Finish classifying the extension property for partial automorphisms (EPPA) and coherent EPPA for the five reducts of the random graph. The open cases were two-graphs, graphs modulo global complementation, and two-graphs modulo global complementation, answering a question of Evans, Hubička, Konečný and Nešetřil.

A six-vertex obstruction with two complementing partial automorphisms, combined with a fixed-point counting lemma for permutation groups, shows that graphs modulo complementation lack EPPA. Rooting two-graphs reduces the two-graph complementation case to the graph case. Coherent EPPA for two-graphs is proved with a modified valuation-function construction that quotients at the level of valuation functions and uses a large block-divisible index set. The main results were formalised in Lean 4, mostly autonomously by AI.

Finite two-graphs have coherent EPPA. Neither graphs modulo global complementation nor two-graphs modulo global complementation have EPPA. This completes the classification for the five reducts.