On the Quantitative Approximation Barrier for Singular PDE Solutions: A 60,000-Fold Error Reduction via Symbolic Atom Discovery

Standard spectral methods exhibit exponential convergence for analytic problems but stall when approximating singular solutions to parametric PDEs. We quantify this classical approximation barrier, demonstrating a $60{,}806\times$ error reduction when the basis is augmented with the correct singular atom (oracle upper bound). In an idealized exponent-scanning surrogate, the recovered exponent achieves mean absolute error $0.006$ under moderate noise, and our sensitivity study shows that such errors still yield large gains when used to augment the basis. We present numerical evidence that, in our specific power-law dictionary, $\ell_1$-minimization (LASSO) successfully recovers singular terms despite near-perfect dictionary coherence ($\mu \approx 0.997$), with a sharp empirical phase transition at small sample sizes, overcoming the limitations of greedy methods. Numerical validation extends to two-dimensional Poisson problems with corner singularities. \end{abstract}