Symmetric chains in noncrossing partition lattices

Every noncrossing partition lattice of a finite well-generated complex reflection group admits a symmetric chain decomposition. We give a computer-assisted proof for the exceptional types $H_4$, $G_{33}$, $G_{34}$, $E_6$, $E_7$, and $E_8$, completing the saturated-chain assertion asked by Henri Mühle. Simion and Ullman, Reiner, Bessis and Corran, and Mühle established the other irreducible cases. Explicit chain certificates are checked against exact reflection-group models; exhaustive rank descent certifies that their graphs are the full absolute-order intervals. Reflection-preserving automorphisms and a constructive product decomposition give every Coxeter element and every reducible group. Lean verifies the general interval-closure, product, and transfer lemmas. The finite group calculations and the published classification and decomposition results remain external inputs.

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