Cosmological Pangaea Object: Stage 5

Abstract

Stage V is the naming of the three sheets. Earlier stages showed that the flag move system of the Cosmological Pangaea Object splits into three connected components, but those components were still known mainly by how they appeared in the computation. This stage asks what each sheet is as geometry. It finds that every sheet lives on sixteen of the twenty-four vertices of the twenty-four-cell and leaves eight vertices out; those three omitted eight-vertex sets are disjoint, cover the whole vertex set, and each is an inscribed sixteen-cell. The three sheets are therefore the three complementary sixteen-cells of the Garden, labeled by the three ways to pair four coordinates. Along the way, Stage V also clears a shadow left by Stage IV: the apparent one-sheet four-valued lift was an artifact of an enumeration-based orientation, not a real distinction among the sheets. With geometric orientations restored, the sheets remain interchangeable, and what they gain is a name. Stage V ends when the three sheets are no longer anonymous components, but complementary sixteen-cells with a geometric address.