The Artin-Cone Atlas Exists but Its Full Category Has Extra Inertia
Entry 426 supplies an fs monoid at every loaded cell and a face localization at every incidence. Associate the Artin cone
The 522 monoid face maps induce representable open face maps between the 215 cones. Every composable pair was checked: whenever two routes have the same loaded endpoint, both composites groupify exactly the endpoint's normal coordinates. Thus all atlas cocycles commute strictly. The resulting diagram, and hence its open-glued 2-colimit, is a canonical logarithmic Artin-cone realization of the fs PC envelope.
There is, however, no equivalence between the full constructible category of this Artin stack and modules on the finite Alexandrov poset. At ,
Since , its rank lies between 15 and 18. The closed stratum of the Artin cone therefore has positive-dimensional torus inertia at every chart. Two rank-one equivariant sheaves carrying respectively the trivial character and a nonzero character have the same underlying stalk module but are not isomorphic on the quotient stack. A finite-poset module has no datum capable of distinguishing them.
Therefore the categorical statement required by the connector must be narrowed. The finite PC/Čech model can correspond only to the sector pulled back from the underlying Kato fan—equivalently, the constructible sector with trivial inertia action—not to all constructible sheaves on the Artin fan.
This distinction matters for six operations. It is not enough to construct the atlas: one must verify that restriction, extension, the normalized blowdown pushforward, the relative dualizing object, and the Thom trace all preserve the trivial-inertia sector. If they do, Entry 424 algebraizes inside that sector. If a pushforward generates a nontrivial torus character, the finite model omits essential logarithmic monodromy.
The executable audit is
research/voevodsky/check_artin_cone_atlas_inertia_gate.py.