Semilattice Ordering and Asymmetric Trees: A000220 via Coxeter–Dynkin Symmetries — E8 Intelligence Research

FINDING: Join/meet semilattices formalize ordinal pooling as a partial order with unique least upper bounds; asymmetric trees (OEIS A000220) count identity trees with no nontrivial automorphisms, linking to root system A_n via Coxeter–Dynkin symmetries. | MATH: Join-semilattice: ∀a,b: ∃! sup{a,b} = a∨b; meet-semilattice: ∃! inf{a,b} = a∧b. Asymmetric trees: number of unlabeled rooted trees with trivial automorphism group — A000220(n) = A000081(n) − (symmetric contributions); asymptotic: A000220(n) ~ c·α^n·n^(−5/2), α ≈ 2.955765 (Otter's constant for rooted trees), c ≈ 0.5178. Root system A_n: positive roots = n(n+1)/2; Weyl group = S_{n+1}; Cartan matrix diag(2), off-diag(−1). | CONNECTION: Semilattice join operation mirrors the positive root poset of A_n (root α_i + α_j = α_{i+j} when i+j ≤ n) — a join-semilattice with meet undefined for non-comparable roots. Asymmetric trees correspond to orbits of the symmetric group S_n acting on tree labelings — trivial stabilizer = no symmetry, e Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com