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We define the notion of a weighted unfolding of quivers with real weights,and use this to provide a categorification of mutations of quivers of finitetypes $H_4$, $H_3$ and $I_2(2n+1)$. In particular, the (un)folding induces asemiring action on the categories associated to the unfolded quivers of types$E_8$, $D_6$ and $A_{2n}$ respectively. We then define the tropical seedpattern on the folded quivers, which includes $c$- and $g$-vectors, and showits compatibility with the unfolding.

Categorifications of Non-Integer Quivers: Types $H_4$, $H_3$ and $I_2(2n+1)$