Weighted Derivative Sums of a Gamma Quotient: Sun's Conjecture and Cyclotomic Specializations

Let f(x) = Γ(x)²/(2Γ(2x)) and set λ_α = 4sin²α for 0 < α < π/2. We establish an elementary parameter identity for a weighted translate of f and derive an explicit formula, valid at every derivative order, for the associated weighted sums. The coefficients satisfy an effective recurrence in ordinary zeta values. The unique unweighted specialization α = π/6 proves Conjecture 4.1 of Zhi-Wei Sun; at the fourth order a depth-two value Gl4,1(π/3) occurs. The construction complements general cyclotomic-multiple-zeta methods for inverse-binomial harmonic sums by supplying a continuous master identity, with concrete specializations at α = π/4 and α = π/3.