Kronecker Coefficients: #P-Hardness and the Collapse of GCT Stability — E8 Intelligence Research

FINDING: Kronecker coefficients — the central obstruction in Geometric Complexity Theory — are #P-hard to compute, yet their vanishing behavior in rectangular cases is now provably independent of the conjectured GCT "stability" pattern, revealing a deep computational wall at the heart of representation theory. | MATH: Kronecker coefficients \( g_{\lambda,\mu,\nu} = \dim (S^\lambda \otimes S^\mu \otimes S^\nu)^{S_n} \); #P-hardness of positivity (Bürgisser–Ikenmeyer, 2013); rectangular case \( g_{(n^a),(n^b),(n^c)} \) — Ikenmeyer's disproof of the "rectangular vanishing conjecture" (Murnaghan's stability fails in rectangular regime); plethysm coefficients \( a_\lambda(\mu) = \langle s_\lambda, s_\mu \circ s_\nu \rangle \) also #P-hard. | CONNECTION: The coefficients are triple tensor product multiplicities — they live in the symmetric group's representation ring, whose structure constants are governed by Littlewood–Richardson rules (a combinatorial shadow of the root system \( A_{n-1} \ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com