Pure-Spinor Polarization and Standard-Model Breaking in the Spin(10) Envelope of Twistor Configuration Geometry

Abstract

The Spin(10) envelope of Twistor Configuration Geometry (TCG) closes the algebraic missing-$SU(2)_R$ gap by embedding the existing $A_3 \oplus A_1$ gauge data into $D_5 \supset D_3 \oplus D_2 \cong A_3 \oplus A_1^L \oplus A_1^R$ [Zhang Spin10 Envelope, DOI:10.5281/zenodo.20091562]. The downstream paper [Zhang Spin10 Downstream v2, DOI:10.5281/zenodo.20115884] splits the remaining residual into $P_{SO(10)}^{\rm br}$ (breaking vacuum + weak-boundary asymmetry) and $P_{\rm fam}$ (family triplication). A direct action-level attack on $P_{SO(10)}^{\rm br}$ via Spin(10)-invariant scalar potential on the TCG-native fields $\mathbf{10}_H \oplus \mathbf{16}_H/\overline{\mathbf{16}}_H$ closes as OBSTRUCTED with theorem-level proof (invariant-potential orbit obstruction): a $G$-invariant potential selects orbits, not named left/right VEV directions, so any alignment requires additional structural input.

This paper investigates a different action-level mechanism: pure-spinor polarization. A nonzero pure chiral spinor $\lambda \in \mathbf{16}$ of $\mathrm{Spin}(10)$ determines a maximal isotropic polarization $W \subset V_{10,\mathbb{C}}$ of complex dimension 5. Proposition 1 (Pure-spinor stabilizer): Over $\mathrm{Spin}(10,\mathbb{C})$, the stabilizer of a pure-spinor line is a parabolic subgroup with Levi factor $GL(5,\mathbb{C})$. On the compact real form, the stabilizer of the corresponding line is $U(5)$-type, while the stabilizer of a normalized representative with phase fixed is $SU(5)$-type [Cartan 1938, Chevalley 1954]. The proposed mechanism reads the Standard Model group as the intersection $G_{\rm SM} = G_{\rm PS} \cap G_\lambda$ where $G_{\rm PS} = SU(4)_C \times SU(2)_L \times SU(2)_R$ is the Pati-Salam subgroup supplied by the TCG Spin(10) envelope, and $G_\lambda \simeq SU(5)$ is the stabilizer of a pure chiral spinor.

Intersection theorem (§3, root-system computation): For a pure-spinor polarization $W = W_3 \oplus W_2$ compatible with the Pati-Salam vector split $V_{10} = V_6 \oplus V_4$, the root-system intersection is $\Phi(A_4) \cap \Phi(D_3 \oplus D_2) = \{e_i - e_j : 1 \le i \neq j \le 3\} \cup \{\pm(e_4 - e_5)\} = A_2 \oplus A_1,$ giving the SM semisimple part $\mathfrak{su}(3)_C \oplus \mathfrak{su}(2)_L$. The leftover Cartan direction inside the $A_4$ Cartan $Y \propto \mathrm{diag}\left(-\tfrac{1}{3}, -\tfrac{1}{3}, -\tfrac{1}{3}, \tfrac{1}{2}, \tfrac{1}{2}\right)$ commutes with the $A_2$ roots and the surviving $A_1$ root $e_4 - e_5$. In the Pati-Salam Cartan it is exactly $Y = T_{3R} + (B-L)/2$ in Pati-Salam normalization. Therefore under the compatibility hypothesis, $\mathfrak{su}(5)_\lambda \cap (\mathfrak{su}(4)_C \oplus \mathfrak{su}(2)_L \oplus \mathfrak{su}(2)_R) = \mathfrak{su}(3)_C \oplus \mathfrak{su}(2)_L \oplus \mathfrak{u}(1)_Y$ up to standard $\mathbb{Z}_6$ finite quotient at the group level.

Compatibility hypothesis (essential): A generic pure-spinor stabilizer is conjugate to $SU(5)$ inside $\mathrm{Spin}(10)$ but need not be aligned with the already-chosen $D_3 \oplus D_2$ Pati-Salam splitting. Deriving the compatible polarization is the residual addressed by this paper.

TCG-native action-level potential (§4): The pure-spinor constraint $\lambda^T C \Gamma^a \lambda = 0$ ($a = 1, \ldots, 10$) lives in the native vector channel $\mathbf{16} \otimes \mathbf{16} \supset \mathbf{10}$ (standard decomposition $\mathbf{16}_s \otimes \mathbf{16}_s = \mathbf{10} \oplus \mathbf{120} \oplus \mathbf{126}$ supplies the symmetric vector part). A minimal positive potential is $V_{\rm pure}(\lambda) = \kappa \sum_{a=1}^{10} \left|\lambda^T C \Gamma^a \lambda\right|^2 + \lambda_0 \left(\lambda^\dagger \lambda - v_R^2\right)^2, \quad \kappa, \lambda_0 > 0,$ whose minima are normalized pure spinors. A related auxiliary-vector realization through a field $H_a \in \mathbf{10}$ shows that the relevant invariant is native to the $\mathbf{10}$ vector channel (with sign-convention caveats; $V_{\rm pure}$ is the primary action-level candidate). The construction uses only the two TCG-native Spin(10) representations $\mathbf{16}$ and $\mathbf{10}$ — no import of standard heavy $SO(10)$ breaking representations $\mathbf{45}$, $\mathbf{54}$, $\mathbf{126}$, $\overline{\mathbf{126}}$, $\mathbf{210}$ [Slansky 1981, Mohapatra 2003].

Status: partial positive, not theorem-level closure. The route reformulates the residual from $P_{SO(10)}^{\rm br,align}: \text{ choose right-handed-neutrino VEV direction in } \mathbf{16}_H$ to a sharper polarization-compatibility target $P_{\rm pol}^{D_5}: \text{ derive a TCG-native pure-spinor polarization compatible with the } D_3 \oplus D_2 \text{ split.}$ The latter points to a specific geometric target: derivation of compatible polarization from the chiral Penrose twistor flag $\mathbb{CP}^1 \subset \mathbb{CP}^2 \subset \mathbb{CP}^3$ underlying TCG. Five open gaps G1-G5 (compatible polarization; chiral twistor origin; vacuum dynamics; family triplication unchanged; $P_{5'}$ normalization unchanged).

The active TCG/FPA postulate ledger is unchanged: $P_0$-$P_4$, $P_{5'}$, $P_6$, $P_7$, $P_{H'}$, $P_{SO(10)}$. $P_{\rm pol}^{D_5}$ is a residual label outside this ledger, NOT a new framework axiom. Same maturity register as the bitwistor pair-channel note [Zhang Bitwistor, DOI:10.5281/zenodo.20111389]: partial positive mechanism reformulation of a named residual, no derivation, no active-ledger change.

DOI

https://doi.org/10.5281/zenodo.20116476