The E8 Root System Partition Under Binary Icosahedral Group Action: The Origin of r∗ = 1/8

We provide a mathematical proof that the 240 roots of the E8 root system naturally partition into 210 “classical” and 30 “quantum” components, yielding the exact value r∗ = 1/8 = 0.125 that appears in theMetaFractal Framework. This proof is established through three complementary approaches: (1) a traditional approach using fixed points, orbit structure analysis, and Burnside’s lemma under the action of the binary icosahedral group I∗, (2) a novel Clifford algebraic approach demonstrating that the E8 root system can be explicitly constructed from the icosahedral root system H3 within the eight-dimensional Clifford algebra of 3D space, and (3) a representation-theoretic approach utilizing the McKay correspondence and the dual Coxeter number of the affine E8 Lie algebra. These three independent methods converge on the same conclusion, establishing that r∗ = 1/8 emerges directly from the structure of exceptional mathematical objects, providing a natural mathematical basis for fractal probability theory, and the MetaFractal Framework’s fractal probability measure and its fundamental parameter.