The cusp fibre W of Alpöge’s complex structure on the six-sphere is the [3.4.6.4] Laves tiling rolled up on a torus T = R2 /Λ, Λ = B0 Z2 , and at the proportions of the Hat that tiling is exactly the substrate of the aperiodic monotile: hexagon centres are components, edge midpoints double curves, hexagon vertices triple points. No monotile tiling descends to T, since an aperiodic tiling is invariant under no lattice. We fold a Hat tiling generated by substitution onto T anyway. Every tile is a union of whole kites, so the fold is well defined, and we compute its fibres. The "mystic" tile (the Turtle, ten kites) lands on exactly six points of the torus in the patches of three and four substitution steps: one per orientation, each anchored on the midpoint of a double curve of W , two orientations per curve. The six classes form one orbit of the rotation group of a hexagon, and six is the number of oriented edges of the dual complex of W , which we show is the torus with one vertex, three edges and two triangles. We prove that at most 72 classes are possible, state six as a conjecture, and show by the boundary statistics of the folded tiling that equal fibres do not mean equal surroundings. The paper also records why the lattice data behind the construction is forced, that its monodromy is achiral and its lattice not self-dual over Z, and three exact contacts with standard objects (a polytope volume, a Kodaira configuration, and central charges).