bshepp/pairwise-poisson-algebras

Pairwise Poisson Algebras of the N-Body Problem

Dataset Description

This dataset contains the first systematic computation of pairwise Poisson bracket algebras arising from the classical and quantum N-body problem. For each configuration of N gravitating (or otherwise interacting) bodies in d spatial dimensions under a pairwise potential V(r), we compute the Lie algebra generated by iterated Poisson brackets of the pairwise interaction terms {H_ij}, starting from the N(N-1)/2 pairwise Hamiltonians.

The dataset spans:

• N = 3 through 50 with exact dimension sequences (L0 through N=50, L1 through N=26, L2 through N=9)
• 576+ exponent sweep values mapping the continuous algebraic landscape for 1/r^n and r^n families (500 at L2 + 76 extended at L3)
• 6 Yukawa (screened Coulomb) mu-sweep values confirming universality for exponentially screened potentials
• 16 distinct potentials with full exact rational structure constant tensors (all 12 non-harmonic L2 algebras proven isomorphic)
• 20 named physical systems from helium atoms to triple black holes, including H₃⁺, ozone, and Yukawa nuclear/plasma systems
• 38 charge configurations testing charge-independence of the algebra (incl. (+1,+q,−1) sweep q=1..20)
• 33 mass ratios spanning 10 orders of magnitude (1 to 10^10) with full singular value spectra
• 19 level-4 convergence samples across different phase-space regions
• 14 atlas configurations with rank distribution statistics across phase space
• 9 CHSH Bell test strata probing quantum-information structure
• 5 scaling formulas with conjecture-falsification-resolution narrative
• 40 tier decomposition rows with S_3 and S_4 representation-theoretic analysis
• 16 contextuality tests showing all algebras are maximally non-commutative (zero commuting pairs)
• 77 convergence trajectories tracking SVD rank convergence across sample counts

No comparable dataset exists on Hugging Face or elsewhere. This is the first machine-readable repository of Lie/Poisson algebra structure constants, bracket dimension sequences, and associated spectral statistics.

What Makes This Unique

Unlike PDE-focused datasets (e.g. Poisson-Gauss for the Poisson equation) or MCQ datasets (e.g. MMLU abstract-algebra), this dataset provides exact algebraic objects -- structure constant tensors over the rationals, dimension sequences, and Killing form signatures -- computed from first principles with verified reproducibility. It bridges classical mechanics, quantum mechanics, random matrix theory, and combinatorics in a single structured resource.

Quick Start

from datasets import load_dataset

# Load dimension sequences and filter to non-universal algebras
ds = load_dataset("bshepp/pairwise-poisson-algebras", "dimension_sequences")
df = ds["train"].to_pandas()
non_universal = df[df["dim_L3"].notna() & (df["dim_L3"] != 116)]
print(non_universal[["N", "potential", "bracket_type", "dim_L3"]])

# Extract a structure constant tensor as a numpy array
import json, numpy as np
ds = load_dataset("bshepp/pairwise-poisson-algebras", "structure_constants")
row = ds["train"][0]
C = np.array(json.loads(row["structure_constants"]), dtype=float)
print(f"Potential: {row['potential']}, shape: {C.shape}, nonzero: {np.count_nonzero(C)}")

# Browse physical systems by category
ds = load_dataset("bshepp/pairwise-poisson-algebras", "physical_systems")
df = ds["train"].to_pandas()
print(df.groupby("category")["matches_universal"].mean())

Mathematical Framework

Given N bodies with positions q_i in R^d and momenta p_i in R^d, interacting pairwise via potential V(r_ij), define pairwise Hamiltonians:

$$H_{ij}=\frac{\mathrm{\mid }{p}_i-p_j{\mathrm{\mid }}^2}{2{\mu }_{ij}}+q_i{q}_j\cdot V(\mathrm{\mid }{q}_i-q_j\mathrm{\mid })H\_\{ij\}\; =\; \backslash frac\{|p\_i\; -\; p\_j|^2\}\{2\backslash mu\_\{ij\}\}\; +\; q\_i\; q\_j\; \backslash cdot\; V(|q\_i\; -\; q\_j|)$$

where mu_ij is the reduced mass. The pairwise Poisson algebra A_V is the Lie algebra generated by all H_ij under the canonical Poisson bracket:

$${\mathcal{A}}_V=\mathrm{L}\mathrm{i}\mathrm{e}(H_{ij}:i\mathrm{\ne }j)\backslash mathcal\{A\}\_V\; =\; \backslash mathrm\{Lie\}(\; H\_\{ij\}\; :\; i\; \backslash neq\; j\; )$$

This algebra is graded by bracket depth (level): level 0 contains the generators H_ij, level 1 contains {H_ij, H_kl}, etc. The dimension sequence [d_0, d_1, d_2, ...] records the cumulative dimension at each level.

Key Results in This Dataset

1. Potential universality: For N=3 with 1/r, 1/r^2, 1/r^3, 1/r^4, log, composite, and polynomial r^n (n≥4) potentials, the dimension sequence is universally [3, 6, 17, 116]. Three exceptional potentials: r^1 → [3,4,5,5] (finite), r^2 → [3,6,13,15] (finite), r^3 → [3,6,17,109] (7 extra relations). Universality extends to fractional exponents (21 tested for 1/r^n, 12 for r^n). At the Lie algebra level, 13 non-harmonic potentials share identical L2 algebraic invariants (Killing signature, derived/lower central series, center dimension). The r^3 algebra matches at L2 but at L3 is not nilpotent, with an oscillating lower central series.

2. N-body scaling: L_0(N) = N(N-1)/2, L_1(N) = N(3N-5)/2, L_2(N) = N(4N^2-9N+3)/2 for N >= 4.

3. Charge independence: The dimension sequence [3, 6, 17, 116] is invariant under arbitrary charge assignments q_i in {-20, ..., +20}.

4. Mass invariance: The dimension sequence is independent of mass ratios across 10 orders of magnitude (m_3/m_1 from 1 to 10^10). At extreme astrophysical ratios (>10^5), numerical conditioning degrades but the algebraic structure persists.

5. Quantum deformation: Replacing the Poisson bracket with the Moyal bracket (quantum commutator) yields [3, 6, 17, 117] for singular potentials (1/r^n, log, composites) -- exactly one extra generator at level 3. Polynomial potentials (r^n) and the three exceptional potentials show no quantum growth. 12 quantum rows total.

6. GUE universality: The log-gas potential (Dyson's GUE eigenvalue dynamics) produces the same algebra [3, 6, 17, 116] as Newtonian gravity, confirming a deep connection to random matrix theory.

7. Classical Bell bound: CHSH-type tests on the algebra show max abs(S) = 1.77, strictly below the classical bound of 2.0 -- the Poisson algebra respects classical locality.

8. Maximal non-commutativity: All 16 tested algebras (dimensions 5 to 109) have ZERO commuting pairs. Every pair of generators has a non-zero Poisson bracket, making KS contextuality tests vacuously non-contextual.

9. S_4 representation decomposition: The N=4 pairwise algebra candidates decompose under S_4 as 941 triv + 976 sign + 2893 std + 2932 sign_std + 1917 hook = 23226 generators. Only 1260 are independent (94.6% syzygies vs 25.6% for N=3).

Dataset Splits

dimension_sequences (~602 rows)

The headline table. One row per (N, d, potential, bracket_type) configuration. Includes flattened dim_L0..dim_L4 integer columns for easy filtering in the HF Dataset Viewer, plus the full JSON dimension_sequence. Includes 12 quantum (Moyal bracket) rows covering the complete quantum classification.

Column	Type	Description
N	int	Number of bodies (3--8)
d	int	Spatial dimensions (1--2)
potential	str	Interaction potential: "1/r", "1/r^2", "1/r^3", "1/r^4", "r^1"--"r^10", "log", "composite(...)", "neural"
bracket_type	str	"poisson" (classical) or "moyal" (quantum)
masses	str/null	JSON-encoded mass list, e.g. '["1","2","3"]' or '"symbolic"'
charges	str/null	JSON-encoded charge list, e.g. '[2,-1,-1]'
external_potential	str/null	External confining potential, e.g. "harmonic_omega_1"
max_level	int	Highest bracket depth computed
dimension_sequence	str	JSON-encoded cumulative dimension list, e.g. '[3, 6, 17, 116]'
new_per_level	str/null	JSON-encoded new dimensions per level
dim_L0..dim_L4	int/null	Individual level dimensions (flattened for filtering)
is_exact	bool	True if computed over QQ (exact arithmetic), False if numerical SVD
computation_method	str	"symbolic_QQ", "symbolic_QQ_m1m2m3", "numerical_svd", etc.
sympy_version	str	SymPy version used
computation_time_s	float	Wall-clock computation time in seconds
physical_system	str/null	Physical interpretation, e.g. "GUE_quantum_log_gas"
source_file	str	Path to originating JSON in the repository

structure_constants (16 rows)

Exact rational structure constants C^k_ij for the bracket algebra (N=3). Most at level 2 (d=1 or d=2), with one level-3 entry (r^3). The tensor satisfies [e_i, e_j] = sum_k C^k_ij e_k. Covers 15 potentials: 1/r through 1/r^4, r^1 through r^10, log, and 3 composites.

Column	Type	Description
potential	str	Interaction potential
algebra_dim	int	Algebra dimension (17 for most, 15 for harmonic)
structure_constants	str	JSON-encoded dim x dim x dim tensor of rational strings
killing_signature	str	JSON-encoded Killing form signature [pos, zero, neg]
is_semisimple	bool	Whether the algebra is semisimple
is_solvable	bool	Whether the algebra is solvable
solvability_length	int/null	Length of derived series
is_nilpotent	bool	Whether the algebra is nilpotent
nilpotency_class	int/null	Nilpotency class
center_dimension	int	Dimension of the center
derived_series	str	JSON-encoded derived series dimensions
lower_central_series	str	JSON-encoded lower central series dimensions

charge_sensitivity (38 rows)

Tests whether the algebra dimension depends on particle charges q_i. Includes flattened dim_L0..dim_L3 for easy filtering.

Column	Type	Description
experiment_key	str	Experiment identifier
label	str	Human-readable label
charges	str	JSON-encoded charge vector
masses	str	JSON-encoded mass vector (physical masses in electron-mass units)
n_samples	int	Number of phase-space samples for numerical SVD
dimension_sequence	str	JSON-encoded dimension sequence
dim_L0..dim_L3	int/null	Flattened dimension at each level
matches_116	bool	Whether level-3 dimension equals 116
physical_system	str	Physical system label
computation_time_s	float	Computation time in seconds

mass_invariance (33 rows)

Sweep of the third mass m_3 from 1 to 10^10 (with m_1 = m_2 = 1), showing dimension invariance through moderate ratios and conditioning degradation at astrophysical extremes.

Column	Type	Description
m3	float	Third body mass
m3_log10	float	log10(m_3)
level_0_dim	int	Dimension at level 0
level_1_dim	int	Dimension at level 1
level_2_dim	int	Dimension at level 2
level_2_gap_ratio	float	Singular value gap ratio at level 2 (rank determinant)
level_2_singular_values	str	JSON-encoded full singular value spectrum
dims	str	JSON-encoded [level_0, level_1, level_2]
elapsed_s	float	Computation time

level4_convergence (19 rows)

Level-4 lower bounds from numerical SVD at increasing sample sizes.

Column	Type	Description
config	str	Phase-space configuration: "global", "euler", "lagrange", "scalene"
n_samples	int	Number of phase-space samples
dimension_sequence	str	JSON-encoded cumulative dimension sequence through level 4
new_per_level	str	JSON-encoded new dimensions per level
d4_lower_bound	int	Lower bound on level-4 dimension
max_gap_ratio	float	Maximum singular value gap ratio
max_gap_index	int	Index of maximum gap
elapsed_seconds	float	Computation time
mu, phi, epsilon	float	Phase-space parametrization (null for global)

spectral_statistics (14 rows)

Rank distributions across phase space from atlas scans.

Column	Type	Description
config	str	Configuration identifier
label	str	LaTeX-formatted label
source_type	str	"targeted_atlas" or "atlas_full_irrational"
n_regions	int	Number of phase-space regions sampled
rank_min	int	Minimum observed rank
rank_max	int	Maximum observed rank
rank_mode	int	Modal rank (most frequent)
n_points	int	Total phase-space points
pct_116	float	Percentage of points achieving rank 116

physical_systems (17 rows)

Named physical systems with their computed dimension sequences, spanning astrophysical (Sun-Earth-Moon, triple black holes) to atomic (helium, lithium ion) to exotic (Penning traps, 2D vortices).

Column	Type	Description
system_name	str	Machine-readable identifier, e.g. "helium", "triple_bh_lisa"
system_label	str	Human-readable name, e.g. "Helium Atom", "Triple Black Hole (LISA)"
category	str	"astrophysical", "atomic", "ion_trap", or "condensed_matter"
dimension_sequence	str	JSON-encoded dimension sequence
dim_L0..dim_L3	int	Flattened dimensions per level
matches_universal	bool	Whether the sequence equals the universal [3, 6, 17, 116]
completed_at	str	ISO timestamp of computation completion
source_file	str	Path to originating JSON

bell_test (9 rows)

CHSH Bell inequality tests on the Poisson algebra. Three phase-space strata (equilateral, pair apparatus, separated) with three observable variants each. Tests whether the algebra structure permits violations of the classical CHSH bound (|S| must stay below 2).

Column	Type	Description
stratum	str	Phase-space stratum: "equilateral", "pair_apparatus", "separated"
variant	str	Observable variant: "variant1", "variant2", "variant3"
max_abs_S	float	Maximum abs(S) observed
max_S	float	Maximum S (signed)
ci_95_lower	float	95% confidence interval lower bound
ci_95_upper	float	95% confidence interval upper bound
optimal_angles_deg	str	JSON-encoded optimal measurement angles
significant_violation	bool	Whether S exceeds classical bound (always False)
n_samples	int	Number of phase-space samples
classical_bound	float	CHSH classical bound (2.0)
tsirelson_bound	float	Tsirelson quantum bound (2*sqrt(2))

scaling_formulas (5 rows)

Closed-form scaling formulas for algebra dimension as a function of N bodies. Documents the meta-scientific narrative: original conjecture, falsification at N=7, and corrected formula.

Column	Type	Description
level	str	Formula identifier: "L0", "L1", "L2_original", "L2_corrected", "L3"
formula_expression	str	Closed-form expression, e.g. "N(4N^2 - 9N + 3)/2"
formula_status	str	"verified", "verified_trivial", "falsified", or "unknown"
leading_term	str	Asymptotic leading term
new_per_level_formula	str	Formula for new generators per level
verified_N_values	str	JSON-encoded list of N values where formula is verified
failed_N_values	str/null	JSON-encoded N values where formula fails
predictions	str/null	JSON-encoded predictions for untested N
data_points	str/null	JSON-encoded known data points
notes	str/null	Context, caveats, and conjectures

tier_decomposition (40 rows)

Clebsch-Gordan decomposition of the candidate generator spaces under S_3 (N=3) and S_4 (N=4) symmetry. Tracks how each irreducible representation contributes at each bracket level, and compares candidate counts to observed independent generators.

Column	Type	Description
N	int	Number of bodies
symmetry_group	str	"S3" or "S4"
level	int	Bracket level (-1 for total)
total_candidates	int	Number of candidate generators at this level
observed_rank	int	Observed independent rank
irrep_name	str	Irreducible representation name
irrep_dim	int	Dimension of the irrep
multiplicity	int	Number of copies of this irrep
contribution	int	Total generators from this irrep (multiplicity x dim)

contextuality (16 rows)

Kochen-Specker contextuality tests on all available structure constant algebras. Tests whether the orthogonality graph (edges between Poisson-commuting pairs) permits KS-uncolorable configurations.

Column	Type	Description
N	int	Number of bodies
d	int	Spatial dimensions
potential	str	Interaction potential
algebra_dim	int	Algebra dimension
n_commuting_pairs	int	Number of Poisson-commuting generator pairs (always 0)
total_pairs	int	Total generator pairs tested
commutativity_fraction	float	Fraction of commuting pairs (always 0.0)
ks_colorable	bool	Whether KS coloring exists (always True)
contextual	bool	Whether algebra is contextual (always False)
pm_constructible	bool	Whether Peres-Mermin square can be built (always False)

convergence_trajectories (77 rows)

SVD rank convergence as a function of sample count, for multiple (N, d, potential, level) configurations. Shows how many phase-space samples are needed for reliable numerical rank determination.

Column	Type	Description
N	int	Number of bodies
d	int	Spatial dimensions
potential	str	Interaction potential
level	int	Bracket level
n_samples	int	Number of phase-space samples
n_candidates	int	Number of candidate generators
rank	int	Numerical rank determined by SVD gap
gap_ratio	float	Best SVD gap ratio
elapsed_s	float	Computation time in seconds

Reproduction

All data was generated from the code at github.com/bshepp/3body.

Requirements

• Python 3.10+
• SymPy >= 1.12 (for exact symbolic computation)
• NumPy, SciPy (for numerical SVD)
• pandas, pyarrow (for Parquet generation)

Rebuilding the Dataset

git clone https://github.com/bshepp/3body.git
cd 3body
pip install pandas pyarrow
python dataset/build_dataset.py
python dataset/validate_dataset.py  # optional: run validation suite

The script reads all JSON result files and produces 12 Parquet tables in dataset/output/.

Reproducing Individual Results

Example: compute the N=3, d=1, 1/r dimension sequence:

from nbody.algebra import NBodyAlgebra

alg = NBodyAlgebra(N=3, d=1, potential="1/r")
result = alg.compute_ranks(max_level=3)
print(result["cumulative_rank"])  # [3, 6, 17, 116]

Citation

If you use this dataset, please cite:

@dataset{sheppeard2026poisson,
  title={Pairwise Poisson Algebras of the N-Body Problem},
  author={Sheppeard, B.},
  year={2026},
  publisher={Hugging Face},
  url={https://huggingface.co/datasets/bshepp/pairwise-poisson-algebras}
}

License

MIT License. See the repository for full terms.