Computing the power spectrum of gravitational radiation from cosmic strings requires evaluating a core integral I(N, α) for arbitrary loop geometries. The integral resisted analytical solution. Numerical methods worked but gave no structural insight. Previous AI attempts produced only partial asymptotic results.
Brenner et al. build a neuro-symbolic system — Gemini Deep Think combined with tree search and automated numerical feedback — that autonomously derives six distinct analytical methods for solving the integral. Not one method. Six. Each approaches the problem from a different direction: different basis expansions, different parameterizations, different handling of the singularity structure.
The most elegant method expands the kernel in Gegenbauer polynomials. This basis naturally absorbs the integrand's singularities — the troublesome features of the integral become structurally manageable in this representation. The resulting asymptotic expansion matches numerical results and connects to the continuous Feynman parameterization framework from quantum field theory.
The structural finding is not that AI can solve integrals. It is that AI can explore the space of methods rather than the space of solutions. A human mathematician who finds one method for a stubborn integral considers the problem solved. The AI searched systematically enough to find six, then identified which was most elegant. The search was over approaches, not answers.
The role of numerical feedback is crucial: the system generates candidate analytical solutions and tests them against numerical computation, discarding failures and refining successes. The feedback loop is what turns generation into discovery — without it, the system would produce plausible-looking mathematics with no guarantee of correctness.