New Intelligence Report
Editor: Dinghui
[New Intelligence Abstract] Just now, the Google Research team used Gemini Deep Think + Tree Search framework to independently crack an unsolved integration problem in theoretical physics—the precise analytical solution for the gravitational radiation power spectrum of cosmic strings. The AI explored 600 candidate paths, found 6 solutions, and the most elegant one made human physicists marvel.
Shocking, the AI scientist is really coming!
Google released a latest paper (March 6), causing a stir.
Gemini Deep Think combined with tree search algorithms has independently cracked an open problem in theoretical physics!
A problem that a top human research team agreed was "so difficult they didn't know where to start" has been solved by this AI system.
Paper link: https://arxiv.org/pdf/2603.04735
This paper is highly groundbreaking!
Simply put, AI has solved a complex mathematical/physics problem that human physicists had previously failed to solve.
This reminds us of the news that Claude helped Knuth solve a graph theory conjecture that went viral.
If Claude cracking the graph theory conjecture in Knuth's paper was a breakthrough for AI in the field of discrete mathematics.
Then this Google paper represents a full-scale offensive of AI in the fields of continuous mathematics and theoretical physics.
One is combinatorics, the other is mathematical physics. The two events happened almost simultaneously, constituting the most iconic "AI scientist" event of March 2026.
AI is blossoming across humanity's core intellectual domains.
Cosmic Strings: An Ultimate Question That Fascinates All Scientists
Cosmic strings are a hypothetical one-dimensional topological defect structure in cosmology, born from phase transitions in the early universe.
When these things vibrate, they radiate gravitational waves outward.
In recent years, Pulsar Timing Arrays (PTA) have for the first time observed a gravitational wave background signal suspected to be from cosmic strings, leading to unprecedented enthusiasm for cosmic string research in the theoretical physics community.
To predict the gravitational wave signals emitted by cosmic strings, one must precisely calculate their gravitational radiation power spectrum.
Specifically, there is a core integral I(N, α)—describing the radiation intensity emitted by the Nth harmonic of a cosmic string loop.
This integral looks simple, but the integration region is a sphere, and the integrand has singularities at the boundaries (when e₁,₂ = ±1), causing standard numerical integration to be unstable.
Using the classic Legendre polynomial expansion? The weight function doesn't match, it explodes.
Past research could only provide asymptotic solutions for large N, or partial results for odd N.
A precise, unified analytical solution has been an unsolved case for many years.
Until Gemini Deep Think stepped in.
One-sentence summary of what problem the paper solved.
AI calculated a precise mathematical formula for gravitational waves emitted by "cosmic strings." To calculate the power of these gravitational waves, physicists needed to solve a very complex mathematical integral formula. This formula contains "singularities" (similar to division by zero in mathematics, where calculations crash), causing traditional numerical methods to often fail.
Over the past few years, human physicists and early AI attempts only found "partial solutions" or "approximate solutions," never finding a unified, precise analytical formula.
Has the Human Scientist's Problem Been Solved by Gemini?
Similar to Claude's 31-step research-style exploration in solving Knuth's problem, Gemini's approach to this problem also resembles a well-trained research team at work.
The Google team didn't let the AI go naked. They built a sophisticated "neuro-symbolic system":
Gemini Deep Think + Tree Search + Automatic Numerical Feedback
All three are indispensable, working in coordination.
Gemini Deep Think acts as the "brain": generating mathematical hypotheses, performing symbolic derivations, and judging which path "looks elegant and feasible."
It doesn't simply brute-force trials, but is instructed to perform deep reasoning chains, anticipating convergence issues in infinite series expansions in advance.
Tree Search is responsible for "systematic exploration": building the entire solution space into a large tree.
Each node represents a mathematical intermediate expression—written in LaTeX, accompanied by automatically generated Python code, letting the computer perform numerical verification.
The search strategy adopted the PUCT algorithm (Predictor + Upper Confidence Bound applied to Trees), which shares the same underlying logic as AlphaGo's gameplay—maintaining a balance between "exploiting existing good paths" and "exploring new possibilities."
Automatic Numerical Feedback is responsible for "quality control": after each derivation step is completed, high-precision numerical calculation is immediately used to verify whether the symbolic result is correct. If they don't match, that path is pruned directly.
This step is most critical: whenever the model proposes an intermediate step, the system automatically executes the corresponding Python code and compares it with high-precision numerical benchmarks. If numerical instability, divergence, or execution errors are found, the system feeds the error information and error to the model, allowing it to self-correct.
Throughout the process, the AI explored a total of approximately 600 candidate nodes.
Over 80% were pruned by the automatic verifier due to "algebraic errors" or "numerical divergence"—including catastrophic cancellation errors, unstable monomial summation, ill-conditioned basis transformations, etc.
Only a few paths survived the layers of screening and ultimately won.
This isn't brute-force searching for answers, but genuine "AI-driven mathematical research."
600 Paths, AI Found 6 Solutions
After systematic exploration, Gemini Deep Think found a total of 6 different solutions, divided into three major categories:
Category 1: Monomial Basis Approaches
The core idea is to expand the function into a power series, then use different techniques to calculate the integral.
Method 1 uses the generating function method, constructing an exponential generating function and solving it using Gaussian integration.
Method 2 uses Gaussian integral lifting, lifting the spherical surface integral into three-dimensional space, converting it into a standard Gaussian integral.
Method 3 is hybrid coordinate transformation, first expanding into a power series, then projecting onto a Legendre basis.
These three methods are mathematically correct, but suffer from numerical instability—when N becomes large, precision loss occurs due to subtraction of large numbers.
Method 1: Generating Function Method
Method 2: Gaussian Integral Lifting Method
Method 3: Hybrid Coordinate Transformation Method
These three methods are based on power series expansion, with solid reasoning.
But there's a fatal weakness: When N→∞, numerical instability occurs with catastrophic cancellation errors.
Category 2: Spectral Basis Approaches
These two methods utilize the Funk-Hecke spherical convolution theorem, working directly in Legendre spectral space.
Method 4: Spectral Galerkin Matrix Method, converting the problem into a tridiagonal linear system of equations.
Method 5: Spectral Volterra Recurrence Method, deriving a forward recurrence relation for the coefficients.
These two methods are numerically stable, with computational complexity of only O(N), a full order of magnitude faster than monomial methods.
Category 3: The Analytic Solution
Method 6: Gegenbauer Method
This is the most elegant method—Gegenbauer Method.
The AI discovered a brilliant approach: choosing Gegenbauer polynomials as the expansion basis, and the weight function of this type of polynomial is exactly (1-t²), which perfectly cancels the singular factor in the integrand's denominator!
This way, the originally troublesome singular integral becomes a completely regular integral.
Through integration by parts and standard identities, the AI derived a precise closed formula, even obtaining a beautiful asymptotic expression.
This is also the champion solution provided by the AI this time.
The Most Elegant Solution Makes Physicists' Hearts Flutter
Gegenbauer polynomials, denoted as Cₗ^(3/2)(t).
This is a family of orthogonal polynomials defined on [-1,1], and its weight function w(t) = 1 - t² can naturally eliminate the singularity of the integrand.
This isn't a coincidence; it's a deep mathematical structure recognized by Gemini.
The specific approach is:
Expand the integrand fN(t) into a linear combination of Gegenbauer polynomials, and use orthogonality to determine the expansion coefficients.
The critical moment arrives—the weight function cancels with the denominator, and the originally troublesome singularity is elegantly "absorbed", leaving behind a completely regular integral.
Subsequently, using the identity Cₖ^(3/2)(t) = Pₖ₊₁'(t) (the relationship between Gegenbauer polynomials and the derivative of Legendre polynomials), as well as integration by parts, the integral further simplifies to a form involving the Fourier transform of Legendre polynomials.
Finally, the result can be precisely expressed using the cosine integral function Cin(z)—a closed analytical expression that requires no numerical approximation and is applicable to any N under arbitrary loop geometry.
The Google team wrote in the paper—the Gegenbauer method is the most elegant of the 6 solutions because it most naturally handles the singularity structure of the integral in a mathematical sense.
Even more stunning: when searching for large N asymptotic behavior, Gemini also independently discovered an intrinsic connection to Feynman parametrization in quantum field theory—this is a deep mathematical unity spanning across physics subfields that even human researchers hadn't anticipated.
Human-Machine Collaboration, Not AI Acting Alone
It's worth noting that the Google team's description of this process is very honest—
The initial 6 solutions were automatically found by the tree search framework. The Gegenbauer method initially provided a precise solution in the form of an infinite tail sum, which was mathematically correct but not concise enough.
To convert it into a truly finite closed form, a human researcher manually intervened, feeding the intermediate results to a larger, more powerful version of Gemini Deep Think, asking it to rigorously verify the existing proof and search for further simplification.
In this human-machine interaction, the advanced model independently discovered an error in the initial formulation of Method 5 (Spectral Volterra Recurrence Method), and after correction, identified the equivalence of Method 5 and Method 6—this allowed the infinite tail sum in Method 6 to be precisely "folded" into a finite form, ultimately obtaining the beautiful analytical solution expressed in terms of cosine integrals.
This was a collaborative relay, not a fully autonomous AI discovery.
But this makes it even more important—it demonstrates a truly feasible paradigm of human-machine collaboration.
The Google team maintained scientific humility in their conclusion:
"We do not claim that this physics problem itself has profound significance, but the fact that the AI system could solve it with ease has important potential for accelerating the scientific discovery process."
But the other side of this statement is equally worth pondering—
The so-called "ease" stands upon 600 explorations and an 80% elimination rate.
This isn't smart luck; this is systematic intellectual search.
For decades, physicists and mathematicians have generally believed that symbolic derivation and theoretical discovery are the sanctuaries most difficult for AI to touch—because this requires true mathematical intuition, the ability to identify "elegance" from a vast solution space.
But the Gegenbauer method tells us: AI is developing something akin to intuition.
It's not random trial and error; it's evaluating the elegance of solutions, identifying the deep beauty of mathematical structures.
This time, it was the gravitational wave spectrum of cosmic strings.
Next time, perhaps it will be deeper equations in string theory, or core integrals in quantum gravity.
Humans pose questions, AI systematically explores structures, and humans complete the final interpretation of meaning—
This new research paradigm is no longer science fiction, but is being written down in black and white by Google through a paper.
"Neuro-Symbolic System": AI's Infrastructure for Scientific Discovery
Worth noting is that the tree search framework used in this paper is not a one-time special tool, but a reusable framework with systematic methodology.
The Google team disclosed in detail in the appendix:
Complete system prompts
Code implementation for evaluation and verification
"Negative Prompting" strategy—this is the key technique that forces AI to explore different solution directions
So-called negative prompting means that after the AI finds a valid solution, it is explicitly told "don't use this method again," forcing it to find another path and continue exploring—this is how the diverse solutions from Method 1 to Method 6 were obtained.
This methodology itself is a transferable research tool.
Today it's used for cosmic strings; tomorrow it can be used for materials science, quantum chemistry, and unsolved conjectures in pure mathematics.
AI Is Knocking on the Door of Theoretical Physics
Looking back at this event, one detail is particularly impressive.
In the field of machine learning, people have long been accustomed to what AI can do: image recognition, text generation, playing chess, writing code...
But deriving symbolic mathematics, independently identifying singularities in mathematical structures and finding elegant methods to eliminate them—this was previously considered almost impossible.
Because mathematical discovery isn't search, it's "insight."
However, the Gemini Deep Think case tells us—"Insight" might be decomposed into:
A sufficiently large search space + sufficiently precise evaluation criteria + sufficiently strong reasoning ability.
When these three are combined, something that looks like "intuition" can emerge.
AI is ready to become the strongest partner for mathematicians, physicists, and all scientists.
This may truly be just the beginning.
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