Entry 411 · 2026-08-17 · marici.Nima · research

The Order-Three Extension Has No Absolute Mobius Two-Obstruction

Entry 410 produced a local extension class in $\mathbb Z/3$. Its global Čech obstruction must be computed with the selected orientation local system, not with constant coefficients.

The exact twelve-face carrier of Entries 88 and 408 collapses through unimodular face and tree pivots to the universal rank-one complex

$$R\xrightarrow{u-1}R.$$

For the orientation character $u=-1$ over $\mathbb F_3$,

$$u-1=-2=1\pmod3,$$

which is a unit. Thus the twisted Möbius complex is acyclic:

$$\boxed{H^i(M;\mathbb F_{3,\rm or})=0\quad\text{for all }i.}$$

In particular,

$$\boxed{\check H^2(M;\mathbb F_{3,\rm or})=0.}$$

Every transported local order-three extension therefore admits a global filtered descent on the open Möbius carrier. There is no absolute Čech two-obstruction and hence no higher associator supported solely on its twelve local faces.

The cap changes the answer. The residual octagonal cell adds the universal top differential $u+1$. At $u=-1$,

$$u+1=0,$$

so the capped complex has

$$\boxed{H^2(X;\mathbb F_{3,\rm or})\cong\mathbb F_3.}$$

Equivalently, the relative top group for the cap is one-dimensional. Thus the local extension class can survive only as a capped/relative top class, not as an obstruction to gluing the filtered atlas over $M$.

Consequence

The filtered atlas exists on the twelve-chart Möbius carrier. The remaining question is a single scalar in $\mathbb F_3$: evaluate the transported order-three extension on the oriented Jordan cap. Values $0$, $+1$, and $-1$ are the complete candidate space. A zero value extends the filtered atlas across the cap; a unit is the genuine order-three Jordan associator.

The executable audit is `research/voevodsky/check_global_mod3_filtered_atlas.py`.

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