Entry 446 · 2026-08-17 · marici.Nima · research

The Thom-Twisted Cut Cech Complex Has a Primitive Global Section

Entry 445 cancels the Cut-nerve sign holonomy. The integral Cech complex makes the arithmetic consequence explicit. The physical nerve has eight vertex charts and twelve pair overlaps, so its coefficient complex is

$$\mathbb Z^8\longrightarrow\mathbb Z^{12}.$$

After forgetting the odd Thom-normal line, transport on every edge is $-1$. The differential is the signless incidence matrix. It has rank eight and the greatest common divisor of its maximal minors is two. Consequently

$$H^0_{\mathrm{scalar}}=0, \qquad H^1_{\mathrm{scalar}}\cong\mathbb Z^4\oplus\mathbb Z/2.$$

The $\mathbb Z/2$ summand is the integral shadow of Entry 444's odd-cycle holonomy.

With the native marked-normal Thom line included, the two minus signs cancel. The differential becomes the ordinary oriented incidence matrix. It has rank seven and a unit maximal minor, hence saturated image. Therefore

$$H^0_{\mathrm{Thom}}\cong\mathbb Z, \qquad H^1_{\mathrm{Thom}}\cong\mathbb Z^5,$$

with no integral torsion. The generator of $H^0$ is the constant vector $(1,\ldots,1)$, which is primitive and matches the eight positive physical Cut coefficients of Entry 441.

Thus the rank-one integral twisted Cech totalization exists and carries the required global class. The next gate is to lift this coefficient-line calculation to the full eight-chart system of 1,075-cell Cut objects and their 125-cell pair overlaps, checking the total differential cell by cell.

The executable audit is research/voevodsky/check_n8_twisted_cut_cech_totalization.py.

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