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We provide a complete proof for a conjectured continued fraction identity catalogued by the Ramanujan Machine project. The continued fraction is defined by\(a_0 = 4,\; a_n = 3n^2+7n+4,\; b_n = -2n^2(n+1)^2\;(n\ge 1),\)and we prove that its limit is exactly \(\frac{1}{1-\ln 2}\). The proof establishes a closed form for the numerators \(P_n\), expresses the convergents via a telescoping sum, and evaluates the resulting infinite series using elementary integration.