A 6-bit vertex splits into two 3-bit halves, giving an 8x8 grid. Make that grid periodic in both directions and it is a torus. The flat vertex codex was always a torus cut open and laid flat — the Improbable Engine closes it.

This makes the engine the third seating of one lattice: the codex reads it flat, one vertex at a time; the star manifold walks it; the torus shares it. The torus seating has no boundary row and no privileged corner — a lattice with no extraction point, which is the festival theme performed by the geometry rather than illustrated.

Every element in the engine holds both seatings at once, and one fold parameter lerps between them — so the torus-to-codex morph carries the whole piece: gems, grid, strands, page-stars, link arcs, hearth threads, walked trails. The seam edges are real on the torus and cut on the codex; they fade as it opens.

The geometry register is inherited from the wider canon: value accrues on walked paths, the boundary seal is taken over edges, and the golden angle drives the ambient drift. Honesty discipline applies throughout — what is architectural, what is conjectural, and what is aesthetic are labelled in the plans that travel with the bundle.