Universal generating function for all cohomologies on the CICY threefold with configuration matrix [P^1|1 1; P^4|1 4]

Establish that for the Calabi–Yau threefold X defined as a general complete intersection of bidegrees (1,1) and (1,4) in P^1 × P^4 (CICY #7885), the rational function specified by the authors 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).

Background

For this CICY, the effective cone has three chambers (two Mori and one Zariski), and the authors construct the proposed generating function by summing Hilbert–Poincaré series from birational models and subtracting a correction term associated with the singular variety appearing on the wall. The conjecture asserts that the resulting single rational function captures not only h^0 but all higher cohomology series via appropriate expansions.

A proof would provide a concrete non-hypersurface example where a universal generating function controls all line bundle cohomologies across chambers in the extended Kähler cone.