Explicit radical forms for the elliptic singular values k6 and k(2/3) via Landen's transformation

We derive explicit radical expressions for two elliptic singular values. The integer singular value k_6 = 2sqrt(3) + sqrt(6) - 2sqrt(2) - 3 (approximately 0.08516) satisfies K'(k_6)/K(k_6) = sqrt(6), and the rational singular value k2/3 = 2sqrt(2) + 2sqrt(3) - sqrt(6) - 3 (approximately 0.84304) satisfies K'(k2/3)/K(k2/3) = sqrt(2/3). Both values share the minimal polynomial k^4 + 12k^3 + 2k^2 - 12k + 1 over Q and lie in the biquadratic field Q(sqrt(2), sqrt(3)). The derivation is entirely algebraic: k_6 is obtained by solving a palindromic quartic via the substitution v = k - 1/k, and k2/3 follows from a single application of Landen's ascending transformation. A sign-exchange symmetry between the two expressions is explained structurally. We further show that applying Landen's transformation to k2/3 yields k8/3, whose minimal polynomial has degree 8 with large coefficients and whose radical form requires nested square roots — establishing that the compact linear form in Q(sqrt(2), sqrt(3)) is a special feature of the pair (k_6, k2/3). These forms do not appear in the standard tables of Borwein-Zucker, Borwein-Borwein, or the Berndt-Ramanujan Notebooks.