Reversing an oriented edge changes the sign of its difference, but does not erase its endpoints. Fix a commutative unital ring \(k\) and two finite incidence presentations \((V,E,s,h)\) and \((V',E',s',h')\). Equip them with weights \(w:E\to k\), \(w':E'\to k\), scalars \(\mu ,\mu '\in k\), and retained subsets \(R\subseteq V\), \(R'\subseteq V'\). Empty carriers, loops and distinct parallel edge labels are allowed.

Write \(C_2=\{+,-\}\) for the abstract group with \(-\cdot -=+\), and \(\sigma :C_2\to k^\times \) for \(\sigma (+)=1\), \(\sigma (-)=-1\). A signed endpoint arrow consists of bijections \(p:V\to V'\), \(q:E\to E'\) and signs \(\varepsilon :E\to C_2\) such that \[\begin {aligned} \varepsilon (e)=+&:\quad s'q(e)=ps(e),\quad h'q(e)=ph(e),\\ \varepsilon (e)=-&:\quad s'q(e)=ph(e),\quad h'q(e)=ps(e),\\ w'q&=w,\qquad \mu '=\mu ,\qquad pR=R'. \end {aligned}\] The signs are abstract orientation data, even when \(1=-1\) in \(k\). An arrow with all signs \(+\) is directed. For the real Gaussian backgrounds, directed arrows are their exact relabelings, with \(\mu =m^2\); positivity of mass then recovers \(m'=m\).

Composition applies the right arrow first. The composite with \((p',q',\varepsilon ')\) is \[\bigl (p'p,q'q,\ e\mapsto \varepsilon '(q(e))\varepsilon (e)\bigr ).\] Identity arrows have identity bijections and all signs \(+\). The inverse of \((p,q,\varepsilon )\) has bijections \(p^{-1},q^{-1}\) and the sign \(\varepsilon (e)\) at \(q(e)\). These operations make the declared objects and arrows a groupoid: composition is associative, identities exist, and every arrow is invertible, as for an action groupoid.

Let \(M=k^V\), \(N=k^E\) and similarly \(M',N'\), with incidence maps \((df)(e)=f(h(e))-f(s(e))\). The induced ordinary linear maps are \[\begin {aligned} (Pf)(p(v))&=f(v),&P&:M\to M',\\ (Ja)(q(e))&=\sigma (\varepsilon (e))a(e),&J&:N\to N'. \end {aligned}\] They are isomorphisms: \(P^{-1}f'=f'\circ p\) and \((J^{-1}a')(e)=\sigma (\varepsilon (e))a'(q(e))\). Evaluation at \(q(e)\), in the two endpoint cases, gives \[d'P=Jd:M\longrightarrow N'.\] Both differences vanish on a loop; the endpoint condition still remembers where that loop is attached.

For a left action of a group \(\Gamma \), take actions \(p_g,q_g\) on the vertex and edge sets. A signed endpoint action additionally requires \[\varepsilon _1(e)=+,\qquad \varepsilon _{gk}(e)=\varepsilon _g(q_k(e))\varepsilon _k(e).\] Thus the arrow assigned to \(gk\) is the composite of those assigned to \(g\) and \(k\). The induced maps satisfy \(P_gP_k=P_{gk}\) and \(J_gJ_k=J_{gk}\). Indeed evaluation at \(p_{gk}(v)\) gives \(f(v)\) on both sides of the first equation; at \(q_{gk}(e)\) the second has coefficient \(\sigma (\varepsilon _g(q_k(e)))\sigma (\varepsilon _k(e))a(e)\). Identity signs and maps give the identity operators. These are linear representations on two distinct carriers; no faithfulness is assumed.

A weaker signed-incidence symmetry requires the same actions, abstract sign cocycle and weight preservation, but only the incidence equation \(dP_g=J_gd\), instead of the endpoint equations. It is an incidence intertwiner and suffices for operator transport. It need not be a graph automorphism; loops and characteristic two distinguish the requirements. The concrete half-turn satisfies the stronger endpoint conditions.

The left-action and equivariance conventions are [milne2025group, Definition 4.1, pp. 57-58]. [milne2025group, Example 7.1(b), p. 101] gives a permutation representation over a field; the ring and signed endpoint construction above is proved directly. The incidence convention in [spielman2009effective, Section 2.1, pp. 4-5] concerns a connected undirected positive-weight graph with a chosen orientation; the endpoint-difference definition here also allows loops, parallel labels and arbitrary coefficients.

Mathlib's Representation.ofMulAction sends the free basis vector at \(v\) to that at \(g\cdot v\). Finsupp.linearEquivFunOnFinite identifies its coefficients with functions on a finite set, yielding inverse precomposition for a group action. Representation.IntertwiningMap bundles an intertwiner between already supplied representations. Neither supplies the endpoint equations or abstract sign cocycle above.