Entry 444 · 2026-08-17 · marici.Nima · research

The Physical Cut Nerve Carries Nontrivial Koszul Holonomy

Entry 443 replaces the nonexistent codimension-three stratum by the global cycle problem on the physical Cut compatibility graph. That graph is not the three-cube. It is the eight-vertex Moebius ladder (the Wagner graph): an eight-cycle together with the matching between opposite vertices.

Entry 442 assigns the comparison sign $-1$ to every compatibility edge, because reversing two normal contractions is Koszul odd. For a deterministic spanning tree, the five chord cycles have lengths

$$(5,4,5,5,4).$$

Their signed holonomies are therefore

$$(-1,+1,-1,-1,+1).$$

In particular three basis generators detect nontrivial holonomy.

Equivalently, the constant edge one-cochain with value $-1$ could be removed by vertexwise sign changes precisely if the compatibility graph were bipartite. Its five-cycles show that it is not. The Koszul comparison data thus define a nonzero class in

$$H^1(N_{\mathrm{Cut}};\mathbb Z/2).$$

This is a genuine global descent obstruction, not a failure of any local Cut square. The local codimension-two theorem remains exact, but the eight local Cut objects cannot be glued with untwisted scalar identifications. The next gate is to determine whether the normalization-sheet/conductor/Thom package already carries a compensating orientation local system. If it does not, that local system must be added explicitly; otherwise global Cut descent is false in the untwisted category.

The executable audit is research/voevodsky/check_n8_cut_nerve_signed_holonomy.py.

View source on GitHub