Wall Deletion in Twistor Configuration Geometry: A (B−L) Quantization Bridge to Pati–Salam Unification

Abstract

We promote the chamber decomposition of TCG from a chamber count to a full type-$A$ Weyl-arrangement structure (postulate P7, named here for the first time). Under P7 the strata $n = 1, 2, 3$ carry root-system data of empty/trivial, $A_1$, and $A_3$ type, with stratum-wise Lie algebra $\mathfrak{su}(2) \oplus \mathfrak{su}(4)$ of total Cartan rank $4$, equal to the Standard Model rank.

A single wall deletion at stratum $n = 3$ — the parabolic Levi reduction induced by deleting one simple root from the $A_3$ Dynkin diagram, realized geometrically as projection along the deleted-root direction in the $A_3$ Cartan subalgebra — reduces $\mathfrak{su}(4)$ to $\mathfrak{su}(3) \oplus \mathfrak{u}(1)$. Combined with the un-deleted $A_1 = \mathfrak{su}(2)$ at stratum $n = 2$, the result is an abstract $\mathfrak{su}(3) \oplus \mathfrak{su}(2) \oplus \mathfrak{u}(1)$, isomorphic as a Lie algebra to the Standard Model gauge algebra but not equal to it as a physical gauge structure (the abelian factor is identified as $(B-L)/2$, not weak hypercharge $Y$).

We compute the abelian generator explicitly: the wall-deleted $\mathfrak{u}(1)$ direction at end-deletion is the fundamental weight $\omega_3 = \tfrac{1}{4}(1, 1, 1, -3)$, with eigenvalues on the $A_3$ fundamental in ratio $1 : -3$, identifying it (up to overall normalization) as the Pati–Salam $(B-L)/2$ generator from $SU(4) \to SU(3)_C \times U(1)_{B-L}$.

The wall-deleted $\mathfrak{u}(1)$ is therefore not Standard Model weak hypercharge $Y$; the construction lands at a sub-algebra of the Pati–Salam unification group $SU(4) \times SU(2)_L \times SU(2)_R$, missing the $\mathfrak{su}(2)_R$ factor required for full Standard Model hypercharge $Y = T_{3R} + (B-L)/2$.

$(B-L)$ charge quantization (a partial resolution of TCG Gap 6) follows automatically from $A_3$ weight-lattice structure: quark $(B-L)/2 = +1/6$, lepton $(B-L)/2 = -1/2$. Full electric-charge quantization at the Standard Model level still requires the missing $T_{3R}$.

We identify Lie-theoretic simple-root deletion, geometric Cartan-projection in TCG, and frame-field constraint in Man’s affine-connection representation as parallel realizations of structure-group reduction; we do not claim one-to-one derivation between them.

Four diagnostic questions are pre-registered, with the missing-$\mathfrak{su}(2)_R$ question (Q4) as the principal new structural gap. The paper does not derive Standard Model phenomenology; it identifies a structural bridge to Pati–Salam unification, conditional on P7.

DOI

https://doi.org/10.5281/zenodo.20045987