Involution-equivariant topological recursion and mirror symmetry for the affine binary dihedral Calabi--Yau threefold

We prove a closed-string remodeling statement for the affine binary dihedral Calabi–Yau orbifold threefold $\mathcal{X}=[{\mathbb{C}}^2\mathrm{/}\mathrm{\Gamma }\times \mathbb{C}]$, where $\mathrm{\Gamma }$ is a binary dihedral subgroup of $SU(2)$. This target lies outside the toric setting of the Bouchard–Klemm–Mariño–Pasquetti remodeling conjecture: the toric mirror curve is replaced by the type-$D_l$ logarithmic Toda curve of Brini–Ma–Strachan, and the Chekhov–Eynard–Orantin topological recursion is replaced by the ${\mathbb{Z}}_2$-equivariant topological recursion of Giacchetto–Kramer–Lewański, run in the sign sector of the Toda-curve involution with the Prym kernel as its two-point input. We identify the equivariant orbifold quantum cohomology Frobenius manifold of $\mathcal{X}$ with the invariant Jacobian Frobenius structure of the Toda curve, and we prove that the B-model $R$-matrix, defined by regularized stationary phase, equals the A-side normalized canonical Givental–Teleman $R$-matrix on the smooth oscillatory chamber; this equality is anchored at the orbifold point through a semistable degeneration of the Toda curve. Comparing the resulting Givental–Teleman and Dunin-Barkowski–Orantin–Shadrin–Spitz graph sums then identifies, after a parity-twisted leaf substitution, the sign-sector recursion with the descendant Gromov–Witten generating functions of $\mathcal{X}$ in the stable range ($2g-2+n>0$ with $n>0$), and identifies the recursion free energies with the equivariant Gromov–Witten free energies of $\mathcal{X}$ for $g\ge 2$.