Mirror symmetry, notably in the work of Mark Gross and Berndt Siebert, identifies involutory "mirror pairs" (X,Y) of (degenerating, polarized) compact Calabi-Yau manifolds and makes predictions relating the symplectic geometry of X with the algebraic geometry of Y. Those predictions range from those of the "open string sector", where homological mirror symmetry (HMS) relates the Lagrangian submanifolds of X to algebraic vector bundles on Y, to those of the "closed string sector" where, for example, counts of holomorphic spheres in X are predicted to equal certain period integrals on Y. I'll report recent work, joint with Nick Sheridan, which says the following: if one can prove a certain fragment of HMS (a fragment which we expect to fit neatly into the Gross-Siebert program) then, without knowing anything more about the geometry of X and Y, one can deduce (i) the full statement of HMS; and (ii) certain algebraic and enumerative claims from closed-string mirror symmetry.