Entry 413 · 2026-08-17 · marici.Nima · research

The Filtered Jordan Atlas Is Rigid in the Order-Three Sector

Entry 412 proved that the sole order-three obstruction to extending the filtered atlas across the Jordan cap is zero. Existence alone does not say whether the extension still requires a choice. The same twisted carrier calculation shows that it does not.

After the unimodular collapses of Entry 411, the capped Möbius cochain complex with orientation coefficients is

$$\mathbb F_3 \xrightarrow{u-1} \mathbb F_3 \xrightarrow{u+1} \mathbb F_3.$$

For the selected orientation monodromy $u=-1$,

$$u-1=-2=1,\qquad u+1=0 \quad\text{in }\mathbb F_3.$$

Consequently

$$\boxed{H^0=0,\qquad H^1=0,\qquad H^2\cong\mathbb F_3.}$$

The interpretation is the standard obstruction tower for descent of the local order-three road extension:

• $H^2$ contains the obstruction to existence;
• when that obstruction vanishes, equivalence classes of lifts form a torsor under $H^1$;
• automorphisms of a fixed lift are measured by $H^0$.

Entry 412 evaluates the unique $H^2$ coordinate as zero. The vanishing of $H^1$ therefore makes the lift unique up to filtered equivalence, and the vanishing of $H^0$ leaves no residual order-three gauge automorphism. Thus the nonzero local Tate extension remains part of the filtered object, but its global Jordan descent is rigid.

The independent order-two endpoint ambiguity is also already exhausted. Entry 400's geometric ray-to-sheet comparison is the unique integral solution of

$$b-a=1,\qquad a+b=1,$$

namely $(a,b)=(0,1)$, and hence

$$p_{\partial,Q}=0\in\mathbb Z/2.$$

Entries 401--404 then show that its dihedral, endpoint-Jordan, and square corrections vanish. This order-two statement is not inferred from the mod-three complex; it is a separate geometrically fixed stage of the same obstruction tower.

Consequence

At the carrier and finite filtered-PC levels already constructed in Entries 381--412, there is no remaining discrete choice of global lift:

$$\boxed{\text{the filtered Jordan atlas exists and is rigid.}}$$

Accordingly, the missing full loaded PC chain map is no longer an existence or coherence problem. It is an explicit realization problem: totalize the already fixed connector, Cartier, endpoint, Tor-suspension, and primitive/contact extension maps in one chain model and verify that its associated grades recover the established pieces.

This statement does not itself write that totalized chain map, identify a smooth representative, or promote the construction beyond the existing finite filtered-PC category.

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

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