A Novel Complex Constant Derived from the Golden Ratio and its Transcendence Conjecture

A research paper by Tristen Harr introduces and analyzes a new complex constant, ΛG1, derived from inverse powers of the golden ratio, ϕ. Defined as ΛG1 = T + iJ, where T = 1/(2ϕ) and J = 1/(2ϕ²), it's proven to be an algebraic number with a magnitude less than one, suitable as an argument for the Polylogarithm function, Lis(z). High-precision numerical evaluations for Dilogarithm (s=2) and Trilogarithm (s=3) suggest Lis(ΛG1) is transcendental for all integers s≥2 and lies outside the field extension Q(π, ln(2), ϕ). This research is partly motivated by potential applications in quasicrystal studies, where the golden ratio is fundamental.