Saturation Theorem Links LR-Positivity to Linear Programming and GCT Lower Bounds — E8 Intelligence Research

FINDING: The Knutson-Tao saturation theorem proves that Littlewood-Richardson coefficients (LR-coefficients) are positive if and only if their scaled versions are positive, reducing LR-positivity to linear programming (P), with direct implications for Geometric Complexity Theory (GCT) lower bounds. | MATH: LR-coefficients \( c_{\lambda,\mu}^{\nu} \) for \( GL_n(\mathbb{C}) \); Saturation: \( c_{\lambda,\mu}^{\nu} > 0 \iff c_{N\lambda,N\mu}^{N\nu} > 0 \) for all \( N \in \mathbb{N} \); Equivalently, the semigroup of triples \( (\lambda,\mu,\nu) \) with positive coefficients is saturated; Deciding positivity is in P via LP (Knutson-Tao, 1999; Bürgisser-Ikenmeyer, 2005). | CONNECTION: The saturation theorem is geometrically realized via **honeycomb patterns** — these are planar tilings by hexagons with edge labels satisfying triangle inequalities, directly encoding the **root system \( A_2 \)** and its Weyl chamber. The honeycomb structure is a discrete manifestation of the **base-60** (s Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com