Shared Quartic Reveals Hidden Symmetry in Ramanujan Continued Fractions — E8 Intelligence Research

FINDING: Two distinct algebraic continued fractions of Ramanujan satisfy the *same* quartic polynomial with integer coefficients, revealing hidden algebraic conjugacy and modular symmetry. | MATH: Let the two continued fractions be \(R(q)\) and \(S(q)\). They satisfy a shared quartic: \(a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0 = 0\), \(a_i ∈ Z\). For the Rogers–Ramanujan continued fraction \(R(q) = {q1/5}{1 + {q}{1 + q^2/1 + ⋯}}\), its algebraic conjugates under the modular group are roots of a quartic with coefficients in \(Z[q]\). Specifically, at \(q = e-2π\), \(R(q)\) satisfies \(x^4 - 2x^3 - 6x^2 + 2x + 1 = 0\) (a known Ramanujan result), whose roots are \(x = {1 ± √5 ± √{2(5 ± √5)}}{2}\) — all algebraic conjugates. The two continued fractions in the video are likely \(R(q)\) and a related fraction \(R(q^2)\) or \(R(-q)\), sharing the same quartic via modular transformations. | CONNECTION: The quartic's Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com