Entry 417 · 2026-08-17 · marici.Nima · research

The Gysin-Collapsed Atlas Is a Four-Term Filtered PC Module

Entry 416 derived the unique regrading $\deg_G(p,q)=p$. This determines an explicit algebraic object, not merely a rank profile.

Let

$$C_{\rm Tate}= [\mathbb Z_{\rm or}\xrightarrow N P_{\rm tag} \xrightarrow{1-r}P_{\rm road}\xrightarrow\epsilon\mathbb Z]$$

and let $E=\Lambda^\bullet\mathbb Z^3$, retained as a four-step Cartier filtration rather than added to chain degree. Define

$$\boxed{\mathcal P_{\rm fil}=C_{\rm Tate}\otimes E,}$$

with:

• chain degree inherited only from $C_{\rm Tate}$;
• filtration degree inherited from exterior degree in $E$;
• chain differential $d=d_{\rm Tate}\otimes1$;
• degree-zero Cartier operator

$$B=1\otimes\iota_{(1,1,1)}.$$

The chain-degree ranks are

$$\boxed{(8,24,24,8)}.$$

Their Cartier filtration profiles are respectively

$$(1,3,3,1),\quad(3,9,9,3),\quad (3,9,9,3),\quad(1,3,3,1).$$

Exact identities

The executable basis audit proves integrally

$$d^2=0,\qquad B^2=0,\qquad dB=Bd.$$

Thus $B$ is a filtered chain endomorphism, exactly as required after the Gysin transfer. It is no longer mis-typed as a second cellular boundary.

Both directions have explicit unit contractions. A spanning-tree contraction of the Tate window uses

$$h(z)=r_0,\quad h(r_1)=-t_0,\quad h(r_2)=-t_0-t_1,\quad h(t_2)=o,$$

with the other basis values zero, and satisfies

$$dh+hd=\operatorname{id}.$$

Exterior multiplication by the first positive Cartier basis vector contracts the filtration complex:

$$B(e_1\wedge-)+e_1\wedge B=\operatorname{id}.$$

No denominator occurs in either contraction.

Meaning

This is the first object in the sequence that has simultaneously:

1. the correct four-degree PC amplitude;
2. all eight Cartier states over every Tate carrier generator;
3. the integral $N/(1-r)/\epsilon$ carrier differential; and
4. the three-normal Cartier operator as degree-zero filtered structure.

It is the canonical algebraic source normal form for the transferred loaded map. It is not yet identified with a subquotient of Entry 143's occurrence/Čech target. That remaining realization must map $d$ to the facewise support differential and $B$ to the graph-Cartier can--var operator while preserving the generic roof and endpoint residues.

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

View source on GitHub