The Denominator Principle: Adjoint Invariants of Minimal Nilpotent Centralizers

For every complex simple Lie algebra 𝔰 of rank at least two and minimal nilpotent element e, we determine the adjoint invariant ring of 𝔰ₑ = 𝔩 ⋉ H(V). A least-denominator theorem constructs one additional generator, whose degree is the top Coxeter exponent h − 1. The degree calculation combines local root-plane orders with the Orlik–Solomon–Terao restriction theorem. The quotient is faithfully flat with complete-intersection fibers and admits a section exactly in type A. In every other type, including E₈, its zero fiber has irreducible support and a square-zero, nonprincipal nilradical. A double cover describes its regular-orbit part. Odd SL₂ extensions supply exact section criteria and explicit local models. A separate tensor calculation determines the unrestricted detector threshold in type C; for the sextonionic algebra the unrestricted minimum remains between 7 and 29.