Pure-Spinor Condensation Obstruction in the Spin(10) Envelope of Twistor Configuration Geometry

Abstract

The compatible-polarization note [Zhang Compatible Pure-Spinor Polarizations, DOI:10.5281/zenodo.20129212] identified a wall-and-weak-compatible pure-spinor polarization $W_+ = (\ell \wedge C) \oplus (\mathbf{2}_L \otimes r_+) \subset V_{10, \mathbb{C}}$ inside the Spin(10) envelope of Twistor Configuration Geometry. That result substantially narrowed the residual polarization postulate: once the P7 wall supplies the color/lepton split $\mathbf{4} = C \oplus \ell$, and once visible $SU(2)_L$ preservation is imposed from the operational weak boundary, the compatible maximal isotropic five-plane is essentially forced.

The remaining G1 question is sharper: can a TCG-native Spin(10)-invariant action on the native fields $\lambda \in \mathbf{16}$ and $H_a \in \mathbf{10}$ force this specific compatible pure-spinor representative as a vacuum, without smuggling in the wall orientation by hand? This note gives a theorem-level negative answer.

The ordinary pure-spinor potential $V_{\rm pure}(\lambda) = \kappa \sum_a |\lambda^T C \Gamma^a \lambda|^2 + \lambda_0 (\lambda^\dagger \lambda - v_R^2)^2$ can force the Cartan equations $Q^a(\lambda) = \lambda^T C \Gamma^a \lambda = 0$ and a nonzero norm, hence produces a pure-spinor condensate. But it cannot distinguish the compatible representative from the rest of the pure-spinor orbit. Three obstructions are isolated.

Theorem 1 (Yukawa-vanishing on pure-spinor locus, §3): the natural Spin(10)-invariant Yukawa coupling $V_{\rm couple}^{\rm hol} = y H_a Q^a(\lambda) + \mathrm{h.c.}$ arising from the bilinear channel $\mathbf{16} \otimes \mathbf{16} \supset \mathbf{10}$ vanishes identically on the pure-spinor locus $Q^a(\lambda) = 0$ (by definition of pure spinor in this representation). So it cannot distinguish $W_+$ from any other pure-spinor representative. A Lagrange-multiplier sharpening shows the vanishing is structural: a multiplier vanishes from the on-shell action precisely when its constraint is satisfied.

Theorem 3 (Single-vector cannot encode wall flag, §4): a single vector field $H_a \in \mathbf{10}$ with a Spin(10)-invariant potential $V_{\rm wall}(H)$ selects only a Spin(10)-orbit of vectors. Generic nonzero $H$ has stabilizer $\mathrm{Spin}(9)$ in the compact real form; null/isotropic $H$ has parabolic stabilizer in the complexified case. Neither stabilizer contains the Pati-Salam wall flag $\mathbf{4} = C \oplus \ell$, the subspace $\ell \wedge C \subset \wedge^2 \mathbf{4}$, or the weak-left-compatible two-plane $\mathbf{2}_L \otimes r_+$. Wall data requires a parabolic/subalgebra-type order parameter, equivalent to a higher SO(10) breaking representation ($\mathbf{45}/\mathbf{54}/\mathbf{126}/\overline{\mathbf{126}}/\mathbf{210}$) explicitly forbidden in the TCG-native discipline. A mixed-invariant loophole-closing remark covers candidates like $|H_a \Gamma^a \lambda|^2$: such terms correlate a vector with the pure-spinor annihilator $W_\lambda$ but still do not supply the missing wall flag.

Theorem 6 (No $\mathbf{10}$ in $\mathbf{16} \otimes \overline{\mathbf{16}}$, §5): for a single chiral Spin(10) spinor $\lambda \in \mathbf{16}$, the Hermitian bilinear representation is $\mathbf{16} \otimes \overline{\mathbf{16}} = \mathrm{End}(\mathbf{16}) = \mathbf{1} \oplus \mathbf{45} \oplus \mathbf{210},$ which contains no $\mathbf{10}$. Therefore $\lambda^\dagger \Gamma^a \lambda H_a$ is not a Spin(10)-invariant vector-channel scalar coupling. The only single-chiral-spinor vector-channel coupling is the holomorphic $\mathbf{16} \otimes \mathbf{16} \supset \mathbf{10}$ bilinear $Q^a(\lambda)$, which Theorem 1 kills.

Corollary (G1 obstruction, §6): under the constraints that the action use only TCG-native fields ($\lambda \in \mathbf{16}$, $H_a \in \mathbf{10}$) and not import standard heavy SO(10) breaking representations ($\mathbf{45}, \mathbf{54}, \mathbf{126}, \overline{\mathbf{126}}, \mathbf{210}$), no natural low-degree $\mathbf{16}+\mathbf{10}$ Spin(10)-invariant action template built from the native vector channel and the pure-spinor constraint — comprising the polynomial invariants $|Q|^2$, $H \cdot Q$, $H^2$, $\lambda^\dagger \lambda$, and the Hermitian bilinears $\lambda^\dagger (\mathbf{1}, \mathbf{45}, \mathbf{210}) \lambda$ — has the compatible pure-spinor representative $W_+$ as a forced vacuum representative.

Status: theorem-level OBSTRUCTION. Pure-spinor condensation is achievable (via $V_{\rm pure}$). Compatible pure-spinor condensation is NOT achievable without an additional structural input encoding the P7 wall and $SU(2)_L$-preserving polarization data.

Residual reformulation: the residual $P_{\rm pol}^{D_5}$ splits cleanly into the compatibility component $P_{\rm pol}^{D_5, \rm compat}: \text{ compatible pure-spinor polarization, substantially narrowed by the compatible-polarization analysis}$ and a new named action-level residual $X_{\rm wall-pol}: \text{ TCG-native dynamical source of the P7 wall + } SU(2)_L\text{-preserving polarization data},$ the latter now obstruction-bounded for the natural $\mathbf{16}+\mathbf{10}$ template. Neither is added to the active framework ledger.

Five failure modes (§7) record where the obstruction does NOT close: multi-field extension at cost of new postulates (F1); twistor-flag-to-polarization route via the chiral Penrose flag $\mathbb{CP}^1 \subset \mathbb{CP}^2 \subset \mathbb{CP}^3$ (G2 of the compatible-polarization note; outside present scope) (F2); boundary or auxiliary BV-BFV structure parallel to the electron-side $P_{\rm BFV}^{\rm sec}$ residual (F3); honest acceptance of compatibility as imposed input (F4); look-elsewhere expansion to higher representations explicitly forbidden by anti-evasion discipline (F5).

The active TCG/FPA postulate ledger is unchanged: $P_0$–$P_4$, $P_{5'}$, $P_6$, $P_7$, $P_{H'}$, $P_{SO(10)}$. $P_{\rm pol}^{D_5, \rm compat}$ and $X_{\rm wall-pol}$ are residual labels outside this ledger, not new framework axioms. Same maturity register as the boundary-superselection obstruction note [Zhang Boundary Superselection, DOI:10.5281/zenodo.20110780]: theorem-level action-level obstruction with named residual; no derivation, no active-ledger change.

Completes a structural parallel. The gauge-side arc (Spin(10) downstream-breaking note + pure-spinor polarization note + compatible-polarization note → present obstruction note) now has the same maturity register as the electron-side arc (bulk-boundary localization note + connected-residues note → boundary-superselection obstruction note), with both arcs supplying action-level obstruction notes that name precise residuals outside the active framework ledger.

DOI

https://doi.org/10.5281/zenodo.20141601