Gamma conjecture II and explicit Stokes data for a quadric projective bundle

We study the connection problem for the quantum differential equation of the Fano fourfold $X={\text{PP}}_{Q^3}(\text{OO}\oplus \text{OO}(1))$. The problem is to identify its canonical asymptotic solutions with exceptional objects and compute the associated Stokes data. We use exact Mellin–Bessel integral representations and uniform vector asymptotics on sectors wider than $\pi$, followed by analytic continuation that preserves the Gamma framing. Cohomological chiral symmetry and mutations of exceptional collections put all eight sections at a common phase. This proves Gamma conjecture II for $X$ and determines a full exceptional collection and its Stokes and central connection matrices at the unit Novikov point and a specified admissible phase. The chiral symmetry also gives a flat symmetric bilinear pairing, an orthogonality identity for the Stokes matrix, and an integral unipotent square root of the monodromy matrix for a full turn. We obtain explicit formulas for the connection data at the opposite phase; on the Gamma lattice, the half-turn acts by tensoring with a square root of the anticanonical bundle. The same spinor integral transform constructs two canonical Gamma sections for ${\text{PP}}_{Q^{2m+1}}(\text{OO}\oplus \text{OO}(1))$, $m\ge 1$, at positive parameters with distinct spectrum. Their sectors have opening greater than $\pi$, and their leading constants are independent of the dimension.