End and Middle Parabolics of A_3 in Twistor Configuration Geometry: Internal Color versus Spacetime Incidence

Abstract

The wall-deletion paper [Zhang Wall Deletion] introduces postulate P7 (Weyl-lift), under which the FPA chamber count $r_3! = 24 = |W(A_3)|$ at the top stratum of Twistor Configuration Geometry is identified with the un-broken $A_3$ root-system data, whose compact real form is $\mathfrak{su}(4)$, the Pati–Salam unification algebra. The wall-deletion paper takes the end-root parabolic Levi reduction $A_3 \to A_2 \oplus \mathfrak{u}(1) = \mathfrak{su}(3) \oplus \mathfrak{u}(1)_{(B-L)/2}$, lands at a Pati–Salam Levi sub-algebra, and identifies the missing $\mathfrak{su}(2)_R$ as the principal new structural gap.

This note investigates whether the second natural parabolic Levi of $A_3$ — deletion of the middle simple root $\alpha_2$, yielding $\mathfrak{sl}_2 \oplus \mathfrak{sl}_2 \oplus \mathfrak{u}(1)$ — could supply the missing internal $SU(2)_R$. The shape resembles $SU(2)_L \times SU(2)_R \times U(1)$, but the computation closes negatively for the internal-$SU(2)_R$ reading. The middle-parabolic abelian generator has eigenvalue pattern $(1, 1, -1, -1)/2$ on the fundamental — a $2{+}2$ split rather than the $3{+}1$ split of $(B{-}L)/2$ — and is not weak hypercharge. The two $\mathfrak{sl}_2$ factors are not internal weak-isospin: the homogeneous space $SL_4/P_{\alpha_2} \cong G(2,4)$ is the twistor-line Grassmannian, and the Levi $GL(2) \times GL(2)$ acts on the tangent factors $\Pi$ and $\mathbb{C}^4/\Pi$ as the two-component left and right Lorentz spinor representations $\mathfrak{sl}_2(\mathbb{C})_L \oplus \mathfrak{sl}_2(\mathbb{C})_R$.

The structural conclusion is that the same complex $A_3$ root datum at the $n=3$ stratum of TCG supports two complementary parabolic Levi readings: internal Pati–Salam (compact form $\mathfrak{su}(4)$) using the end-root parabolic for the $3{+}1$ color/lepton split, and external Lorentz/twistor incidence (complexified $\mathfrak{sl}_2 \oplus \mathfrak{sl}_2$) using the middle-root parabolic for the $2{+}2$ Lorentz spinor split. The Grassmannian $G(2,4)$ appearing here is the same one appearing in P5’ [Zhang EW] (line-deformation bundle base) and in $P_H'$ [Zhang Hadronic] (Plücker ambient $\mathbb{P}(\wedge^2\mathbf{4}) \cong \mathbb{CP}^5$). The middle-parabolic identification makes the Lie-algebraic origin of the Grassmannian explicit.

The internal $SU(2)_R$ gap of the wall-deletion paper is not closed by middle deletion and remains open. Four candidate sources for $SU(2)_R$ are surveyed without endorsement: chiral doubling of the $n=2$ $A_1$; embedding in a larger unification algebra such as $\mathfrak{so}(10)$ via $\mathrm{spin}(6) \oplus \mathrm{spin}(4)$; a spinor-internal correspondence at a deeper level; or acceptance that TCG lands at the Pati–Salam Levi sub-algebra.

The note records a structural observation; it does not introduce a new postulate. The active TCG/FPA postulate ledger remains P0–P4, P5’, P6, P7, $P_H'$.

DOI

https://doi.org/10.5281/zenodo.20090828