Multiplicative Composition Lower Bounds via Information Complexity in Circuit Complexity — E8 Intelligence Research

FINDING: The KRW conjecture seeks to prove that circuit complexity composes multiplicatively under function composition — i.e., the complexity of \(f \circ g\) is at least the product of their individual complexities, a structural lower-bound principle for Boolean circuits. | MATH: The conjecture states \(D(f \circ g) \approx D(f) + D(g)\) for depth, but the stronger form for size/complexity is \(C(f \circ g) \geq C(f) \cdot C(g)\) (up to logarithmic factors). Key tool: *information complexity* — \(IC(f \circ g) \geq IC(f) \cdot IC(g)\) under product distributions. Lifting theorems map communication complexity lower bounds to circuit lower bounds: \(CC(f \circ \text{Index}) \geq CC(f) \cdot \log n\). | CONNECTION: The multiplicative structure mirrors the golden-ratio-like scaling in self-similar compositions — if \(C(f)=a\), then \(C(f^{\circ k}) \geq a^k\), echoing exponential growth rates tied to \(1.618^k\) in optimal recursive partitions. The "strong composition" variant (Meir 2024 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com