The Connector Six Operations Preserve the Kato Sector
Entry 427 separated the finite connector category from the full equivariant coefficient category of its Artin-cone realization. For the operations actually used by the connector, the Kato-pulled sector is closed.
There are two reasons. First, each of the 522 incidence maps is a strict face localization
Restriction of a zero inertia character remains zero. Extension by zero has the same zero character on the open face and the zero object elsewhere. Consequently restriction, extension by zero, support cones, tensor products, internal Homs, and duals preserve the Kato sector.
Second, the normalized blowdown of Entry 423 was deliberately given the pulled-back structure sheaf. At every expanded source cell its monoid chart is literally the chart of its old-face image. The induced map on chart tori is the identity. Its fibers are singletons except for the finite contractible V-tree, so left and right Kan extension use only finite sums, products, and the integral V-tree differential. None can generate an inertia character from zero-character inputs. Thus
all remain in the Kato-pulled sector for the marked coefficient packet.
The relative dualizing object also has zero torus character. Its sign under reflection, computed in Entry 424, belongs to the discrete permutation of the two exceptional endpoints; it is not a character of a chart torus. Therefore the primitive Thom trace is a morphism inside the same sector. Rotation and the Čech assembly preserve this conclusion because their transition maps are again strict face maps.
Hence the complete finite six-operation connector of Entries 422–424 algebraizes on the fs/Kato diagram of Entry 426 and embeds into the Artin-cone presentation through its Kato-pulled sector.
This is intentionally not a claim that arbitrary pushforwards in the full equivariant or quasi-coherent category preserve trivial inertia. Such functors may retain stabilizer representations or character twists. The result applies to the explicit strict face maps, finite normalized blowdown, and dualizing packet used here.
The remaining comparison is generic: after localizing , verify that the Kato-sector Thom trace becomes the raw DNC generic trace with the same primitive sign and no additional unit.
The executable audit is
research/voevodsky/check_kato_sector_six_operations.py.