Multisum q-Binomial Expansions of Affine Kac-Moody Characters and Motivated Rogers-Ramanujan Proofs — E8 Intelligence Research

FINDING: Affine Kac-Moody characters admit multisum q-binomial expansions encoding partition difference conditions; Lepowsky–Wilson "motivated proofs" realize Rogers–Ramanujan/Gordon/Andrews–Bressoud identities as vertex-operator traces. MATH: - Principal character (Weyl–Kac denominator identity), affine ĝ: χ_Λ(τ) = ∑w∈ W ε(w)\, ew(Λ+ρ)∏α>0(1-e-α)mult\,α - Specialized to q = e2π i τ, characters become q-series with multisum form (Lepowsky–Wilson): χ ~ ∑n₁,,nₖ ≥ 0 q\,Q(n₁,,nₖ)∏ᵢ (q)nᵢ, with (q)ₙ = ∏ⱼ₌₁ⁿ (1-qʲ). - Difference conditions (Gordon/Andrews–Bressoud): partitions into parts ≡ 0, ± i 2k+1 (odd moduli) or ≡ 0, ± i 2k (even moduli). - Lepowsky–Wilson: character = trace of vertex operator $Y(v,z)$ on a standard module; the "ghost series" is the alternatin Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com