Compute the Lie algebra generated by pairwise interaction Hamiltonians under the Poisson bracket. For any three bodies interacting via any singular potential, the cumulative dimension at each bracket depth is exactly [3, 6, 17, 116].
This holds for Newton's gravity, Coulomb's law, Calogero-Moser, log-potentials, cubic interactions, and composites. It is invariant under changes of mass ratio, spatial dimension (d=1,2,3), and charge configuration.
Singular potentials (1/r, 1/r², log r, ...) all produce the universal sequence and generate an infinite-dimensional algebra (GK dim = ∞).
Regular potentials (r², harmonic oscillator) saturate: [3, 6, 13, 15, 15, ...], a finite algebra of dimension 15.
The mechanism: a universal cubic chain rule du/dq ~ u³ drives infinite growth when the potential is singular at r=0.
The pattern extends beyond N=3:
| N | Level 0 | Level 1 | Level 2 |
|---|---|---|---|
| 3 | 3 | 6 | 17 |
| 4 | 6 | 14 | 62 |
| 5 | 10 | 25 | 145 |
| 6 | 15 | 39 | 279 |
| 7 | 21 | 56 | 476 |
d(0) = C(N,2). Closed-form L2 formula: N(4N²−9N+3)/2 for N≥4.
A critical finding: the 1D Calogero-Moser system (an exactly integrable system with known action-angle variables) produces the same [3, 6, 17, 116] sequence.
This means the sequence is not a non-integrability certificate. It is a deeper algebraic invariant: a structural property of the singularity class of the potential, independent of dynamical integrability.
Replacing the Poisson bracket with the Moyal bracket (ℏ-deformation), the quantum algebra gains exactly one extra dimension: [3, 6, 17, 117]. The 117th generator is a negative semi-definite octupole (Legendre P₃) with 1/r¹⁴ collision divergence, and it is not conserved. This +1 appears for all singular potentials; polynomial potentials show no quantum correction.
The headline result [3, 6, 17, 116] has been reproduced end-to-end in a second computer algebra system: Wolfram Language 14.3, using an independent rank algorithm (MatrixRank over Rationals on a SparseArray) for both 1/r and 1/r² potentials. The harmonic closure [3, 6, 13, 15, 15] is also confirmed through L=4 in the same oracle. The headline numbers now stand on two unrelated CAS implementations and two unrelated rank algorithms.
Reproduction script: mathematica/poisson_n3_d2.wl · CPU time ≈ 40 s in Mathematica 14.3.
The universal sequence is now an entry in the Online Encyclopedia of Integer Sequences:
A395423: Dimensions of the Poisson-bracket Lie algebra generated by the pairwise interaction Hamiltonians of a 3-body system with a singular pairwise potential.
Initial terms: 3, 6, 17, 116. Conjectured lower bound a(4) ≥ 5604. Four follow-on candidate sequences (N=4 singular, N=3 harmonic, L₂ closed-form, A095794 cross-reference) staged in OEIS/candidates/ for submission once the headline entry clears editorial review.
Treating SGD-with-momentum on a 3-layer linear network as a Hamiltonian system, the pairwise gradient-product algebra produces [3, 6, 17, 119], matching physics through level 2, then diverging by exactly 3 generators at level 3. Sweeping the coupling operator (gradient, Fisher, Hessian, gradient·sum, …) reveals seven distinct universality classes at L=3: 119, 116, 115, 111, 104, 87, 47.
The physical class 116 sits inside this neural ladder (between Class A (119) and Class B (115)), making the three-body algebra a literal benchmark inside the broader landscape of pairwise-Hamiltonian algebras. MSE and L1 collapse to the same class; cross-entropy's quartic term breaks it down to dim 62.
Empirical depth-scaling formula (extras at level 1, exact for L=3,4,5): extrasL,1 = C(L,2)·(L−3). Full write-up: read the article →
Full timeline, dimension sequence tables, work plan, systems catalog, papers, conjectures, and infrastructure details.
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