GraviGUT unification with revisited Pati-Salam model

We propose a graviGUT unification scheme based on the simple orthogonal group $\mathrm{SO}(1,9 ,\mathbb{C})$ that resolves the chiral duplication of weak isospin in Pati--Salam models. In the conventional $SU(4)\times SU(2)_+\times SU(2)_-$ framework, the unobserved second chiral $SU(2)$ is typically removed by ad hoc high-energy scale breaking. Here we instead \emph{geometrize} it: one $SU(2)$ factor is identified with a chiral half of the Lorentz group, so it belongs to gravity rather than to an additional weak force. This identification becomes natural inside $\mathrm{SO}(1,9 ,\mathbb{C})$, where the algebra decomposes as $\mathfrak{so}(1,3)_{\mathbb{C}}\oplus\mathfrak{so}(6)_{\mathbb{C}}\oplus(\text{coset})$. We construct a parity-symmetric chiral action that, upon breaking \emph{dynamically} selects one chirality: the surviving Yang--Mills factor is identified with $SU(2)_+$, while the opposite chirality persists as the gravitational chiral connection. These lead to concrete phenomenological handles, including graviton and weak-boson vertices with the other fundamental forces in $SU(3)$ and $U(1)$ and parity-sensitive gravitational-wave signatures, that distinguish the $\mathrm{SO}(1,9,\mathbb{C})$ construction from both traditional Pati--Salam and larger, less economical unifications.