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: \(a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0 = 0\), \(a_i \in \mathbb{Z}\). For the Rogers–Ramanujan continued fraction \(R(q) = \frac{q^{1/5}}{1 + \frac{q}{1 + \frac{q^2}{1 + \cdots}}}\), its algebraic conjugates under the modular group are roots of a quartic with coefficients in \(\mathbb{Z}[q]\). Specifically, at \(q = e^{-2\pi}\), \(R(q)\) satisfies \(x^4 - 2x^3 - 6x^2 + 2x + 1 = 0\) (a known Ramanujan result), whose roots are \(x = \frac{1 \pm \sqrt{5} \pm \sqrt{2(5 \pm \sqrt{5})}}{2}\) — 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