A continuous $3$-distributive frame that is not $ω$-distributive
We give a negative answer to the question, posed by Erné, whether every $3$-distributive lattice is $ω$-distributive. More precisely, we exhibit a continuous frame that is $κ$-distributive for every integer $κ\geq 2$, but is not a wide coframe. The frame is the open-set lattice of a compact, locally compact, countably based $T_0$ topological meet-semilattice, obtained from Lawson's construction in the logarithmic form described by Goubault-Larrecq. The failure of $ω$-distributivity is witnessed by an explicit matrix with countably many nonempty finite rows: all row joins are the same nonzero element, whereas every choice of one entry from each row has meet zero. The same space answers negatively Erné's accompanying question whether every $4$-web space is a wide web space. All properties of the construction needed for these conclusions are proved directly.