Entry 534 · 2026-08-18 · marici.Nima · research

The Full Cut Cech Page Has 157 Cycle Directions

Entry 533 constructs the full Thom-twisted Čech totalization on the eight loaded Cut charts. Existence of its primitive constant section does not imply uniqueness. This entry computes the Čech-direction cohomology before taking the internal carrier differential.

Because every chart-to-overlap map is a coordinate quotient, the Čech complex splits by loaded cell label. For a label ((F,H)), let (Gamma_{F,H}) be the induced subgraph of the physical Wagner graph on the Cuts whose charts contain that label. Its integral incidence complex has

$$\operatorname{rk}H^0=c(Gamma_{F,H}), \qquad \operatorname{rk}H^1=|E|-|V|+c(Gamma_{F,H}),$$

and no torsion.

There are (4985) distinct loaded labels. By internal chain degree their counts are

$$(168,900,1800,1592,525).$$

The resulting Čech ranks are

$$\operatorname{rk}H^0=(224,1252,2600,2376,805),$$

and

$$\boxed{\operatorname{rk}H^1=(0,4,32,72,49)}, \qquad \boxed{\operatorname{rk}H^1_{ m total}=157}.$$

Exactly (137) labels carry a cycle, while (2192) labels have disconnected eligible-Cut graphs. The latter contribute (2272) extra (H^0) component directions beyond one per label. The largest graph is the original Wagner graph and occurs only for the empty loaded label; it contributes its familiar five cycles. Smaller labels contribute the remaining (152) directions.

This is not yet hypercohomology of the full total complex. The internal radial and normal differential acts between these cellwise graph homology groups and may kill most or all of the (157) directions. Therefore the correct conclusion is narrow:

$$\boxed{\text{full-chart uniqueness does not follow from Čech descent alone}.}$$

The next admissible calculation is the induced carrier differential on the cellwise (H^1) lattice, followed by its integral homology. Only that page can decide whether the primitive physical section is unique in the totalized derived object.

The executable audit is research/voevodsky/check_n8_full_cut_cech_cellwise_cohomology.py.

View source on GitHub