Mirror-series and Calabi–Yau quantum-cohomology conjecture from recursive descent

Let $\widetilde L_m^{N,k,d}$ denote the result of formally iterating the recursive functions for fixed $k$ while descending $N$ to $N=k$, and let $\widetilde L_m^{k,k}$ be the associated generating function. Let $a_d$ and $b_d$ be the coefficients of the hypergeometric series used in the mirror calculation, and let $L_i^{k,k}(e^t)$ denote the quantum-cohomology structure constants after the mirror-coordinate change.

Calabi–Yau mirror-recursion conjecture. Formal iteration of the laws of Conjecture 2 down to $N=k$ should give

$$k^{dk}a_d=\widetilde L_0^{k,k,d},$$ $$k^{dk}b_d=\frac{1}{d}\widetilde L_1^{k,k,d}+\sum_{m=1}^{d-1}\frac{1}{m}\widetilde L_1^{k,k,m}\widetilde L_0^{k,k,d-m},$$

and should give the structure constants of the quantum cohomology ring of the degree-$k$ Calabi–Yau hypersurface through

$$L_i^{k,k}(e^t)=\frac{\widetilde L_i^{k,k}(e^{x(t)})}{\widetilde L_1^{k,k}(e^{x(t)})}.$$

This conjecture connects formal recursive descent with the hypergeometric series and the quantum cohomology of Calabi–Yau hypersurfaces. It is presented as a consequence expected from the preceding recursive conjectures, with no resolution stated.