Entry 409 · 2026-08-17 · marici.Nima · research

The Primitive Verdier Quotient Types the Crosscap Holonomy

The full occurrence-resolved road object at a cut $D$ is

$$P_D=\mathbb Z^3,\qquad \epsilon_D=(1,1,1):P_D\to\mathbf1,$$

with contact kernel $A_2=\ker\epsilon_D$. Each chart restriction selects one road section $s_i:\mathbf1\to P_D$.

In the full category, no $s_i$ is an inverse to $\epsilon_D$: $s_i\epsilon_D$ has rank one, while $P_D$ has rank three. Its defect

$$\operatorname{id}_{P_D}-s_i\epsilon_D$$

has image in $A_2$. Thus the obstruction to inversion is exactly the two-dimensional contact sector, not a denominator or a missing sign.

Let

$$\mathcal D_{\rm prim} = \mathcal D_{\rm occ}/\langle A_2\rangle$$

be the Verdier quotient by the thick contact subcategory. Then

$$\epsilon_Ds_i=\operatorname{id}_{\mathbf1},\qquad s_i\epsilon_D=\operatorname{id}_{P_D} \quad\text{in }\mathcal D_{\rm prim}.$$

Moreover $s_i-s_j$ factors through $A_2$, so all three roofs define the same inverse. Cyclic road transport merely permutes these representatives. This gives integral, choice-independent equivalences in the primitive associated grade without constructing a forbidden $C_3$-equivariant section in the full lattice category.

Entry 406 selected the orientation character on the primitive line. Consequently the now-typed rank-one transition system has multiplicative holonomy

$$\boxed{\operatorname{Hol}(\gamma)=-1}$$

on the Möbius core. Entry 408 gives $[\partial O]=2[\gamma]$, hence

$$\boxed{\operatorname{Hol}(\partial O) =\operatorname{Hol}(\gamma)^2=+1.}$$

The additive and multiplicative descriptions therefore agree: the primitive additive period is one on the core and two on the boundary, while its orientation exponential is minus one and plus one respectively.

Scope

This is the first typed multiplicative holonomy, but only in $\mathcal D_{\rm prim}$, the primitive associated-grade quotient. It does not promote the road legs to equivalences in the full occurrence-resolved PC category, where $A_2$ is real QTDS contact data. A full multiplicative atlas would require a recollement or filtered enhancement that retains $A_2$ while exposing the primitive quotient, rather than killing it.

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

View source on GitHub