Theorem. Pairing isometries transport incidence operators [fdip-003D]
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October 29, 2026

Let \(k\) be a commutative unital ring. Give \(M,N,M',N'\) chosen perfect symmetric pairings, and let \(d:M\to N\), \(d':M'\to N'\) be ordinary linear maps. Suppose the linear isomorphisms \(P:M\to M'\), \(J:N\to N'\) preserve these pairings and satisfy \(d'P=Jd\). Let \(W:N\to N\), \(W':N'\to N'\) be self-adjoint maps with \(W'J=JW\). For a fixed \(\mu \in k\), put \[L=d^\dagger Wd,\qquad Q=L+\mu I_M,\] and define \(L',Q'\) by the primed data. Then \[\begin {aligned} (d')^\dagger &=Pd^\dagger J^{-1}:N'\to M',\\ L'&=PLP^{-1},\qquad Q'=PQP^{-1}. \end {aligned}\] If \(Q\) is an isomorphism, so is \(Q'\), and its inverse response satisfies \[G'=(Q')^{-1}=PQ^{-1}P^{-1}=PGP^{-1}.\] Consequently \(Qf=j\) implies \(Q'(Pf)=Pj\).

Proof.

October 29, 2026

For \(z'\in M'\), pairing preservation gives \(b_{M'}(Px,z')=b_M(x,P^{-1}z')\). The uniqueness of the adjoint therefore gives \(P^\dagger =P^{-1}\), and similarly \(J^\dagger =J^{-1}\). From \(d'=JdP^{-1}\), the adjoint product law gives \((d')^\dagger =Pd^\dagger J^{-1}\). Thus \[\begin {aligned} L'&=Pd^\dagger J^{-1}W'JdP^{-1}\\ &=Pd^\dagger WdP^{-1}=PLP^{-1}. \end {aligned}\] Here \(J^{-1}W'J:N\to N\) equals \(W\); every composition is typed. The same scalar \(\mu \) gives the assertion for \(Q\). Multiplication in either order proves that \(PQ^{-1}P^{-1}\) is the inverse of \(Q'\). Finally \(Q'Pf=PQf=Pj\).

For signed endpoint arrows, the finite function modules have perfect counting duality, including empty carriers. Reindexing gives \[\begin {aligned} b_{M'}(Pf,Pg)&=\sum _{v\in V}f(v)g(v),\\ b_{N'}(Ja,Jb)&=\sum _{e\in E}\sigma (\varepsilon (e))^2a(e)b(e) =b_N(a,b). \end {aligned}\] Multiplication by transported weights obeys \(W'J=JW\), by evaluation at \(q(e)\). Thus the theorem applies without inverting any weight. It also applies to the weaker signed-incidence symmetries. The weighted edge pairing is \(b_W(a,b)=b_N(a,Wb)\); its adjoint of \(d\) into the counting vertex pairing is \(d^\dagger W\), whereas \(d^\dagger \) uses the counting edge pairing. The expression \(d^\dagger Wd\) does not require \(b_W\) to be perfect.

The inverse hypothesis cannot be replaced by a unit mass scalar. Over \(\mathbb Z\), a unit-weight edge on two vertices with \(\mu =1\) has \(Q=\begin {pmatrix}2&-1\\-1&2\end {pmatrix}\). Solving \(Q(x,y)=(1,0)\) would require \(x=2y\) and \(3y=1\), which is impossible in \(\mathbb Z\). This \(Q\) is not even surjective.

For automorphisms the conjugacies are commutation identities. They concern symmetric bilinear pairings over fixed coefficients. A grading is preserved only if \(P\) preserves its graded pieces; a star is preserved only if \(P\) commutes with that star. A nontrivial coefficient involution and Hermitian adjoint need their own conventions.

The proof uses the chosen duality construction in Proposition [fdip-001P], with the Hom convention of [stacks2017stacks, Section 10.10], Tag 0581. For real graph incidence, the energy convention is [spielman2009effective, Section 2.1, pp. 4-5]; the displayed ring transport is a direct deduction, not a positivity theorem. Mathlib's Matrix.reindexLinearEquiv and Matrix.reindexRingEquiv support its coordinate reindexing specialization, without supplying the endpoint or pairing hypotheses.