Universal generating function for all cohomologies on tuned (2,4) hypersurfaces in P^1×P^3

Establish that for smooth Calabi–Yau hypersurfaces X ⊂ P^1 × P^3 defined by f = x_0^2 f_0 + x_1^2 f_2 (with f_0 and f_2 general quartics), a single rational function encodes all line bundle cohomology series via expansions at (0,0), (0,∞), (∞,0), and (∞,∞), respectively yielding CS^0(X, O_X), CS^1(X, O_X), CS^2(X, O_X), and CS^3(X, O_X), with the stated signs.

Background

Specializing the defining polynomial to f = x_0^2 f_0 + x_1^2 f_2 changes the birational geometry: isolated flopping curves combine into a rigid divisor, altering the cohomology pattern. The authors conjecture that despite this non-generic complex-structure tuning, a single rational generating function still captures all cohomology dimensions when expanded at different points.

Confirming this would demonstrate robustness of the universal generating-function principle under significant complex-structure specializations that alter the birational behavior and cohomology support regions.