The Doublet Tower of Cl(4,2): Universal Nilpotency, Projector Identities, and the sl(8,R) Parabolic from Conformal Null Vectors

Abstract

The conformal geometric algebra Cl(4,2) ~ R(8) carries a natural Z_2-grading defined by the dilatation bivector D=\e_4e_0, which satisfies $D^2=1$ and splits the eight-dimensional spinor module into two four-dimensional eigenspaces $S=S_+\oplus S_-$. We prove that the ad_D-eigenvectors of Cl(4,2), those elements with eigenvalue~$\pm1$ under the Lie bracket $[D,\cdot]$, are exhausted by 16 doublets of the form $T^\pm_J=n_\bullet e_J$, where $n_\bullet \in {n_\infty,n_o}$ is a null vector and $e_J$ is a product of Lorentz-sector basis vectors indexed by $J\subseteq\{1,2,3,5\}$. These doublets satisfy universal algebraic identities: $(T^\pm_J)^2=0$ (nilpotency) and $T^+_J\,T^-_J=(-1)^{k(k+1)/2+1}\,η_J 2Π_+$ (single-projector proportionality), where $k=|J|$, $η_J=\prod_{j\in J}η_{jj}$ is the spectator metric, and $Π_+=\tfrac12(1+D)$. The contracted sums $\sum_{|J|=k}T^+_J T^-_J$ are shown to be controlled by the elementary symmetric polynomials $S_k$ of the Lorentz signature values $(+1,+1,+1,-1)$, with the vanishing of $S_2=0$ reflecting the $(3,1)$ signature of physical spacetime. The even-grade doublets (belonging to Cl^+(4,2) ~ Cl(4,1)) contribute $8Π_+$ to the unweighted grand total, while the odd-grade doublets cancel exactly. The 32 doublet eigenvectors generate, under the commutator bracket, the 63-dimensional Lie algebra sl(8,R) in which the conformal algebra so(4,2) embeds as a 15-dimensional subalgebra. All identities, including a closed form for the commutators across the Levi interface, are proved in every conformal extension Cl(p+1,q+1); there the commutator span misses precisely the center of the algebra, a parity dichotomy invisible at the Lorentz signature. Two applications are developed: the top rung of the tower computes the Dirac versus Majorana dichotomy of rank-five pseudoscalars ... [shortened]