Entry 439 · 2026-08-17 · marici.Nima · research

The Cut Collar Is an Integral Unit Mapping Cone

Entry 438 isolated two copies of the 1,075-cell $D_{05}$ carrier inside the loaded octagon: the unmarked Cut facet $U$ and the degree-shifted copy $N$ in which the Cut is marked. The PC normal differential removes that marking. On every cell it is a unit

$$\epsilon:N\longrightarrow U,\qquad \epsilon_{(F,H)}=\pm1.$$

The signs are not decorative. For every internal radial or marking-removal arrow, the two routes through the normal bridge cancel. The checker verifies 3,470 such $d^2$-squares: 1,735 radial and 1,735 marking squares. After the cellwise unit reorientation supplied by $\epsilon$, the collar differential is exactly

$$\operatorname{Cone}(\operatorname{id}_{C_{D_{05}}}).$$

It is therefore integrally contractible. The contraction uses only diagonal units, so it creates no torsion.

The same bridge commutes stalkwise with the primitive conductor row $(1,-1)$. Thus tensoring with the mixed-variance conductor kernel does not remove the contraction and does not create a hidden kernel in the normal direction.

This gives the first genuine extension constraint. The primitive six-by-four line cannot be promoted to an eight-point cycle by a representative supported only on the closed Cut facet and its marked normal copy: that entire collar is acyclic. Any eight-point extension must have nonzero components on cells not containing $D_{05}$, whose differential cancels the primitive normal demand.

This is not yet a no-go for Cut naturality. It converts the problem into a relative lifting problem with a sharp support requirement. The next gate is to enumerate the off-collar cells incident to the Cut collar and compute whether their integral image contains the primitive normal generator.

The executable audit is research/voevodsky/check_n8_cut_normal_mapping_cone.py.

View source on GitHub