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 $\hat{\mathfrak{g}}$: $\chi_\Lambda(\tau) = \dfrac{\sum_{w\in W} \varepsilon(w)\, e^{w(\Lambda+\rho)}}{\prod_{\alpha>0}(1-e^{-\alpha})^{\mathrm{mult}\,\alpha}}$ - Specialized to $q = e^{2\pi i \tau}$, characters become $q$-series with multisum form (Lepowsky–Wilson): $\chi \sim \sum_{n_1,\dots,n_k \ge 0} \dfrac{q^{\,Q(n_1,\dots,n_k)}}{\prod_i (q)_{n_i}}$, with $(q)_n = \prod_{j=1}^n (1-q^j)$. - Difference conditions (Gordon/Andrews–Bressoud): partitions into parts $\not\equiv 0, \pm i \pmod{2k+1}$ (odd moduli) or $\not\equiv 0, \pm i \pmod{2k}$ (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