Entry 414 · 2026-08-17 · marici.Nima · research

The Rigid Filtered Atlas Has a Canonical Integral Total Normal Form

Entry 413 proves that the filtered Jordan atlas exists and has no residual order-three or endpoint-parity choice. The first explicit realization step is therefore forced: totalize the geometric Tate carrier of Entry 115 with the normalized three-normal Cartier packet.

Write

$$C_{\rm Tate}= [\mathbb Z_{\rm or}\xrightarrow{N}\mathbb Z^3 \xrightarrow{1-r}\mathbb Z^3\xrightarrow{\epsilon}\mathbb Z]$$

and let $K_{\rm Cart}$ be the exterior Koszul packet on the three positively oriented Cartier units. In bidegree $(p,q)$, set

$$\mathcal T_{p,q}=C_{{\rm Tate},p}\otimes K_{{\rm Cart},q}, \qquad d_{\rm Tot}=d_{\rm Tate}+(-1)^p d_{\rm Cart}.$$

The total chain ranks are the convolution

$$(1,3,3,1)*(1,3,3,1) =\boxed{(1,6,15,20,15,6,1)}.$$

Writing every basis and every block explicitly gives

$$\operatorname{rank}(d_1,\ldots,d_6) =\boxed{(1,5,10,10,5,1)}.$$

All matrix entries lie in $\{0,\pm1\}$, and the Koszul sign makes every mixed square anticommute, so $d_{\rm Tot}^2=0$.

Integral contraction

Choose the first positively oriented Cartier basis vector $e_1$. Exterior multiplication by $e_1$ contracts the Cartier complex because

$$\iota_{(1,1,1)}(e_1\wedge-) +e_1\wedge\iota_{(1,1,1)}=\operatorname{id}.$$

On a Tate-degree-$p$ tensor, the total contraction is

$$H=(-1)^p(1\otimes e_1\wedge-).$$

The checker verifies basiswise over $\mathbb Z$ that

$$d_{\rm Tot}H+Hd_{\rm Tot}=\operatorname{id}.$$

Thus the 64-generator normal form is split exact with an explicit integral contraction. No division by two or three and no rational projector occurs.

This contraction does not erase the local order-three road extension. It applies after the positively oriented Cartier unit has been evaluated in the normalized finite packet. The filtered road recollement still records the nonsplit $A_2$-to-primitive extension described in Entries 409--413.

Boundary of the result

The result supplies the unique integral block-and-sign normal form that the full loaded PC realization must carry. It is not yet the desired occurrence- resolved map into Entry 143's extended Čech target. The remaining construction is now concrete: totalize the already proved normalized-blowdown connector of Entries 396--398 with the endpoint/Tor suspension of Entries 399--400 and the Cartier comparison, then compare its 64 normalized generators with the corresponding occurrence/Čech summands.

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

View source on GitHub