Information Complexity: Geometric Bounds and Computable Limits in Communication — E8 Intelligence Research

FINDING: Information complexity provides exact, computable bounds for communication problems, revealing a geometric structure in the space of protocols. | MATH: Information complexity \(IC(f,\mu) = \inf_{\pi} I(XY; \Pi | \mu)\) — the minimum mutual information between inputs and transcript over all protocols; exact bounds via "information cost" duality: \(IC(f,\mu) = \sup_{\text{priors}} \ldots\) (Braverman's computability result); connection to amortized communication complexity: \(\lim_{n\to\infty} \frac{1}{n} CC(f^n) = IC(f)\) — a rate-like limit, analogous to Shannon entropy \(H(X)\) as the fundamental compression limit. | CONNECTION: The infimum over protocols forms a convex polytope in the space of information-cost vectors; the duality (sup over priors) mirrors Legendre-Fenchel transforms — the same geometry as root systems (e.g., \(A_n\) lattice) where distances are measured by mutual information, not Euclidean norm. The ratio \(IC/CC\) for specific functions (e.g., equality, in Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com