Interpretable Analytic Calabi-Yau Metrics via Symbolic Distillation

The pointwise determinant ratio [ R_psi(z)equiv log!left(fracdet gRF(z;psi)det gFS(z)right) ] measures how the Ricci-flat metric on the Dwork quintic departs from the Fubini--Study baseline. We ask whether this scalar observable can be described compactly in terms of a small number of projective invariants, and whether the same scaffold remains usable across complex-structure moduli. Using Donaldson's k=10 balanced metric as an algebraic teacher and symbolic regression on sampled points, we find that, within the restricted moduli-only feature class studied here, two low-order symmetric features, the power sum p2=sumi |zi|4 and the cubic elementary symmetric polynomial σ3=e3, already capture most of the teacher variation. A degree-3 polynomial in (p2,σ3) achieves held-out test R2=0.946, while adding the remaining low-order symmetric generators changes this by less than 10-3. Within the same two-feature space, symbolic regression identifies a five-term rational-polynomial expression that matches the k=10 teacher with R2=0.9994. Refitting the same functional scaffold across psi∈[0,0.8] keeps the mean determinant-ratio proxy langle R_psirangle within 0.01% of the local teachers on the sampled point clouds and yields smoothly varying fitted coefficients over the studied range. The holomorphic Yukawa coupling kappa111=5 is reproduced as a normalization check only. Taken together, these results provide a compact symbolic description of one metric-derived scalar observable on the Dwork family, while remaining bounded by the finite-k teacher used for distillation rather than establishing a closed-form Ricci-flat metric.