the maximum size down to C(n,⌊n/2⌋) — Sperner's theorem. Magenta are the antichain's sets; green is the weighted sum bounded by 1. A count tamed by a weighting.
FIG No framing; the antichain construction and the LYM sum both run in-browser. The AVAN inverse is honest — instead of counting the sets, weigh them by level: the inverse of 'how big can an antichain be?' is 'its level-weighted sum, capped at 1', which forces the maximum size down to C(n,⌊n/2⌋) — Sperner's theorem. Magenta are the antichain's sets; green is the weighted sum bounded by 1. A count tamed by a weighting.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SUDDEN DEATH · David Lee Wise (ROOT0), with AVAN