Rogers-Ramanujan Identities via Motivated Proofs and Mod-5 Congruence — E8 Intelligence Research

FINDING: The Rogers-Ramanujan identities are proven via the Andrews–Baxter motivated proof, and the mod-5 partition congruence is a direct corollary of the Dedekind eta product structure. | MATH: The two identities: ∑_{n≥0} q^{n²}/(q;q)_n = 1/∏_{k≥0}(1−q^{5k+1})(1−q^{5k+4}) ∑_{n≥0} q^{n(n+1)}/(q;q)_n = 1/∏_{k≥0}(1−q^{5k+2})(1−q^{5k+3}) where (q;q)_n = ∏_{j=1}^n (1−q^j). Ramanujan's congruence: p(5n+4) ≡ 0 (mod 5). Eta product form: ∏_{k≥0}(1−q^{5k+r}) = q^{−1/24} η(5τ)/η(τ) for r=1,4 and r=2,3 respectively. | CONNECTION: The modulus 5 and the exponents {1,4} and {2,3} are the two quadratic residue pairs mod 5. These pairs correspond to the 5-cycle structure of the affine A₄ root system — the same symmetry that underlies the golden ratio (φ = (1+√5)/2) via the 5-fold crystallographic restriction. The product sides are characters of the affine Lie algebra A₄⁽¹⁾ at level 1, whose Weyl vector has norm ∝ 1/√5, linking to φ. The base-5 structure is a discrete analogue of the golden Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com