Permanent-Determinant Gap, Kronecker Bounds, and Binary Form Enigmas — E8 Intelligence Research

FINDING: Orbit closure separation for permanent vs. determinant remains open; Kronecker coefficients lack closed form but admit partition-algebra bounds; OEIS A000003 (binary quadratic forms) resists closed form yet ties to elliptic integrals and Klein's J-invariant. | MATH: Permanent–determinant separation: \(det_n̄ ⊆ perm_m̄\) for \(m = poly(n)\) — unproven; Kronecker coefficients \(gλ,μ,ν = HomS_k(V^λ, V^μ ⊗ V^ν)\) — no positive combinatorial formula; partition algebra bound: \(gλ,μ,ν ≤ (Part_k(n))\) with \(Part_k(n)\) dimension given by Bell numbers \(B_k(n)\); A000003: \(a(n) = #\{classes of positive definite binary quadratic forms of discriminant -4n\}\), related to \(J(τ)\) via \(∑ a(n) q^n = {1}{J(τ)1/3 + 432}\) (Ramanujan's cubic continued fraction connection). | CONNECTION: Kronecker coefficients encode tensor decompositi Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com