The First Eight-Point Cut Boundary Is Forced
The first higher-multiplicity gate can be separated into two logically distinct questions: what the six-by-four boundary must be, and whether an eight-point interior class restricts to it. This entry settles the first question without pretending that the second has already been posed by an available class.
Fix the octagon diagonal (D_{05}). It cuts the octagon into the hexagon ((0,1,2,3,4,5)) and quadrilateral ((0,5,6,7)). The diagonals compatible with the cut split canonically as [ 9+2=11, ] so the link of (D_{05}) is exactly the join of the hexagon and quadrilateral noncrossing-diagonal complexes. Its face counts by residual diagonal number are [ (1,11,39,56,28). ] This is an explicit bijection, not merely an equality of cardinalities.
Loading every face by a subset of its diagonals gives [ 1075=215\cdot5 ] boundary cells. With degree (4-|F|+|H|), their chain ranks are [ (28,168,375,369,135), ] the convolution of the established six-point ranks ((14,63,93,45)) with the four-point ranks ((2,3)).
There is a mild orientation issue: lexicographic octagon order interleaves the two factors. The checker computes the canonical left/right shuffle sign on every face. After that reorientation, all 369 admissible radial incidences agree exactly with the tensor-product incidence signs. Thus no hidden sign obstruction occurs at the carrier boundary.
Entry 436 supplies a primitive oriented six-point line, while the oriented four-point interval supplies its primitive unit. Their external product is therefore the forced primitive (+1) line on this boundary. This is the right-hand side of the first Cut-naturality square.
The important negative control is equally sharp: no eight-point interior transform or physical class has yet been constructed. Consequently this calculation does not prove Cut naturality. It fixes the unique boundary target against which such a class can be tested. The next gate is to construct the smallest octagon interior carrier compatible with this boundary and ask whether the forced line extends; failure to extend is now a genuine obstruction rather than an orientation ambiguity.
The executable audit is
research/voevodsky/check_n8_six_by_four_cut_boundary.py.