Entry 418 · 2026-08-17 · marici.Nima · research

Cartier Filtration Is Coefficientwise and Not the Face-Circle Grading

Entry 417 constructed the four-term filtered module and proposed placing its eight Cartier states over the fixed occurrence/Čech target. A literal identification of Cartier exterior degree with Entry-105's normal-circle number is impossible.

For a facewise generator $(S,H)$, normal-circle number is $|H|$ and chain degree is $3-|S|+|H|$. After removing the two endpoint packets, the bigraded ranks of $F_K/F_V$, with rows indexed by chain degree and columns by face-circle number, are

$$\begin{array}{c|rrrr} &0&1&2&3\\ \hline 0&12&0&0&0\\ 1&21&36&0&0\\ 2&9&42&36&0\\ 3&1&9&21&12 \end{array}$$

and their row sums are $(12,57,87,43)$.

The Gysin-collapsed source has Cartier filtration profiles

$$\begin{array}{c|rrrr} &0&1&2&3\\ \hline 0&1&3&3&1\\ 1&3&9&9&3\\ 2&3&9&9&3\\ 3&1&3&3&1. \end{array}$$

The literal identification fails in bidegrees

$$(0,1),(0,2),(0,3),(1,2),(1,3),(2,3).$$

In particular, degree-zero Tate states carry all four Cartier levels, while a degree-zero facewise PC generator necessarily has $H=\varnothing$.

This is not a new obstruction. It proves that the two gradings encode different geometry:

• $H\subseteq S$ records original/Borel--Moore normal circles already supported on a PC face;
• the Cartier exterior degree records independent multi-Rees conormal filtration transported by extraordinary purity.

They must not be identified.

Correct target type

The realization target must be treated as

$$\boxed{ \mathcal E_{\partial,Q}^{\rm BM,\check C} \widehat\otimes \Lambda^\bullet N_{\rm Cart}^{\vee}, }$$

where the exterior packet is coefficientwise and Gysin shifted, so it does not alter cellular chain degree. On this object:

• the Entry-143 differential acts on the PC/Čech factor;
• the Cartier Bockstein acts on the external coefficient factor;
• the two operations commute formally before the geometric graph-Cartier comparison is imposed.

The coefficientwise target has enough capacity in every bidegree. This is again only a typing and capacity theorem. The remaining nonformal step is to identify the external Cartier action with the actual graph-Cartier can--var operation using Entry 131's local purity maps and the three-connector Čech assembly.

Thus the next gate is not a search for 64 target cells. It is one global operator comparison:

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

on the assembled connector, including the generic $Q$ roof and endpoint restrictions.

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

View source on GitHub