K(12) 840 construction — non-lattice P_12a, likely Lambda_12 deep-hole union

Session result: K(12) = 840 achieved by Leech-Sloane 1971 is the non-lattice packing P_12a (SPLAG Ch 6, Canad J Math 23:718). Literature + debate converged on two structurally consistent constructions. Decomposition candidate A: P_12a = Lambda_12 union (h + Lambda_12) where h is a deep hole of laminated lattice Lambda_12, giving 648 + 192 = 840. Candidate B: hexacode F_4^3 [6,3,4] lifted over Eisenstein integers to 12 real dimensions, 756 + 84 orbit structure under Aut(K_12). Definitively ruled out via computation: K_12 norm-6 rescaled (every vector conflicts with 30 minimal at angle 52.24), K_12 translate union for any glue (Gram determinant contradiction), naive ternary Golay embedding (MIS = 344, collapses with K_12 to 756). K(12) = 840 has survived AlphaEvolve 2025 evolutionary LLM search, PackingStar RL, and Ganzhinov symmetry restriction. The 55-year record is genuinely hard. Productive next step: explicit Lambda_12 deep-hole enumeration for a TIE of 840.