Doubly-Infinite Rogers-Ramanujan Family from A_{2n}^{(2)} Characters — E8 Intelligence Research

FINDING: Rogers-Ramanujan identities admit a doubly-infinite family indexed by (m,n) whose product sides are specialised characters of affine Kac-Moody algebra A_{2n}^{(2)}, generalising Andrews-Gordon. MATH: - Classical RR (two identities): Σ_{k≥0} q^{k²}/(q;q)_k = 1/((q;q^5)_∞(q^4;q^5)_∞) Σ_{k≥0} q^{k²+k}/(q;q)_k = 1/((q²;q^5)_∞(q³;q^5)_∞) - Andrews-Gordon: Σ q^{k₁²+...+k_{m-1}²+k_i+...+k_{m-1}} / ∏(q;q)_{k_i} = ∏_{j≠0,±i mod 2m+1} 1/(1-q^j) - A_{2n}^{(2)} family: product side = specialised affine character at level ℓ; sum side = q-binomial/multisum over partitions with difference conditions. - Key structural constant: modulus 2m+1 (odd), and 5 = 2·2+1 for classical RR. CONNECTION: - Modulus 5 in classical RR connects to φ = (1+√5)/2 via q → e^{-2π/5} evaluations (Ramanujan's continued fraction, Hardy-Ramanujan). - Affine A_{2n}^{(2)} root systems are crystallographic (type A, rank 2n+1); Weyl group symmetries and level-ℓ characters encode lattice/VOA structure. - Differen Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com