The Physical Octagon Cut Nerve Has Dimension One
Entry 442 proposed a codimension-three test comparing all six orders of three compatible physical Cut restrictions. Direct enumeration falsifies the premise of that proposal: among the eight physical distance-three octagon diagonals, there is no pairwise-compatible triple.
The full physical compatibility nerve has eight vertices and twelve edges. Every vertex has degree three, the graph is connected, and it has no two-simplices. Hence
This also corrects the phrase "on a complete quadrangulation" in Entry 442. Two compatible physical Cuts already quadrangulate the octagon into three quadrilaterals. A third physical Cut must cross one of them, so there is no codimension-three physical boundary stratum on which an restriction law could be evaluated.
Thus Entry 442 exhausts local higher-intersection coherence for the physical Cut family: its anticommuting pair squares are the top-dimensional local conditions. The next nonvacuous global test is different. Because the compatibility graph has five independent cycles, one must compose the pairwise comparison maps around a cycle basis and check whether their signed holonomy is trivial. This tests descent over the Cut nerve rather than a nonexistent triple intersection.
The executable audit is research/voevodsky/check_n8_physical_cut_nerve_dimension.py.