Centralizers of nilpotent wedges in real orthogonal Lie algebras so(p,q): a signature-resolved Levi-radical map (Geometry of the vacuum, floor −1)

Abstract

For a rank-2 nilpotent (a wedge) in a real orthogonal Lie algebra so(p,q), the centralizer decomposes as a reductive part in semidirect product with a nilradical — a framework classical over algebraically closed fields (Springer-Steinberg; Collingwood-McGovern). We give the signature-resolved real form of this decomposition for the enumerated signatures n = p+q at most 6: the non-abelian part is exhausted by four classes, each a three-dimensional engine in semidirect product with a nilpotent module, falling into a dichotomy between the affine type and the Jacobi type. The governing mechanism is obtained for all n in block-symbolic form (by solving a bracket equation, not by fitting). A signature biconditional — for the soft (rank-1) family, the reductive Levi is compact if and only if q = 1 — follows from a definiteness condition. The explicit isomorphisms are machine-verified in Lean 4 / mathlib (kernel-checked; axioms propext, Classical.choice, Quot.sound; no native_decide). Three statements — the signature-resolved dichotomy table, the all-n centralizer form, and the soft-family compact-Levi biconditional — are absent from the literature surveyed (the nearest, Djokovic-Lemire-Sekiguchi 2001, gives the closure order, not centralizer structure). No physical reading of any object is made or implied.