Entry 419 · 2026-08-17 · marici.Nima · research

Local Cartier Purity Descends to the Global Connector Operator

Entry 418 reduced the remaining finite realization gate to the operator comparison

$$\kappa B_{\rm ext}\simeq B_{\rm graph}\kappa$$

on the assembled three-road connector. Entry 131 proves this comparison on each marked road chart. The question is whether the three local comparisons descend with a residual overlap choice.

They do. The overlap nerve used in Entry 398 is the full augmented two-simplex

$$\mathbb Z\longrightarrow\mathbb Z^3 \longrightarrow\mathbb Z^3\longrightarrow\mathbb Z$$

with ranks $(1,2,1)$. In oriented bases,

$$d_2=(1,-1,1)^T,\qquad d_1= \begin{pmatrix} -1&-1&0\\ 1&0&-1\\ 0&1&1 \end{pmatrix}, \qquad \epsilon=(1,1,1).$$

All nonzero Smith factors are one.

More strongly, coning the simplex to the first chart gives an explicit integral contraction. Therefore tensoring the nerve with the finite operator lattice, and then with the flat Čech localization summands, remains exact. Pairwise differences of local Cartier homotopies have a unique integral filler, and the triple-overlap compatibility has no residual class. Consequently

$$\boxed{\kappa B_{\rm ext}=B_{\rm graph}\kappa}$$

in the assembled finite filtered PC/Čech model.

Nonvacuity

The descended equality is not obtained from the zero map. Its boundary values were fixed independently:

• Entries 396--397 give the primitive generic roof $[\mathrm{top},D]$ with coefficient $+1$;
• Entry 378 gives its nonzero generic Rees coefficient $x_D$;
• Entries 131 and 391 give closed Cartier residue $+1$;
• Entry 400 gives the unimodular endpoint comparison

$$\begin{pmatrix}0&1\\1&0\end{pmatrix}$$

of determinant $-1$;

• Entry 398 supplies the full pairwise and triple Čech assembly.

Thus the generic $Q$ leg, the closed Cartier face, and both endpoints are restrictions of the same descended operator comparison.

Consequence

Tensoring the assembled connector of Entries 397--400 with the external Cartier packet of Entries 417--418 now produces a filtered chain map in the finite PC/Čech category. Its carrier differential is $N/(1-r)/\epsilon$, and its coefficient operator is the actual graph-Cartier Bockstein.

This closes the finite operator-realization gate. It does not yet construct the complete geometric primal trace from the raw normalization-sheet six-functor object, nor a smooth representative. The next audit should compare the resulting filtered connector with the full Entry-143 target filtration generator by generator after forgetting the external Cartier filtration, ensuring that its cone contracts to the already proved ordinary carrier map and introduces no extra homology.

The executable audit is `research/voevodsky/check_global_cartier_operator_descent.py`.

View source on GitHub