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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 $\\sum_{k\\ge1} \\lambda_\\alpha^{k-1} f^{(r)}(k)$. The coefficients satisfy an effective recurrence in ordinary zeta values. The unique unweighted specialization $\\alpha = \\pi/6$ proves Conjecture 4.1 of Zhi-Wei Sun; at the fourth order a depth-two value $\\mathrm{Gl}_{4,1}(\\pi/3)$ occurs. 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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 $\\sum_{k\\ge1} \\lambda_\\alpha^{k-1} f^{(r)}(k)$. The coefficients satisfy an effective recurrence in ordinary zeta values. The unique unweighted specialization $\\alpha = \\pi/6$ proves Conjecture 4.1 of Zhi-Wei Sun; at the fourth order a depth-two value $\\mathrm{Gl}_{4,1}(\\pi/3)$ occurs. 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