Cyclically Compatible Deformations of the Braid Arrangement

We prove a characteristic-polynomial shift formula for two-sided extensions of cyclically compatible deformations of the braid arrangement. For a nonnegative integer matrix $M=(m_{ij})$ with zero diagonal, let $\mathcal{A}_M$ be the arrangement \[ x_i-x_j=s,\qquad 1\le i<j\le n,\quad s\in[-m_{ij},m_{ji}]_{\mathbb{Z}}. \] Given $α,β\in\mathbb{N}^n$, define its two-sided extension $\mathcal{A}_M(α,β)$ by replacing this interval with \[ [-m_{ij}-α_i-β_j,\, m_{ji}+α_j+β_i]_{\mathbb{Z}}. \] Call $M$ cyclically compatible if all pairwise distinct $a,b,c$ with $1\le a,b,c\le n$ satisfy \[ m_{ac}\le m_{ab}+m_{bc}+1. \] Under this condition, for the reduced characteristic polynomial $\widetildeχ(\mathcal{A},t) =\dfrac{χ(\mathcal{A},t)}{t}$, we have \[ \widetildeχ(\mathcal{A}_M(α,β),t) = \widetildeχ(\mathcal{A}_M,t-|α|-|β|). \] The proof uses the finite-field method and a cyclic-gap enumeration formula. We also establish redistribution invariance, study weak-sum perturbations, and give applications to Shi, uniform interval, graphical, and Ferrers-type deformations.