Imagine three stars pulling on each other through gravity. Predicting their paths is one of the oldest unsolved problems in physics; even Newton struggled with it. There is no simple formula, and the motion is famously chaotic.
We identified a hidden pattern in the mathematics of this system. When you write down all the ways the stars' positions and momenta can interact through their gravitational forces, those interactions generate a kind of "algebraic language." The words in this language combine to form new words, and the number of independent words at each level of complexity follows a specific sequence:
3 → 6 → 17 → 116
What makes this remarkable is its universality. Whether you use Newton's gravity, an inverse-square force, or an entirely different force law: the same numbers appear. Whether the stars live in 1D, 2D, or 3D: the same numbers. Whether the masses are equal or wildly different: still the same numbers.
There is one striking exception: a spring-like force (the simple harmonic oscillator) gives a different, much smaller sequence (3 → 6 → 13 → 15) that stops growing. Every other force we tested produces the universal pattern. This singular-vs-regular dichotomy is itself part of the discovery.
This suggests there is a deep structural fingerprint of the three-body problem that nobody knew about before.
In physics, a force between two objects depends on the distance between them. For gravity, the force goes as 1/r², where r is the distance. The potential energy tells you how much stored energy the pair has; for gravity it is proportional to 1/r.
With three bodies, there are three pairs, so three pairwise potentials: H₁₂, H₁₃, and H₂₃. These are our starting "words" (3 of them).
There is an operation in physics called the Poisson bracket. It's like a rule for combining two quantities to get a new one. When you apply this rule to every pair of our 3 starting words, you get new expressions. After simplifying and removing duplicates, you end up with 6 independent words at level 1.
Bracket the level-1 words with the originals and you get 17 at level 2; bracket again and you get 116 at level 3. The number of truly independent expressions at each level is what we call the dimension sequence:
dim = [3, 6, 17, 116]
We checked this by representing each expression as a vector of numbers, then using a technique from linear algebra (the SVD, or singular value decomposition) to count how many vectors are linearly independent. The gap between the "real" singular values and the numerical noise is enormous (ratios above 10¹⁰), so the count is rock solid.
The three-body problem is formulated in Hamiltonian mechanics. Each body has position coordinates qᵢ and conjugate momenta pᵢ. The state lives in a phase space of dimension 2dN, where d is the spatial dimension and N = 3.
The Poisson bracket of two functions f, g on phase space is:
{f, g} = Σᵢ (∂f/∂qᵢ · ∂g/∂pᵢ − ∂f/∂pᵢ · ∂g/∂qᵢ)
This bracket is antisymmetric and satisfies the Jacobi identity, making the smooth functions on phase space into a Lie algebra under {·,·}.
We start with the three pairwise Hamiltonians H₁₂, H₁₃, H₂₃ and build the Lie algebra they generate:
- Level 0: {H₁₂, H₁₃, H₂₃} — dimension 3
- Level 1: span of all {Hᵢⱼ, Hₖₗ} — dimension 6
- Level 2: brackets of level-1 with level-0 — dimension 17
- Level 3: next bracket layer — dimension 116
The dimension is computed numerically. We substitute random phase-space points into the symbolic expressions, building a matrix where each row is a generator evaluated at a sample point. The SVD of this matrix reveals the numerical rank: singular values above 10⁻⁶ count as "real," and the gap ratio σₖ/σₖ₊₁ exceeds 10¹⁰ at the transition, giving an unambiguous rank determination.
A key technical trick: we introduce auxiliary variables uᵢⱼ = 1/rᵢⱼ so that Poisson brackets of potentials like 1/r produce polynomial expressions (via chain rule), avoiding messy radicals.
The universality result — the same [3, 6, 17, 116] for 1/r, 1/r², 1/r³, log(r), all masses, and all dimensions d = 1, 2, 3 — is checked by running the computation across thousands of parameter configurations in the Interactive Atlas on this site.
The configuration space is (ℝᵈ)ᴺ \ Δ (collision set removed) with the canonical symplectic structure on T*((ℝᵈ)ᴺ \ Δ). The pairwise Hamiltonians Hᵢⱼ = mᵢmⱼ V(rᵢⱼ) are elements of the Poisson algebra C∞(T*M).
We compute the Lie subalgebra 𝔤 = Lie({Hᵢⱼ}) generated under the Poisson bracket, with the natural filtration:
𝔤₀ ⊂ 𝔤₁ ⊂ 𝔤₂ ⊂ 𝔤₃ ⊂ ⋯
where 𝔤ₖ = 𝔤ₖ₋₁ + [𝔤ₖ₋₁, 𝔤₀]. The dimension sequence dim(𝔤ₖ) = [3, 6, 17, 116] is the central result.
Key observations and open questions:
- Universality: The sequence is independent of the potential V(r) (for non-degenerate potentials), the spatial dimension d, mass ratios, and charge configurations. This suggests a purely combinatorial or representation-theoretic origin tied to the S₃ symmetry of the pair graph.
- Degenerate case: The harmonic potential V(r) = r² causes the algebra to close at dimension 15. This is the only known closure and corresponds to the well-known superintegrability of the harmonic oscillator.
- N-body scaling: For N = 4 bodies the sequence is now exact through level 3:
[6, 14, 62, 1260](Apr 2026). Closed-form formulas are known for the lower levels:L₀(N) = N(N−1)/2(verified through N = 50),L₁(N) = N(3N−5)/2(verified through N = 26), andL₂(N) = N(4N²−9N+3)/2for N ≥ 4 (the original cubic conjecture was falsified at N = 7 and superseded). Level-3 growth remains super-exponential and lacks a closed form. - S₃ filtration: The generators at each level decompose into irreducible representations of S₃ (permutation of body labels). This decomposition is detailed in Paper 2.
- Bracket-tensor spectral statistics: Eigenvalues of the level-2 structure-constant tensor follow GUE-to-GSE level repulsion for the 1/r algebra and Poisson statistics for the harmonic r² algebra: the Bohigas–Giannoni–Schmit conjecture manifested in the algebra itself. The Jacobi identity also fails on the truncated 1/r structure constants, confirming the algebra is genuinely infinite-dimensional.
- Calogero–Moser connection: The 1/r² (Calogero–Moser) potential is exactly integrable yet produces the same sequence as Newtonian gravity. The dimension sequence is therefore not a non-integrability certificate; it is an algebraic invariant of the singularity class. Paper 4 develops this.
- Quantum and machine-learning extensions: Replacing the Poisson bracket with the Moyal bracket adds exactly one generator at level 3 for every singular potential, giving
[3, 6, 17, 117]. SGD-with-momentum on a 3-layer linear network defines a Hamiltonian whose pairwise gradient-product algebra gives[3, 6, 17, 119]; varying the coupling produces seven distinct universality classes. The GUE log-gas (Dyson model for Riemann-zeta-zero spacings) lands on the universal[3, 6, 17, 116].
The computational pipeline uses SymPy for symbolic Poisson brackets, auxiliary variable substitution uᵢⱼ = 1/rᵢⱼ for polynomiality, NumPy/SciPy for high-precision SVD, and mpmath for extended-precision verification. All code is open source in the linked GitHub repository.
Independent cross-CAS verification (April 21, 2026): The headline [3, 6, 17, 116] for both 1/r and 1/r² has been reproduced end-to-end in Wolfram Language 14.3, using an independent rank algorithm (MatrixRank over Rationals on a SparseArray) operating on independently constructed bracket matrices. The harmonic closure [3, 6, 13, 15, 15] is also confirmed through L=4 in the same oracle. The result therefore stands on two unrelated CAS implementations and two unrelated rank algorithms.