Enumerative geometry and modularity in two-modulus K3-fibered Calabi-Yau threefolds

Motivated in part by the modular properties of enumerative invariants of K3-fibered Calabi-Yau threefolds, we introduce a family of 39 Calabi-Yau mirror pairs $(X,Y)$ with h1,1(X)=h2,1(Y)=2, labelled by certain integer quadruples $(m,i,j,s)$ with m≤ 11. On the A-model side, X arises as a complete intersection in a projective bundle over a Fano fourfold Vₘ[i,j], and admits a Tyurin degeneration into a pair of degree m Fano threefolds Fₘ[i]∪ Fₘ[j] intersecting on an anticanonical K3 divisor of degree $2m$. On the B-model side, Y is fibered by Mₘ-polarized K3-surfaces of Picard rank 19, and determined by a branched covering of P¹, consistent with the Doran-Harder-Thompson mirror conjecture. When $s=0$, Y itself acquires a Tyurin degeneration, and correspondingly X acquires a fibration by degree $2m$ K3 surfaces, such that the two K\"ahler moduli control the size of the K3-fiber and base P¹. While the mirror pairs with m≤ 4 can be realized as complete intersections in products of projective spaces or as hypersurfaces in toric varieties, the examples with m≥ 5 are intrinsically non-toric. We obtain uniform formulae for the genus 0 and 1 topological free energies near the Tyurin degeneration (mirror to the large base limit), exhibiting modular properties under the Fricke-extended congruence group Γ₀(m)⁺. We use these results to compute the vertical Gopakumar-Vafa and Noether-Lefschetz invariants and check that their generating functions satisfy the expected modular properties. We also compute generating series of Gopakumar-Vafa invariants with fixed non-zero base degree and exhibit their modular properties.