Compatible Pure-Spinor Polarizations from P7 Wall Data in the Spin(10) Envelope of Twistor Configuration Geometry

The Spin(10) envelope of Twistor Configuration Geometry (TCG) closes the algebraic missing-SU(2)_R gap at postulate-envelope level, but it leaves the breaking-vacuum residual: why does the right-handed factor hide while the observed low-energy weak boundary is left-handed? The previous pure-spinor polarization note (Zhang 2026, DOI:10.5281/zenodo.20116476) reformulated this problem as an intersection mechanism: a pure spinor in the chiral 16 has an SU(5)-type stabilizer, and a compatible SU(5) stabilizer intersects the Pati-Salam subgroup in the Standard Model group. Paper #30 named the remaining compatibility question as the residual P_{pol}^{D_5} but did not derive it.

This paper attacks the compatibility part of P_{pol}^{D_5} using only existing TCG data. We show that the existing P7 end-wall deletion (Wall Deletion, Zhang 2026, DOI:10.5281/zenodo.20045987) supplies the lepton line ℓ ⊂ 4, hence the color/lepton split 4 = C ⊕ ℓ. Under the Spin(10) vector branching 10 → (6, 1, 1) ⊕ (1, 2_L, 2_R) with 6 = ∧²4, this datum canonically supplies the color part ℓ ∧ C of a maximal isotropic five-plane in V_{10,C}. Proposition 1 (§3) proves ℓ ∧ C ⊂ ∧²4 is a 3-dimensional maximal isotropic subspace for the wedge-pairing quadratic form, with the conjugate ∧²C playing the corresponding role for the conjugate orientation.

If one additionally imposes preservation of the visible SU(2)_L factor — a compatibility input, not derived here; it is the same physical observation that P_{5'} (g_{2,W}^2 = 4/(3π)) targets the left-handed weak factor only — the weak part of any compatible maximal isotropic plane is forced to be 2_L ⊗ r for a line r ⊂ 2_R; all such r are gauge-equivalent under SU(2)_R (§4 Lemma). The resulting polarization

  W_+ = (ℓ ∧ C) ⊕ (2_L ⊗ r_+)

(or its conjugate W_-) is a maximal isotropic 5-plane in the complexified Spin(10) vector representation, hence determines a projective pure spinor (§5 Theorem). Its stabilizer intersection with Pati-Salam is the Standard Model group:

  SU(5)_{W_+} ∩ (SU(4)_C × SU(2)_L × SU(2)_R) ≃ S(U(3) × U(2))

up to the standard finite quotient, with hypercharge Y = T_{3R} + (B-L)/2 in the (B-L)/2-symmetric normalization (§6). The proof is made explicit via a determinant reduction (§6 Proposition 5): for g_4 = diag(A, a) ∈ SU(4)_C preserving the C ⊕ ℓ split and g_R ∈ SU(2)_R preserving r_+ with phase b, the SU(5)_{W_+} stabilizer condition imposes a² b² = 1, reducing the two U(1) phases to a single U(1)_Y. The charge decomposition

  W_+ ≅ (3, 1)_{-1/3} ⊕ (1, 2)_{+1/2}

matches the SU(5) fundamental 5 under the Standard Model subgroup. (The matter content remains in the chiral spinor 16 of Spin(10), whose SU(5) ⊃ G_SM decomposition is the usual 5̄ ⊕ 10 ⊕ 1 packaging one full generation; the 5 here is the polarization subspace, not a matter multiplet.)

Verdict: partial positive — the compatibility part of P_{pol}^{D_5} is substantially constrained by P7 wall data and visible SU(2)_L preservation. The compatible polarization is no longer arbitrary: it is determined by the end-wall lepton line, the color three-plane, and the requirement of preserving the observed left-handed weak factor, up to the expected SU(2)_R gauge choice and conjugate orientation. The remaining residual is no longer a choice of right-handed-neutrino VEV direction, but the action-level existence of a pure-spinor condensate in this compatible orbit. The construction uses only the two TCG-native Spin(10) representations (10 and 16); it does not import standard heavy SO(10) breaking representations (45, 54, 126, 126-bar, 210).

Five explicit gaps remain (§7): G1 pure-spinor condensation (action-level); G2 compatibility as boundary condition or theorem; G3 conjugate orientation W_+ vs W_-; G4 breaking scale and P_{5'} value; G5 family triplication. Active TCG/FPA postulate ledger UNCHANGED: P_0–P_4, P_{5'}, P_6, P_7, P_{H'}, P_{SO(10)}. P_{pol}^{D_5} remains as a residual label outside the active framework ledger, NOT as a new framework axiom — sharpened but not closed. Same maturity register as Papers #28 (Bitwistor Pair Channels, DOI:10.5281/zenodo.20111389) and #30 (Pure-Spinor Polarization, DOI:10.5281/zenodo.20116476): partial positive mechanism progress on a named residual, no derivation, no active-ledger change.