Condensation, Not Orientation: Why a Native Spin(10) Action Cannot Select the Wall-Compatible Pure Spinor

Yesterday’s compatible-polarization note substantially narrowed the gauge-side residual $P_{\rm pol}^{D_5}$. The P7 wall postulate already supplies the lepton line $\ell \subset \mathbf{4}$ and the color/lepton split $\mathbf{4} = C \oplus \ell$. Visible $SU(2)_L$ preservation forces the weak half of any compatible polarization to be $\mathbf{2}_L \otimes r$. Together these constrain the compatible pure-spinor representative to the form $W_+ = (\ell \wedge C) \oplus (\mathbf{2}_L \otimes r_+) \subset V_{10,\mathbb{C}},$ up to the expected $SU(2)_R$ gauge choice and conjugate orientation. The remaining G1 question was 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 representative as a vacuum, without smuggling in the wall orientation by hand?

Today’s note gives a theorem-level negative answer. The framework can produce a pure-spinor condensate. It cannot select the wall-compatible one.

The natural action template, and where it fails

The natural template is

$V(\lambda, H) = V_{\rm pure}(\lambda) + V_{\rm wall}(H) + V_{\rm couple}(\lambda, H),$

with the schematic pure-spinor potential of the pure-spinor polarization note,

$V_{\rm pure}(\lambda) = \kappa \sum_a |Q^a(\lambda)|^2 + \lambda_0 (\lambda^\dagger \lambda - v_R^2)^2, \quad Q^a(\lambda) = \lambda^T C \Gamma^a \lambda,$

forcing the pure-spinor locus $Q^a = 0$ and the normalization $\lambda^\dagger \lambda = v_R^2$. So far so good — a pure-spinor condensate is achievable.

But $V_{\rm pure}$ chooses an orbit, not a representative within it. The nonzero projective pure spinors of a given chirality form a single homogeneous variety with $U(5)$-type stabilizer (parabolic with Levi $GL(5,\mathbb{C})$ over the complex group). Choosing $\lambda \in \mathbf{16}$ rather than $\overline{\mathbf{16}}$ distinguishes the $S_+$ and $S_-$ chiral pure-spinor families, but it does not label the P7 wall line $\ell$, the color plane $C$, or the visible $SU(2)_L$ factor. Something more is needed to select $W_+$ from the rest.

The note proves that nothing more is available within the TCG-native $\mathbf{16}+\mathbf{10}$ discipline.

Three independent obstructions

Theorem 1 — Yukawa-vanishing. The natural Spin(10)-invariant vector-channel coupling $V_{\rm couple}^{\rm hol}(\lambda, H) = y H_a Q^a(\lambda) + \bar{y} H_a^\dagger \overline{Q^a(\lambda)}$ arising from the bilinear channel $\mathbf{16} \otimes \mathbf{16} \supset \mathbf{10}$ vanishes identically on the pure-spinor locus, because $Q^a(\lambda) = 0$ is the pure-spinor constraint by definition. The coupling is the quantity the pure-spinor potential just forced to vanish. It cannot distinguish $W_+$ from any other pure-spinor representative.

A Lagrange-multiplier sharpening makes this even sharper: if $H_a$ is treated as a multiplier enforcing $Q^a = 0$, it vanishes from the on-shell action precisely when the constraint is satisfied. The vanishing is structural, not a tuning artifact.

Theorem 3 — Single-vector cannot encode the wall flag. A single vector field $H_a \in \mathbf{10}$ with a Spin(10)-invariant potential $V_{\rm wall}(H)$ can select only a Spin(10)-orbit of vectors. In the compact real form a generic nonzero vector has stabilizer $\mathrm{Spin}(9) \subset \mathrm{Spin}(10)$; in the complexified discussion, null or isotropic vector choices have parabolic stabilizers. Neither of these contains the data needed for the compatible polarization:

$\mathbf{4} = C \oplus \ell, \qquad \ell \wedge C \subset \wedge^2 \mathbf{4}, \qquad \mathbf{2}_L \otimes r_+ \subset \mathbf{2}_L \otimes \mathbf{2}_R.$

That is flag data — a Pati-Salam-decomposition-plus-weak-orientation choice — and it is not vector-orbit data. To encode the flag dynamically one would need an order parameter transforming as a projector, an adjoint, or a higher tensor — exactly the kind of higher SO(10) breaking representation ($\mathbf{45}/\mathbf{54}/\mathbf{126}/\overline{\mathbf{126}}/\mathbf{210}$) that the TCG-native discipline explicitly forbids.

A mixed-invariant loophole-closing remark covers the natural candidate $|H_a \Gamma^a \lambda|^2$. Such terms can correlate $H$ with the pure-spinor annihilator $W_\lambda$ — for example, forcing $H$ to lie in or be orthogonal to $W_\lambda$ — but they still do not supply the missing wall flag. They correlate a vector with a pure spinor; they do not generate Pati-Salam-flag-plus-weak-orientation data.

Theorem 6 — No $\mathbf{10}$ in $\mathbf{16} \otimes \overline{\mathbf{16}}$. The last hope is a Hermitian variant $V_{\rm couple}^{\rm Herm} = y_2 \lambda^\dagger \Gamma^a \lambda H_a,$ which need not vanish on the pure-spinor locus. But this expression is not a Spin(10)-invariant scalar coupling for a single chiral $\mathbf{16}$. The Hermitian bilinear representation is

$\mathbf{16} \otimes \overline{\mathbf{16}} = \mathrm{End}(\mathbf{16}) = \mathbf{1} \oplus \mathbf{45} \oplus \mathbf{210}$

(Slansky 1981 standard decomposition; dimension check: $16^2 = 256 = 1 + 45 + 210$). There is no $\mathbf{10}$ in this decomposition. So $\lambda^\dagger \Gamma^a \lambda H_a$ is not a valid Spin(10)-invariant vector-channel scalar coupling. The only single-chiral-spinor vector-channel coupling available is the holomorphic $Q^a(\lambda)$, which Theorem 1 has already killed.

The corollary, and what it means

Combining the three obstructions: under TCG-native discipline (only $\mathbf{16}+\mathbf{10}$ fields, no import of $\mathbf{45}/\mathbf{54}/\mathbf{126}/\overline{\mathbf{126}}/\mathbf{210}$), no natural low-degree $\mathbf{16}+\mathbf{10}$ Spin(10)-invariant action template — comprising the polynomial invariants $|Q|^2$, $H \cdot Q$, $H^2$, $\lambda^\dagger \lambda$, and the Hermitian bilinears in $\mathbf{1} \oplus \mathbf{45} \oplus \mathbf{210}$ — has the compatible representative $W_+$ as a forced vacuum.

Pure-spinor condensation: achievable. Compatible pure-spinor condensation: 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:

$P_{\rm pol}^{D_5} \leadsto P_{\rm pol}^{D_5,\rm compat} + X_{\rm wall-pol}.$

Here $P_{\rm pol}^{D_5,\rm compat}$ is the compatibility component substantially narrowed by the compatible-polarization analysis: once P7 and visible $SU(2)_L$ are supplied, the compatible polarization is constrained to the $W_+$ form. And $X_{\rm wall-pol}$ is the new named action-level residual:

$X_{\rm wall-pol}$: a TCG-native dynamical source of the P7 wall + $SU(2)_L$-preserving polarization data.

Today’s theorem says that $X_{\rm wall-pol}$ is not supplied by a single $\mathbf{10}$-vector coupled to a single chiral $\mathbf{16}$ pure-spinor condensate.

Neither is in the active framework ledger:

$P_0, \ldots, P_4, \quad P_{5'}, \quad P_6, \quad P_7, \quad P_{H'}, \quad P_{SO(10)} \quad \text{(unchanged)}.$

This is important. The result is not a hidden positive derivation. It is an obstruction note: it names the missing action-level datum and keeps it outside the active ledger. The 2026-05-01 framework closure verdict is respected — the purpose is to record a sharp negative result at the action-level layer, not to continue an open-ended search by renaming residual assumptions as derived content.

Five failure modes

The obstruction is deliberately narrow. Five logically possible routes are recorded:

• F1. Multi-field extension. A second vector or additional tensorial order parameters could in principle encode more than a single vector orbit. But such an extension adds new structural assumptions; unless those fields are independently derived from the TCG ledger, the obstruction is merely displaced.
• F2. Twistor-flag-to-polarization. The wall data might come not from a bulk Spin(10) Higgs potential but from the chiral Penrose twistor flag $\mathbb{CP}^1 \subset \mathbb{CP}^2 \subset \mathbb{CP}^3$. This is the G2 future route named in the compatible-polarization note. It is outside the present paper’s scope, and invoking it here would conflate the internal gauge sector with Lorentz/twistor geometry — which is exactly what the anti-evasion discipline forbids in this context.
• F3. Boundary or auxiliary structure. The wall-compatible polarization could be encoded in a boundary action or auxiliary BV–BFV sector rather than a bulk Higgs potential. This would be structurally parallel to the electron-side $P_{\rm BFV}^{\rm sec}$ residual from the boundary-superselection obstruction note.
• F4. Compatibility as explicit input. The honest fallback is to retain compatibility as an imposed but sharply narrowed residual ($P_{\rm pol}^{D_5,\rm compat}$), with $X_{\rm wall-pol}$ as a named action-level source still missing.
• F5. Look-elsewhere expansion. Adding higher SO(10) breaking representations would solve a different problem (conventional GUT model-building) and would collapse the distinction between native TCG derivation and standard unification machinery. Explicitly forbidden.

Where the structural arc stands now

This note completes a structural parallel across two of the three arcs:

The gauge-side arc has matched the electron-side arc’s two-layer structure: narrowing via structural-motivation closure, then theorem-level obstruction at the action level with a precisely named residual outside the active ledger. The framework’s structural picture is now its most precise version yet, and the active TCG/FPA postulate ledger remains exactly what it was before the gauge-arc work began.

The shared open layer is still the same: action-level / dynamical-completion / vacuum-mechanism construction. On the gauge side, the live future route is now G2 — derive the compatible polarization from the chiral Penrose twistor flag, outside the bulk Spin(10) action that today’s theorem has bounded.

The paper, Pure-Spinor Condensation Obstruction in the Spin(10) Envelope of Twistor Configuration Geometry, is on Zenodo (DOI 10.5281/zenodo.20141601; CC-BY-4.0). Ten pages, three theorems, one corollary, five failure modes, twelve references. Refinement trail: Tier-3 G1 prompt to GPT-5.5 Pro under strict anti-evasion guards → GPT G1 verdict OBSTRUCTED with three independent theorem-level no-gos → GPT draft paper → Claude house-style consistency pass + independent review → GPT fresh-session review (verdict: publish with minor precision revisions) → Claude application of six round-2 precision edits → Claude final review.

Condensation, not orientation. The action selects orbits; orientations require structure the action does not supply.