Compact-Image Obstructions for a Hyperbolic Grading in Sp(32,32): Neutral, Grading-Even, and Extremal Order Parameters

Abstract

Let a connected real reductive group G act faithfully on a finite-dimensional real vector space E, and let A_Z be a one-parameter subgroup with unbounded represented image. This paper isolates three compact-image obstructions for stabilizers of order parameters. A vector fixed by A_Z retains the entire unbounded flow in its stabilizer. An operator commuting with the infinitesimal generator retains the same flow in its centralizer. Under real-diagonalizability and an explicitly detected sign-matched nilpotent witness, a maximal or minimal Z-weight vector retains a corresponding unipotent one-parameter subgroup. In each case the represented stabilizer has noncompact closure and therefore preserves no positive-definite inner product on E.For Sp(32,32), the hypotheses are verified directly in a quaternionic matrix model. The grading has adjoint eigenvalues -2, 0, and 2; the two nonzero eigenspaces each have real dimension 2080, are abelian, and consist of square-zero matrices. The same block calculation works for Sp(n,n) for every n >= 1. The paper also separates continuous vector neutrality from discrete operator parity and proves that noncommutation of a compact-reducing involution does not by itself imply pure oddness.The contribution is a scoped synthesis, explicit fixed-grading calculation, and falsifier taxonomy, not a new classification theorem for real Lie groups. It is finite-dimensional and algebraic. It does not establish a physical Hilbert space, vacuum, compactification, interacting unitarity, or dynamical stability. The written proofs are accompanied by exact SageMath and property-based certificates and a narrow Lean 4 kernel, with their scope and dependencies disclosed.