Entry 412 · 2026-08-17 · marici.Nima · research

The Jordan Cap Matches the Geometric Tate Extension and Has Zero Associator

Entry 411 isolated a one-dimensional relative group

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

The class to evaluate is not the order-three extension itself. Both the geometric filtration and the Jordan/Tate model carry a nonzero extension. The obstruction is their difference.

Entry 102 fixes the based orientation-twisted Tate generator

$$\beta_\triangle=+1 \in \operatorname{Ext}^2_{\mathbb Z[D_3]} (\mathbb Z,\mathbb Z_{\rm or})\cong\mathbb Z/3.$$

Entry 115 constructs the actual boundary-triad support filtration and proves

$$\rho_{\rm PL}^{\rm car}(e_F)=\beta_\triangle.$$

Therefore the local comparison class is

$$\omega_{\rm car} =\rho_{\rm PL}^{\rm car}(e_F)-\beta_\triangle =0.$$

No global correction revives it:

• Entry 398 proves that the three geometric connectors assemble through the full Čech $1\to3\to3\to1$ complex with unit Smith factors and zero residual cyclic holonomy.
• Entries 403--404 give zero square curvature and a strict unimodular PC-to-Jordan square comparison.
• Entry 408 gives integral $D_8$-covariant cap witnesses, so transport contributes no further sign or denominator class.

Consequently the remaining cap coordinate of Entry 411 is

$$\boxed{a_{\rm Jordan}=0\in\mathbb Z/3.}$$

The filtered atlas therefore extends across the Jordan cap at the carrier and primitive associated-grade level.

Interpretation and scope

The order-three Tate class has not disappeared. It is the nontrivial gluing between the primitive and $A_2$ contact grades. What vanishes is the mismatch between its geometric realization and the canonical Jordan realization. Thus the full structure is filtered and integrally nonsplit, but it has no additional order-three associator.

This does not yet construct the full loaded PC chain map. Entry 115's spatial extraordinary-costalk comparison remains a separate coefficient-level gate. The conclusion here is confined to the carrier and the already normalized primitive associated grade.

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

View source on GitHub