Example. What zero incidence and scalar signs forget [fdip-003C]
Example. What zero incidence and scalar signs forget [fdip-003C]
The signed endpoint equations retain information that an incidence operator alone can lose. Take vertices \(\{0,1\}\) and one loop \(e\) attached at \(0\). Its incidence map is zero over every ring. Swapping the vertices, fixing \(e\) and choosing sign \(+\) therefore intertwines incidence and preserves any weight on \(e\). Nevertheless \(s(q(e))=0\ne 1=p(s(e))\); the same obstruction holds for sign \(-\), since both endpoints of the loop are \(0\). This is a signed-incidence symmetry, but not a signed endpoint arrow.
A separate loss concerns composition. Let \(C_3=\{1,g,g^2\}\) fix a single loop and its vertex. Assign abstract signs \(\varepsilon _1=+\), \(\varepsilon _g=\varepsilon _{g^2}=-\). Every endpoint and zero-incidence equation holds separately. But the action cocycle at \(g\cdot g=g^2\) would require \[\varepsilon _{g^2}=\varepsilon _g\varepsilon _g=+,\] contrary to the assigned \(-\). Thus these arrows do not define a \(C_3\)-action. Over a ring of characteristic two the scalar maps all become the identity because \(-1=1\); even their representation equations then conceal the failed abstract cocycle. Orientation data remain subject to the abstract \(C_2\) law.
By contrast, one edge \(e:0\to 1\) admits the vertex swap with fixed edge label and sign \(-\). Its endpoints reverse as required, and using this arrow twice gives sign \(+\) and identity bijections. For fields, \[(dPf)(e)=f(0)-f(1)=-(df)(e)=(Jdf)(e).\] It is a valid signed symmetry. Parallel labels are still distinct: an edge bijection may permute them, but never identifies two labels. Each sign must satisfy its own endpoint rule and the composition law.
These calculations use the endpoint-difference construction of Definition [fdip-003B].
[spielman2009effective, Section 2.1, pp. 4-5] allows an arbitrary orientation of each edge. The calculations explain its extension in the signed endpoint construction.
[milne2025group, Definition 4.1, p. 57] gives the group-law requirement. Neither source is being asserted to contain these loop or characteristic-two examples.