Finite matrix formal kernel and inverse series

For a formal indeterminate $z$, let $\mathbb{C}\mathrm{⟦}z\mathrm{⟧}$ be the commutative ring of sequences $(a_0,a_1,\dots )$ written ${\sum }_{n\ge 0}{a}_n{z}^n$, with coefficientwise addition and Cauchy-product multiplication. No convergence of a numerical series is part of this definition. For $d\ge 1$, matrices over $\mathbb{C}\mathrm{⟦}z\mathrm{⟧}$ use the usual finite matrix addition and multiplication.

Use the finite graph phase data of Finite-graph U(1) phases for matrix kernels and the QED $U(1)$, Clifford, and natural-unit conventions of Charged gauge data and SI-natural conventions. Let $K_0(\zeta )\in G{L}_d(\mathbb{C})$ be an invertible base matrix and let $V(\zeta )\in M_d(\mathbb{C})$ be its finite link-hopping matrix. The admissible covariant families satisfy $K_0({\zeta }^{\theta })=D_{\theta }{K}_0(\zeta )D_{\theta }^{-1},V({\zeta }^{\theta })=D_{\theta }V(\zeta )D_{\theta }^{-1}\mathrm{.}$ The scalar base $K_0=m{I}_d$, $m\in \mathbb{C}\setminus \{0\}$, is an explicit admissible case. Define $K(z;\zeta ):=K_0(\zeta )+zV(\zeta ),S(z;\zeta ):={\sum }_{n=0}^{\mathrm{\infty }}(-z{K}_0(\zeta {)}^{-1}V(\zeta ){)}^n{K}_0(\zeta {)}^{-1}\mathrm{.}$ The coefficient of each power $z^n$ in $S$ is one finite matrix. The item defines this formal candidate; the following lemma proves its two-sided inverse and covariance properties. Define $\det A$ for a finite matrix $A$ over $\mathbb{C}\mathrm{⟦}z\mathrm{⟧}$ by the Leibniz formula $\det A={\sum }_{\sigma \in S_d}\mathrm{sgn}(\sigma ){\prod }_{j=1}^d{A}_{j,\sigma (j)}$.

The draft carrier for Charged gauge data and SI-natural conventions is present and fixes the $(-+++)$ spacetime convention, the gamma-matrix convention, the natural charge $g$, and $D_a={\mathrm{\partial }}_a-ig{a}_a$. The current qft-mg01 manifest lists its draft items; this definition remains conditional on their mathematical conventions. The finite graph angle $\theta$ is an independent real vertex tuple. If a real vertex tuple $\alpha$ is separately assigned, one may set ${\theta }_x=g{\alpha }_x$ numerically, but no graph-to-spacetime embedding or evaluation of a spacetime gauge function at a vertex is supplied. This definition asserts no continuum kernel, convergence, or loop-distribution claim.