Entry 406 · 2026-08-17 · marici.Nima · research

The Primitive Cousin Normalization Selects the Nonzero Mobius Atlas Class

Entry 405 proved that endpoint parity alone cannot distinguish the zero atlas cocycle from the primitive Möbius cocycle. The missing datum was the crosscap comparison

$$\Psi_{\rm crosscap}: \operatorname{Fib}_{\partial,Q}\longrightarrow L_{\rm or}.$$

The required map is already determined when the later geometric construction is combined with the earlier primitive Cousin normalization.

The two primitive lines

Entry 142 computes the only homology of the unsplit endpoint pullback:

$$H_1(C_{\partial}^{\rm coeff})\cong\mathbb Z_{\rm or}.$$

Entries 91--92 compute the primitive Möbius quotient:

$$L_{\rm or}\cong\mathbb Z_{\rm or}.$$

On both lines, rotations act by $+1$ and reflections by $-1$. Therefore every equivariant homomorphism is multiplication by an integer $n$. Character matching alone leaves $n$ undetermined, exactly as Entry 92 warned.

Entry 94 supplies the missing nonordinary map. Its normalization--conductor Cousin/Gysin symbol lands in the tag resolution before road incidence, and its Verdier-dual primitive branch has unit value on every retained polarized occurrence. It is integral and primitive, so

$$|n|=1.$$

The positive Cousin, polarity, and coaction convention fixes the sign

$$n=+1.$$

Entries 392--400 subsequently construct the spatial connector and endpoint mapping fiber with that same retained positive normalization; Entry 404 provides the faithful $D_8$-equivariant square-sector identification. Thus the associated crosscap comparison is

$$\boxed{\Psi_{\rm crosscap}=+1: \mathbb Z_{\rm or}\xrightarrow{\sim}\mathbb Z_{\rm or}.}$$

Atlas value

Let $\gamma$ be the primitive Möbius core and $\omega$ its primitive dual cocycle. The geometrically normalized endpoint generator maps to $\gamma$ with coefficient one:

$$\langle\Theta,\gamma\rangle=1.$$

Hence the selected additive atlas class is

$$\boxed{\Theta=\omega,}$$

not the zero solution. Since the outer octagon is twice the core,

$$\langle\Theta,\partial O\rangle=2.$$

This is fully compatible with Entry 400:

$$2\bmod2=0=p_{\partial,Q}.$$

The even endpoint parity records only the capped mod-two shadow and therefore did not reveal that the integral crosscap weight is primitive and nonzero.

Scope

This selects the additive Möbius/Jordan atlas class at the primitive associated-grade and spatially normalized connector level. It does not turn the noninvertible residue/Gysin cospans into transition automorphisms, and it does not construct multiplicative holonomy. The result is a nonzero additive higher-coherence class, not an obstruction to the already proved polarity homotopy or square comparison.

The next question is interpretive and structural: determine whether the primitive class $\omega$ is exactly the capped rectangular Jordan fundamental-formula cell, or whether their equality requires one further top-dimensional comparison homotopy.

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

View source on GitHub