Mirror symmetry for gCICY threefolds (I): a $[3,-1]$ block and a non-Gorenstein toric phase

We construct and analyze a mirror family for the generalized complete intersection Calabi-Yau threefold $X_{+}=[{\mathbb{P}}^1\mathrm{\mid }3-1;{\mathbb{P}}^4\mathrm{\mid }23]$, with $(h^{1,1},h^{2,1})=(2,46)$. An auxiliary-variable presentation turns the negative-degree section into a variation of toric GIT. Crossing one wall flops 16 disjoint $(-1,-1)$-curves and yields a Calabi-Yau threefold $X_{-}$, a nef complete intersection in a toric $\mathbb{Q}$-Fano eightfold, which carries a genus-one fibration of index 4. Because this ambient space is not Gorenstein, the Batyrev-Borisov construction does not apply. We use the Hori-Vafa equations only to select a 2-parameter Laurent family, and construct a smooth projective crepant compactification $Y$ of the Laurent model with $(h^{1,1},h^{2,1})=(46,2)$; for very general parameters the Mordell-Weil group is $\mathbb{Z}\mathrm{/}4\mathbb{Z}$. The periods of $Y$ satisfy a rank-6 Picard-Fuchs system with two maximally unipotent boundary points, whose indicial algebras are isomorphic to the rational even cohomology rings of $X_{-}$ and $X_{+}$. The resulting genus-zero predictions agree with independent counts of 176 vertical lines and 100 vertical conics on the genus-one fibration $X_{-}\to {\mathbb{P}}^2$.