Elementary unimodality for natural unit interval graphs

We prove that every elementary coefficient of the chromatic quasisymmetric function of a natural unit interval graph is a nonnegative, palindromic, unimodal polynomial. This establishes the elementary-unimodality assertion of Shareshian and Wachs. More precisely, if e is the number of edges and c(q) is an elementary coefficient, then v⁻ᵉc(v²) is a nonnegative integral combination of irreducible sl₂ characters. Coefficientwise positivity alone does not give this conclusion: the character decomposition must have a common grading center. Building on existing triangular quantum-torus charts and their section formalism, we construct explicit rigid quiver potentials whose Jacobian algebras have nilpotent arrow ideals. This nilpotence forces the relevant cohomological support to be projective. Purity and hard Lefschetz then supply centered wall characters. A semistability bound makes comparisons across mutations finite in each relevant degree. These comparisons identify the selected symmetric polynomial as a chamber value of a section of the original diagram, whose coefficients inherit the centered-character property. The argument applies to all partitions and imposes no connectedness assumption or restriction on the number of vertices.