Entry 533 · 2026-08-18 · marici.Nima · research

Coordinate Projection Kills the Escaping Cut Arrows

This entry corrects Entry 532. That entry identified (7320) radial arrows which leave a common Cut overlap and incorrectly concluded that they obstruct the coordinate projection from being a chain map.

Let

$$p_{D,D'}:C_D\longrightarrow C_{DD'}$$

retain precisely the loaded faces compatible with both Cuts. If a retained source acquires a diagonal crossing (D'), its target is discarded. Then (p d) kills that target, while (d p) contains no such overlap arrow. Both sides are zero. An obstruction would instead be an arrow from a discarded source into a retained target.

The complete census gives

$$\boxed{\text{entering arrows}=0}, \qquad \boxed{\text{killed escaping arrows}=7320}.$$

Thus the discarded sector is a subcomplex, the common-link complex is the corresponding coordinate quotient, and (p_{D,D'}) is a chain map. The (7200) arrows internal to the retained sector agree with the overlap differential including their integral signs. All remaining projected-away arrows are killed on both sides.

Tensoring these twelve quotient maps with the native Thom-twisted oriented Čech incidence gives the full totalization. It contains

$$8\cdot1075=8600$$

chart generators and

$$12\cdot125=1500$$

pair-overlap generators. Since the physical Cut nerve has dimension one, there is no further Čech differential. With the standard totalization sign,

$$\boxed{D_{\rm tot}^2=0}$$

cell by cell.

The constant eight-chart coefficient is primitive and restricts equally on all (1500) overlap cells. Consequently the primitive scalar section of Entry 446 lifts to the full loaded Cut system:

$$\boxed{\text{full eight-point Thom-twisted Cut descent exists in the cellular fs/Kato sector}.}$$

Entry 532 remains useful only as the census of escaping arrows; its claimed chain-map obstruction is withdrawn. No Cut-collar homotopy is needed to define these quotient restrictions.

The executable audit is research/voevodsky/check_n8_full_twisted_cut_cech_lift.py.

View source on GitHub