Integral Stokes comparison for the completed fourfold mirror

Establish the comparison between the full derived direct image of the fourfold potential and the iterated rank-eight de Rham module, including the fourfold Brieskorn lattice, its higher residue pairing, and its integral rapid-decay homology, in order to identify the integral Stokes data.

Background

The appendix constructs an algebraic de Rham transform for the coupled mirror potential and proves that its resulting complex meromorphic differential module has rank eight and is isomorphic to the scalar system governing the quantum connection. The residue pairing is also identified with the Frobenius pairing of small quantum cohomology.

However, these algebraic and Frobenius-level identifications do not establish the integral mirror comparison. In particular, the paper does not construct the relevant fourfold Brieskorn lattice, higher residue pairing, or integral rapid-decay homology, nor does it identify the Barnes or oscillatory contours with primitive integral thimbles. Establishing this comparison would provide the missing integral interpretation of the Stokes data.

References

It also leaves open the comparison of the full derived direct image of the fourfold potential with the iterated module in top degree. A complete comparison must include the fourfold Brieskorn lattice, its higher residue pairing, and its integral rapid-decay homology. Theorem~\ref{aux:residueiso} identifies the $z=0$ Frobenius algebra, while Theorem~\ref{aux:transformtheorem} identifies a complex meromorphic connection. The integral Stokes comparison remains to be established.

— Gamma conjecture II and explicit Stokes data for a quadric projective bundle  (2610.13331 - Li, 9 Oct 2026) in Remark aux:complexboundary, Appendix, subsection “The exponential de Rham transform”