Solving an Open Problem in Theoretical Physics using AI-Assisted Discovery

Abstract: This paper demonstrates that artificial intelligence can accelerate mathematical discovery by autonomously solving an open problem in theoretical physics. We present a neuro-symbolic system, combining the Gemini Deep Think LLM with a systematic Tree Search (TS) framework and automated numerical feedback, that successfully derived novel, exact analytical solutions for the power spectrum of gravitational radiation emitted by cosmic strings. Specifically, the agent evaluated the core integral $I(N,α)$ for arbitrary loop geometries, directly improving upon recent AI-assisted attempts \cite{BCE+25} that only yielded partial asymptotic solutions. To substantiate our methodological claims regarding AI-accelerated discovery and to ensure transparency, we detail system prompts, search constraints, and intermittent feedback loops that guided the model. The agent identified a suite of 6 different analytical methods, the most elegant of which expands the kernel in Gegenbauer polynomials $C_l^{(3/2)}$ to naturally absorb the integrand's singularities. The methods lead to an asymptotic result for $I(N,α)$ at large $N$ that both agrees with numerical results and also connects to the continuous Feynman parameterization of Quantum Field Theory. We detail both the algorithmic methodology that enabled this discovery and the resulting mathematical derivations.

• The paper’s main contribution is an exact analytic solution for evaluating cosmic string gravitational radiation integrals using AI-assisted symbolic discovery.
• It utilizes a neuro-symbolic system combining a deep LLM with spectral methods to overcome numerical instabilities, achieving efficient O(N) scaling.
• The work derives a closed-form asymptotic formula for the cosmic string harmonic power spectrum, validated by benchmarks against direct numerical integration.

AI-Assisted Resolution of an Open Analytical Problem in Cosmic String Radiation

Problem Statement and AI-Driven System Architecture

The paper addresses the analytical evaluation of the integral $I(N, \alpha)$ central to the calculation of the power spectrum of gravitational radiation from cosmic string loops, a problem that had previously resisted a comprehensive closed-form solution for general $N$ and loop geometries. Prior progress, such as in (Bubeck et al., 20 Nov 2025), was limited to partial or asymptotic characterizations. The difficulty stems from the integral's structure, involving singularities at $e_{1,2} = \pm 1$ and a denominator $(1-e_1^2)(1-e_2^2)$ that complicates both numerical and series-based methods.

To tackle this, the authors developed a neuro-symbolic system that combines the Gemini Deep Think LLM with a Tree Search (TS) framework augmented by high-precision automated numerical verification. The pipeline entails proposing symbolic manipulations and basis expansions (e.g., Legendre, Chebyshev, Jacobi, Gegenbauer polynomials), scoring these candidates based on agreement with reliable numerics, and employing negative prompting to diversify methodological exploration. This search process autonomously identified six distinct analytical approaches—including an exact, efficiently computable solution—thus both demonstrating and advancing AI-assisted mathematical discovery.

Analytical Methodologies Identified

Monomial Expansion Approaches (Methods 1–3)

For all methods, the integral is reduced to a convolution over $S^2$ of the form

$$I(N, \alpha) = \int_{S^2} d\Omega\; f_N(\mathbf{u}\cdot\mathbf{z})\,f_N(\mathbf{u}\cdot\mathbf{a}),$$

where $f_N(t) = \frac{1 - (-1)^N \cos(N\pi t)}{1-t^2}$. Expanding $f_N(t)$ into a power series yields double sums that depend on moments $J_{k,l}(\alpha)$, evaluated either via a generating function, a Gaussian integral lift, or hybrid projection onto the Legendre basis. While mathematically tractable, all three methods are plagued by numerical instability (especially as $N$ increases), due to cancellation in large alternating sums and poorly matched orthogonality between basis and weight.

Spectral Legendre-Based Approaches (Methods 4–5)

Methods 4 and 5 leverage the Funk-Hecke convolution theorem to exploit the integral's rotational symmetries and decompose $f_N(t)$ into a Legendre series:

$$I(N, \alpha) = 4\pi \sum_{j=0}^\infty \frac{C_{2j}^2}{4j+1} P_{2j}(\cos\alpha),$$

where $C_{2j}$ are the Legendre coefficients of $f_N(t)$. The Spectral Galerkin method (Method 4) constructs a tridiagonal, symmetric positive definite linear system exploiting three-term recurrence relations, yielding stable solutions in $O(N)$ time. The Spectral Volterra approach (Method 5) derives a forward recurrence for $C_{2j}$ based on the Legendre differential equation and integration by parts, also resulting in stable, efficient $O(N)$ scaling.

Exact Gegenbauer Polynomial Solution (Method 6)

The most robust analytic advancement comes from expanding $f_N(t)$ in the Gegenbauer polynomials $C_{l}^{(3/2)}(t)$, which are orthogonal under the $1-t^2$ weight—naturally regularizing the singular denominator. This yield

$$f_N(t) = \sum_{m=0}^\infty b_{2m}\,C_{2m}^{(3/2)}(t),$$

with coefficients $b_{2m}$ obtainable via standard identities involving spherical Bessel functions. The subsequent transformation back to the Legendre basis and summation allows the exact, analytic construction of $I(N,\alpha)$, free from numerical instabilities of prior techniques.

Figure 1: Agreement of the analytic Gegenbauer solution (Method 6) with direct numerical integration validates the derivation across the full range of $N$ and $\alpha$.

Numerical Stability and Computational Efficiency

Comprehensive benchmarks (for $N=20$) contrast absolute errors and runtime across all methods:

Figure 2: The spectral (Legendre-based and Gegenbauer) methods achieve errors at the numerical noise floor with orders-of-magnitude speedup over monomial methods, which diverge at large $N$; Method 2 completely fails due to catastrophic cancellation.

For small $N$, all methods exhibit comparable accuracy; for large $N$, only the spectral/Gegenbauer-based approaches remain stable. The Galerkin (Method 4) and the Gegenbauer (Method 6) approaches are especially efficient, with the former slightly outperforming the latter in wall-clock speed.

Large-$N$ Asymptotics and Closed-Form Results

A significant theoretical advance is achieved by asymptotically analyzing the spectral coefficients and summing the resultant series via distributional limits and QFT-inspired Feynman parameterization techniques. This analysis yields the closed-form asymptotic formula for the cosmic string harmonic power spectrum:

$$P_n(\alpha) \approx \left[ \frac{128G\mu^2}{\pi^2 n^2 \sin^2\alpha} \right] \bigg[ \gamma + \ln(n\pi\sin\alpha) + \cos\alpha \ln\left(\tan\frac{\alpha}{2}\right) \bigg]$$

where $\gamma$ is Euler’s constant. This formula captures both the leading logarithmic dependence on $N$ and essential subleading corrections as $\alpha$ varies.

The numerical validation demonstrates perfect convergence of the asymptotic formula to the exact infinite-series result in the $N \to \infty$ limit and elucidates the nature of subdominant corrections:

Figure 3: The asymptotic large-$N$ formula exhibits fast convergence to the exact result, illustrating parity oscillations as $N$ is reduced.

Implications and Speculation on AI-Assisted Scientific Discovery

This investigation constitutes a rigorous demonstration of advanced neuro-symbolic AI systems solving intricate mathematical-physics problems previously unsolved by human researchers. Notably, the use of negative prompting to force methodological diversity and the deployment of automated numerical validation significantly increase both the breadth and rigor of discovery.

The practical upshot is an exact, efficient, and stable analytic solution to a class of challenging integrals relevant to gravity wave backgrounds from topological defects, with direct applications to cosmological theory and stochastic gravitational wave backgrounds. Theoretically, the approach showcases the integration of LLM-based reasoning with robust algorithmic search, verification, and human-AI synergies for mathematical creativity. The potential for scaling such methods to broader domains in both applied mathematics and physical theory appears considerable.

Conclusion

The paper presents a comprehensive AI-driven resolution of a longstanding open problem in theoretical physics: the analytic computation of the cosmic string gravitational radiation spectrum. By systematizing prompt engineering, algorithmic control, and numerically verifiable reasoning within a neuro-symbolic architecture, the authors uncover stable, exact, and computationally efficient solutions. The advancement of an explicit closed-form asymptotic result and demonstration of AI-guided mathematical insight set the stage for further developments in AI-assisted scientific discovery and the automation of advanced research workflows.