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 k₆ = 2sqrt (3) + sqrt (6) - 2sqrt (2) - 3 (approximately 0. 08516) satisfies K' (k₆) /K (k₆) = 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 k⁴ + 12k³ + 2k² - 12k + 1 over Q and lie in the biquadratic field Q (sqrt (2), sqrt (3) ). The derivation is entirely algebraic: k₆ 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 (k₆, k₂/₃). These forms do not appear in the standard tables of Borwein-Zucker, Borwein-Borwein, or the Berndt-Ramanujan Notebooks.