Bitwistor Pair Channels and the Baryon-State Gap in $P_{H'}$

Abstract

The hadronic extension $P_{H'}$ of Twistor Configuration Geometry (TCG) reads the Lenz observation $m_p/m_e \approx 6\pi^5$ [Lenz 1951] through the Pati—Salam representation-volume invariant $\mathcal{L}_H = \dim(\wedge^2 \mathbf{4})! \cdot \mathrm{Vol}_{FS}(\mathbb{P}(\wedge^2 \mathbf{4})) = 6! \cdot \pi^5/5! = 6\pi^5$ [Zhang Hadronic, DOI:10.5281/zenodo.20102322]. The hadronic-extension paper classified this as a phenomenological boundary condition rather than a theorem, and a pre-registered second-observable audit (recorded in the v2 revision) closed negatively: no additional hadronic or structural observable survives the strict grammar $X_R = \dim(R) \cdot \pi^{\dim(R)-1}$. Therefore $P_{H'}$ is a single-anchor structural reading of the Lenz ratio. Four subgaps remain open: G1 (why $\wedge^2 \mathbf{4}$ appears at all); G2 (how a two-index pair channel connects to a three-quark proton); G3 (why the full $S_6$ representation-slot measure); G4 (why the ratio is normalized by the electron mass).

This companion note investigates whether bitwistor geometry supplies a natural baryonic interpretation of the two-index sector. The verdict is a partial positive for G1 and G2; G3 and G4 are not addressed.

Proposition (Two bitwistor volumes, §2). With Fubini—Study normalization $\int_{\mathbb{CP}^1} \omega_{FS} = \pi$: $\mathrm{Vol}_{FS}(\mathbb{P}(\wedge^2 T)) = \mathrm{Vol}_{FS}(\mathbb{CP}^5) = \frac{\pi^5}{5!}, \qquad \mathrm{Vol}_{FS}(G(2,4)) = \frac{\pi^4}{12},$ where the simple-bitwistor locus $G(2,4) \subset \mathbb{CP}^5$ is the Klein quadric, a smooth degree-2 hypersurface of complex dimension 4 [Griffiths—Harris 1978]. The Lenz invariant requires the full projective bitwistor space; the decomposable locus would change the relevant volume and no longer yield $6\pi^5$.

Off-shell pair-channel hypothesis (§5). The hadronic Lenz boundary condition $P_{H'}$ is controlled by the full off-shell projective bitwistor pair-channel $\mathbb{P}(\wedge^2 \mathbf{4})$, not only by the decomposable simple-bitwistor locus $G(2,4)$. “Off-shell” here means unconstrained by the Plücker simplicity relation $B \wedge B = 0$, not off-shell in the dynamical QFT propagator sense. The quantum-mechanical reading is that the state space of an antisymmetric two-particle sector is the full projective Hilbert space $\mathbb{P}(\wedge^2 \mathbf{4})$, with the decomposable locus consisting of simple-wedge classical states and generic non-simple points representing superpositions or entangled antisymmetric pair states.

Pati—Salam decomposition and baryon projection (§3). Under $SU(4)_C \to SU(3)_C \times U(1)_{B-L}$, $\mathbf{4} = (\mathbf{3}, +1/3) \oplus (\mathbf{1}, -1)$ and $\wedge^2 \mathbf{4} = (\bar{\mathbf{3}}, +2/3) \oplus (\mathbf{3}, -2/3)$ — antisymmetric two-quark color sector plus quark—lepton mixed sector. The baryonic relevance enters through $\mathbf{4} \otimes \wedge^2 \mathbf{4} \cong \wedge^3 \mathbf{4} \oplus \mathbf{20}$ with $\wedge^3 \mathbf{4} \cong \bar{\mathbf{4}} = (\mathbf{1}, +1) \oplus (\bar{\mathbf{3}}, -1/3)$, where $(\mathbf{1}, +1)$ is the color-singlet three-quark baryon channel. The pair-channel reading is $qqq \sim q + (qq)$, with the $(qq)$ pair governed by $\wedge^2 \mathbf{4}$ and the $q + (qq)$ projection mapping into the color-singlet baryon channel via $\mathbf{4} \otimes \wedge^2 \mathbf{4} \to \wedge^3 \mathbf{4}$.

Spin(10) consistency (§4). The Spin(10) vector branches as $\mathbf{10} \to (\mathbf{6}, \mathbf{1}, \mathbf{1}) \oplus (\mathbf{1}, \mathbf{2}, \mathbf{2})$ with $\mathbf{6} = \wedge^2 \mathbf{4}$, so the bitwistor pair-channel sits inside the Spin(10) envelope as the electroweak-singlet $SU(4)_C$ component.

Three explicit caveats: (i) twistor-versus-Pati—Salam real-form distinction — “bitwistor” is used at the level of the complex $A_3$ representation $\wedge^2 \mathbb{C}^4$; the external Penrose-twistor and internal Pati—Salam $SU(4)_C$ interpretations are different physical readings of the same complex representation data inside the TCG architecture; spacetime twistor indices are NOT identified with internal color-lepton indices in the standard field-theoretic sense; (ii) “off-shell” means unconstrained by Plücker simplicity, NOT off-shell in the QFT propagator sense; (iii) $6! \neq |W(SU(4))| = 4!$ — the labeled-slot factor is FPA-style, not forced by bitwistor representation theory alone (G3 not closed).

Six failure modes (§7): F1 simple-bitwistor collapse, F2 slot-measure obstruction, F3 baryon projection without mass control, F4 electron normalization gap, F5 one-anchor limitation (audit verdict respected), F6 flavor/isospin specificity gap (paper does not distinguish proton from neutron, $\Delta$, $\Lambda$, $\Sigma$).

The active TCG/FPA postulate ledger is unchanged: $P_0$—$P_4$, $P_{5'}$, $P_6$, $P_7$, $P_{H'}$, $P_{SO(10)}$. The bitwistor pair-channel reading is the structural content of the existing $P_{H'}$ postulate, not a new framework axiom. $P_{H'}$ status under audit verdict is also unchanged: still a single-anchor phenomenological structural reading; the bitwistor construction does not reopen broad hadronic scanning. Same maturity register as electron-side companions (Bulk—Boundary Localization, Connected Boundary Residues, Boundary Superselection Obstruction): clarification, not derivation.

DOI

https://doi.org/10.5281/zenodo.20111389