bshepp/pairwise-poisson-algebras

Pairwise Poisson Algebras: Neural Networks vs Physics

Dataset Description

This dataset contains the first systematic computation of pairwise Poisson bracket Lie algebras for both neural network training dynamics and physical N-body systems. SGD with momentum is a Hamiltonian system; the pairwise interactions between weight layers generate a Lie algebra — and we discover that neural networks produce richer algebraic structures than any physical system.

Neural Network Results

• 21 neural network algebra configurations sweeping depth (L=2-5), width (k=1-3), coupling type (12 variants), loss function (MSE, cross-entropy, L1), and activation (linear, tanh, ReLU)
• Seven universality classes discovered at L=3:
  • A: [3, 6, 17, 119] — gradient, fisher, gradient_abs, l1-loss, hessian_plain (5 configs)
  • B: [3, 6, 17, 115] — directional (kinetic-gradient) coupling
  • C: [3, 6, 17, 111] — gradient_sum (diagonal-only gradient)
  • D: [3, 6, 17, 104] — gradient_cubic (cubic gradient potential)
  • E: [3, 6, 17, 87] — natural_gradient (Fisher-rescaled gradient)
  • F: [3, 6, 17, 62] — cross-entropy loss (nonlinear loss structure)
  • G: [3, 5, 11, 47] — hessian, symmetric, hessian_full, loss_power (4 configs) — only class that diverges at level 1
• Within-class invariances: The [3, 6, 17, 119] gradient class is invariant under width (k=1,2,3), loss (MSE = L1), and activation (linear = tanh = ReLU, through level 2) — confirmed AT level 3 for width.
• Depth scaling diverges from physics at L>=4: Neural L=3 matches physical levels 0-2 then breaks by +3 at level 3. Neural L=4 gives [6, 20, 164] vs physical [6, 14, 62] — breaks by +6 at level 1 already. Pattern: extras at level 1 = C(L,2)*(L-3).

Physical Benchmark (Universality Reference)

• 685 dimension sequences for N=3..50, 16+ potentials, quantum/classical
• 576+ exponent sweep values across the continuous algebraic landscape
• 20 named physical systems from helium atoms to triple black holes — all producing [3, 6, 17, 116]
• 16 exact rational structure constant tensors (all non-harmonic L2 algebras proven isomorphic)
• 38 charge, 33 mass configurations confirming invariance across 10 orders of magnitude

Total: 993 rows across 13 Parquet tables.

What Makes This Unique

This dataset bridges machine learning and mathematical physics by providing exact algebraic invariants computed from first principles. Neural network training dynamics generate Lie algebras that can be directly compared to those of physical systems — revealing at least seven distinct universality classes as a function of the coupling structure chosen, with some classes strictly richer than any physical interaction law and others strictly less rich.

Quick Start

from datasets import load_dataset

# Load neural network algebras and compare coupling types
ds = load_dataset("bshepp/pairwise-poisson-algebras", "neural_algebras")
df = ds["train"].to_pandas()
for _, row in df[df["n_layers"] == 3].iterrows():
    print(f"{row['coupling_type']:12s} {row['loss_function']:15s} "
          f"width={row['width']} -> {row['dimension_sequence']}")

# Compare neural vs physical: how many extra generators?
import json
nn = load_dataset("bshepp/pairwise-poisson-algebras", "neural_algebras")["train"].to_pandas()
phys = load_dataset("bshepp/pairwise-poisson-algebras", "physical_systems")["train"].to_pandas()

nn_gradient = nn[(nn["coupling_type"] == "gradient") & (nn["n_layers"] == 3)
                  & (nn["activation"] == "linear") & (nn["loss_function"] == "mse")]
nn_dims = json.loads(nn_gradient.iloc[0]["dimension_sequence"])
phys_dims = json.loads(phys.iloc[0]["dimension_sequence"])
print(f"Neural:  {nn_dims}")
print(f"Physics: {phys_dims}")
print(f"Extra generators at level 3: {nn_dims[3] - phys_dims[3]}")

# Browse all dimension sequences
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"]])

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. Neural networks break physical universality: Gradient-product coupling on a 3-layer linear network gives [3, 6, 17, 119] — 3 extra generators vs the physical universal 116. The neural and physical algebras have the same level-0/1/2 structure (3, 6, 17) but diverge at the highest polynomial-degree stratum of level 3. Neural generators peak at degree 34; physical at degree 10. Within each degree stratum, neural has fewer syzygies (linear dependencies), producing 3 extra independent directions.

2. Seven neural universality classes at L=3 (not three): The choice of pairwise coupling yields distinct algebras. Diagonal gradients (gradient_sum) give [3, 6, 17, 111]; kinetic coupling (directional) gives 115; cubic gradients give 104; Fisher-rescaled gradients give 87; Hessian/symmetric/loss-squared give [3, 5, 11, 47]; standard gradient and several variants give 119.

3. Within-class invariances (for the 119 class): The algebra is independent of weight dimension (width k=1,2,3 all give [3, 6, 17, 119] at level 3), loss function (MSE = L1 = 119), and activation function (linear, tanh, ReLU all give [3, 6, 17] through level 2) — within the gradient-product universality class.

4. Depth-scaling asymmetry: At L=3, neural and physical algebras agree through level 2 (both [3, 6, 17]) and diverge only at level 3. But at L=4 they already diverge at level 1 (neural 20 vs physical 14), and at L=5 even more so (neural 45 vs physical 25). The extra dimensions at level 1 follow the formula C(L,2)*(L-3): 0, 6, 20 for L=3, 4, 5.

5. Physical potential universality: For N=3 with 1/r, 1/r^2, 1/r^3, 1/r^4, log, composite, Yukawa, and polynomial r^n (n≥4) potentials, the dimension sequence is universally [3, 6, 17, 116]. Only three exceptional potentials break this: r^1 → [3,4,5,5], r^2 → [3,6,13,15], r^3 → [3,6,17,109].

6. 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.

7. Charge/mass independence: The sequence [3, 6, 17, 116] is invariant under arbitrary charges and mass ratios spanning 10 orders of magnitude.

8. Quantum deformation: Moyal bracket yields [3, 6, 17, 117] for singular potentials — exactly one extra generator. Physics adds 1 generator upon quantization; neural networks add 3 (gradient class) or as many as depth^2 for deeper networks.

9. GUE universality: The log-gas potential (Dyson's GUE eigenvalue dynamics) produces the same algebra as Newtonian gravity: [3, 6, 17, 116].

10. Classical Bell bound and non-commutativity: All tested algebras have zero commuting pairs and respect the CHSH classical bound (max |S| = 1.77 < 2).

Dataset Splits

neural_algebras (21 rows)

Poisson bracket algebras arising from neural network training dynamics. SGD with momentum on a linear network is a Hamiltonian system; the pairwise interactions between weight layers generate a Lie algebra. This table sweeps across network depth (L=2--5), width (k=1--3), twelve coupling types, loss function (MSE, cross-entropy, L1), and activation function (linear, tanh, ReLU).

Column	Type	Description
n_layers	int	Number of weight layers (network depth)
width	int	Weight dimension per layer (1=scalar, >1=vector)
activation	str	Activation function: "linear", "tanh_taylor", "relu_taylor"
loss_function	str	Loss function: "mse", "cross_entropy", "l1"
coupling_type	str	Pairwise coupling: gradient, hessian, symmetric, fisher, gradient_sum, gradient_abs, hessian_plain, hessian_full, directional, natural_gradient, loss_power, gradient_cubic
max_level	int	Highest bracket depth computed
dimension_sequence	str	JSON-encoded cumulative dimension list
new_per_level	str	JSON-encoded new dimensions per level
dim_L0..dim_L3	int/null	Flattened dimensions per level
matches_physical_116	bool	Whether level-3 dimension equals physical universal 116
n_generators	int	Total candidate generators
stabilized	bool	Whether algebra growth has stabilized
is_exact	bool	Whether computed with exact arithmetic
computation_method	str	"numerical_SVD" or "exact_gaussian_elimination_QQ"
computation_time_s	float	Computation time in seconds

Seven distinct universality classes at L=3:

Class	Dims	Members
A	[3, 6, 17, 119]	gradient, fisher, gradient_abs, hessian_plain, l1-loss
B	[3, 6, 17, 115]	directional (kinetic-gradient)
C	[3, 6, 17, 111]	gradient_sum (diagonal only)
D	[3, 6, 17, 104]	gradient_cubic
E	[3, 6, 17, 87]	natural_gradient
F	[3, 6, 17, 62]	gradient + cross-entropy loss
G	[3, 5, 11, 47]	hessian, symmetric, hessian_full, loss_power

Within-class invariances (Class A, gradient coupling with MSE or L1):

• Width invariance: k=1, 2, 3 all give [3, 6, 17, 119] at L=3
• Loss invariance: MSE = L1 = [3, 6, 17, 119]; cross-entropy BREAKS invariance → Class F (62)
• Activation invariance: linear = tanh = ReLU give [3, 6, 17] through level 2

Depth scaling: L=2 → [1,1,1,1]; L=3 → [3,6,17,119] (matches physics at L0-L2); L=4 → [6,20,164] (diverges from physics [6,14,62] already at L1 by +6); L=5 → [10,45,210] (diverges at L1 by +20). Level-1 extras follow C(L,2)*(L-3) exactly.

dimension_sequences (~685 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 13 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.