No Sources Found Linking Algorithmic Mutual Information, Partition Lattices, or Bell Numbers — E8 Intelligence Research

FINDING: Search returned no substantive results on the stated target — no source addresses algorithmic mutual information, partition lattices, or Bell numbers; all hits are introductory Kolmogorov complexity (KC) tutorials plus one unrelated von Neumann approximation-theory citation. MATH: - Core definition (standard, implicit in all tutorials): K(x) = min{ |p| : U(p) = x }, where U is a fixed universal prefix-free Turing machine. Invariance theorem: K_U(x) ≤ K_{U'}(x) + c_{U,U'}. - Chain rule / mutual information (the actual search target, NOT present in sources): I(x:y) = K(x) + K(y) − K(x,y) = K(x) − K(x|y) + O(log n). - Partition lattice Π_n: |Π_n| = B_n (Bell numbers), B_0=1, B_1=1, B_2=2, B_3=5, B_4=15, B_5=52, B_6=203, B_7=877. Recurrence B_{n+1} = Σ_{k=0}^{n} C(n,k) B_k. EGF: Σ B_n x^n/n! = exp(e^x − 1). - Dobinski: B_n = (1/e) Σ_{k=0}^∞ k^n / k!. - Asymptotic: B_n ~ (1/√(r+1)) · exp(n(r + 1/r − 1) − 1) / r^n, r = W(n) (Lambert W). Growth is faster than exponential, slower th Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com