Doubly-Infinite Rogers-Ramanujan Family from A₂ₙ⁽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 A₂ₙ⁽²⁾, generalising Andrews-Gordon. MATH: - Classical RR (two identities): Σk≥0 qk²/(q;q)_k = 1/((q;q^5)_∞(q^4;q^5)_∞) Σk≥0 qk²+k/(q;q)_k = 1/((q²;q^5)_∞(q³;q^5)_∞) - Andrews-Gordon: Σ q^{k₁²+...+kₘ₋₁²+k_i+...+kₘ₋₁} / ∏(q;q)k_i = ∏j≠0,±i mod 2m+1 1/(1-q^j) - A₂ₙ⁽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 A₂ₙ⁽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