The Spin(10) Envelope of Twistor Configuration Geometry: A Postulate-Equivalent Completion of the SU(2)

Abstract

The wall-deletion paper [Zhang Wall Deletion, DOI:10.5281/zenodo.20045987] introduces the Weyl-lift postulate $P_7$ and shows that the un-broken $A_3$ root datum at the top stratum of Twistor Configuration Geometry carries the Pati–Salam algebra $\mathfrak{su}(4)$ , while end-root parabolic Levi reduction yields the Pati–Salam Levi sub-algebra $\mathfrak{su}(3)_C \oplus \mathfrak{u}(1)_{(B-L)/2}$ . Combined with the $n=2$ stratum’s $A_1 \cong \mathfrak{su}(2)_L$ , the framework supplies four of the five rank generators of full Pati–Salam $\mathfrak{su}(4)_C \oplus \mathfrak{su}(2)_L \oplus \mathfrak{su}(2)_R$ . The single missing factor is $\mathfrak{su}(2)_R$ . The parabolic note [Zhang Parabolic, DOI:10.5281/zenodo.20090828] closes middle-root deletion of $A_3$ as a candidate source: the two $\mathfrak{sl}_2$ factors are the left and right Lorentz spinor algebras of complexified spacetime via $SL_4 / P_{\alpha_2} \cong G(2,4)$ , not internal weak-isospin. Chiral doubling of the $n=2$ stratum’s $A_1$ likewise fails: the two simple-root $A_1$ ‘s of $A_2$ do not commute, with closure equal to $\mathfrak{sl}_3 = A_2$ .

The cleanest available completion is the Spin(10) envelope: the regular maximal subalgebra branching $D_5 \supset D_3 \oplus D_2$ , with $D_3 \cong A_3$ and $D_2 \cong A_1 \oplus A_1$ , gives in compact form $\mathfrak{so}(10) \supset \mathfrak{so}(6) \oplus \mathfrak{so}(4) \cong \mathfrak{su}(4)_C \oplus \mathfrak{su}(2)_L \oplus \mathfrak{su}(2)_R$ exactly. The chiral spinor $\mathbf{16}$ branches under Pati–Salam as $(\mathbf{4}, \mathbf{2}, \mathbf{1}) \oplus (\bar{\mathbf{4}}, \mathbf{1}, \mathbf{2})$ , supplying one Standard Model family in all-left-handed Weyl notation including a right-handed neutrino. The vector $\mathbf{10}$ branches as $(\mathbf{6}, \mathbf{1}, \mathbf{1}) \oplus (\mathbf{1}, \mathbf{2}, \mathbf{2})$ , with $\mathbf{6} = \wedge^2 \mathbf{4}$ recovering the antisymmetric Pati–Salam representation of $P_{H'}$ [Zhang Hadronic, DOI:10.5281/zenodo.20077931].

We state the completion as a new postulate $P_{SO(10)}$ (with stronger spinorial reading $P_{SO(10)}^{\rm spin}$ as preferred motivation) and emphasize that this is postulate-equivalent, not a derivation: the existing $P_7$ machinery is parabolic-Levi-based, while the Spin(10) step is a maximal-subalgebra envelope completion outside $P_7$ ‘s scope. No new numerical constant relation is proposed; the chiral spinor $\mathbf{16}$ is used as structural motivation, not as a representation-volume observable. $P_{SO(10)}$ extends the gauge-structure ledger, not the dimensionless-constant formula grammar of the framework’s empirical body. Six explicit gaps are listed (G1–G6). The active TCG/FPA postulate ledger is updated to read $P_0$ – $P_4$ , $P_{5'}$ , $P_6$ , $P_7$ , $P_{H'}$ , $P_{SO(10)}$ .

DOI

https://doi.org/10.5281/zenodo.20091562