Recursive law for cubic curves on Calabi–Yau hypersurfaces

Let $M_k^k$ be a degree-$k$ Calabi–Yau hypersurface in $CP^{k-1}$, and let $L_m^{N,k,d}$ denote the structure constants used in the paper for rational curves of degree $d$. For $m\geq 2$, the proposed recursion relates the degree-three constants on $M_k^k$ to constants on the corresponding Fano hypersurface in one higher ambient dimension.

Cubic Calabi–Yau recursion conjecture. The recursive law is

$$\begin{aligned} L_m^{k,k,3}={}&\frac{1}{18}\bigl(4L_{m-2}^{k+1,k,3}+10L_{m-1}^{k+1,k,3}+4L_m^{k+1,k,3}-10L_0^{k+1,k,3}-4L_1^{k+1,k,3}\\ &+12L_{m-1}^{k+1,k,2}L_m^{k+1,k,1}+12L_m^{k+1,k,2}L_m^{k+1,k,1}+6L_m^{k+1,k,2}L_{m+1}^{k+1,k,1}\\ &+6L_{m-1}^{k+1,k,2}L_{m-1}^{k+1,k,1}+12L_{m-1}^{k+1,k,2}L_m^{k+1,k,1}+12L_m^{k+1,k,2}L_m^{k+1,k,1}\\ &+18(L_m^{k+1,k,1}-L_1^{k+1,k,1})^2(L_m^{k+1,k,1}+2L_1^{k+1,k,1})\\ &-\frac16L_{m-1}^{k+1,k,2}L_m^{k+1,k,1}-\frac16L_m^{k+1,k,2}L_m^{k+1,k,1}-\frac34L_0^{k+1,k,2}L_m^{k+1,k,1}-\frac34L_1^{k+1,k,2}L_m^{k+1,k,1}\\ &-\frac5{12}L_1^{k+1,k,2}L_1^{k+1,k,1}-\frac5{12}L_0^{k+1,k,2}L_1^{k+1,k,1}-\frac13L_1^{k+1,k,2}L_2^{k+1,k,1}\\ &-3L_1^{k+1,k,1}\frac12\bigl(L_{m-1}^{k+1,k,2}+L_m^{k+1,k,2}-L_0^{k+1,k,2}-L_1^{k+1,k,2}+2(L_m^{k+1,k,1}-L_1^{k+1,k,1})^2\bigr)\bigr). \end{aligned}$$

This is proposed because the usual specialization method cannot control reducible cubic curves on the Calabi–Yau hypersurface. The source presents it as a conjectural recursive formula; no resolution is supplied.