Interpretable Analytic Calabi-Yau Metrics via Symbolic Distillation

Abstract

The pointwise determinant ratio \[ R_ψ(z)\equiv \log\!\left(\frac{\det g_{\mathrm{RF}}(z;ψ)}{\det g_{\mathrm{FS}}(z)}\right) \] 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 $p_2=\sum_i |z_i|^4$ and the cubic elementary symmetric polynomial $σ_3=e_3$, already capture most of the teacher variation. A degree-3 polynomial in $(p_2,σ_3)$ achieves held-out test $R^2=0.946$, 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 $R^2=0.9994$. Refitting the same functional scaffold across $ψ\in[0,0.8]$ keeps the mean determinant-ratio proxy $\langle R_ψ\rangle$ 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 $κ_{111}=5$ 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.