The Carrier Differential Reduces Cut Cech Ambiguity to Five Top Cycles
Entry 534 found (157) integral cycle directions on the cellwise Čech page. This entry computes the differential induced on that page by every radial and normal arrow of the loaded Cut carriers.
For each induced Cut graph, choose a spanning forest. Its non-tree edges give an integral basis of the graph-incidence cokernel. Project each carrier arrow to the target graph and reduce it against the target forest. This constructs the induced matrices without choosing a quotient after seeing their ranks.
The chain groups, in internal degrees zero through four, have ranks
The four differentials have rational ranks
and their consecutive products vanish exactly. Therefore the rational homology is
Thus (152) of the (157) cellwise Čech directions are removed by the ordinary loaded-carrier differential. Only five top-degree directions survive. Projection of the top kernel to the five Wagner-cycle coordinates of the empty loaded label has full rank five. The survivors are therefore not a new bulk family: rationally they are the transported form of the five global cycles already present in the physical Cut nerve.
This result still does not prove uniqueness of the descended physical line. It instead isolates the exact residual ambiguity:
The remaining integral test has two parts. First compute Smith data for the induced matrices and their kernel/image quotients to exclude finite torsion or nonprimitive projection onto the Wagner lattice. Then evaluate the physical line's edge comparison on those five cycles. If all five evaluations vanish, the physical section is unique despite ambient derived (H^1); otherwise it forms a torsor over the detected cycle lattice.
The executable audit is
research/voevodsky/check_n8_cut_cech_h1_carrier_homology.py.