Dashboard Research Tracker Data Explorer Interactive Atlas
Three-Body Poisson Algebra

Pairwise Poisson Algebras of the N-Body Problem

[ 3, 6, 17, 116, ≥5604† ]
3 Papers drafted 21 gap items tracked 16 configs complete Last updated: Mar 27, 2026
N=3 Dimension Sequence
3, 6, 17, 116
d(4) ≥ 5,604 · GK dim = ∞
N=4 Dimension Sequence
6, 14, 62
Through Level 2 · Gap 3.4×10¹¹
Completed Atlas Configs
15
+ 4 in progress · 6 not started
Gap Work Items
1 blocked 0 in progress
20 not started · 0 complete

Universality Evidence Matrix

Parameter Tested Values N=3 N=4
Mass ratios 0.001 to 10⁶ (25+ configs) ✓ Invariant ✓ (3 configs)
Spatial dimension d d = 1, 2, 3 ✓ Invariant ✓ Invariant
Potential type 1/r, 1/r², 1/r³, log(r), composite ✓ Invariant Not tested
Charge sign All-attract, all-repulse, mixed ✓ Invariant Not tested
Charge magnitude |q| = 1, 2, 3 ~ Sensitive at L3 Not tested

Structural Dichotomy

Singular potentials (1/r, 1/r², 1/r³, log, Yukawa) [3, 6, 17, 116] → ∞
Regular potentials (r²) [3, 6, 13, 15, 15] → finite
Mechanism Universal cubic chain rule du/dq ~ u³
Integrability detection NOT a certificate — CM integrable, same sequence

Internal Structure (Paper 2)

Level-3 tier decomposition 52 + 44 + 16 + 4 = 116
Scaling exponents ε⁰, ε¹, ε², ε³ (integer-quantized)
S₃ isotypic decomposition 24A + 28A' + 52E (E-fraction = 2/3)
Null generators (40 total) 32 syzygies + 8 true zeros (momentum)

Spectral Knee Index — 1/r² (Calogero-Moser)

SV index where the steepest consecutive log-drop occurs across the shape sphere (100×100 grid). Dark bands trace the S₃ symmetry locus — sharper spectral elbows near symmetric configs.

Knee index heatmap for 1/r² Calogero-Moser potential
Gap Work Plan — 21 Items Across 4 Phases
▶ Phase 1: Free Post-Processing Zero Compute
1.1
Spectral Depth Mining — Interior SV Landscapes
Plot SV #50, #80, #100, #110 as shape-sphere heatmaps. Data exists in sv_spectra.npy.
Not Started
1.2
Spectral Decay Rate Map
Log-slope between SV #80–#115 at each grid point. Reveals near-degenerate generators.
Not Started
1.3
Spectral Clustering on Shape Sphere
k-means/hierarchical clustering (k=3,4,5) on normalized 156-dim SV vectors.
Not Started
1.4
Clebsch-Gordan Predictions vs Full Atlas
Extend CG doublet count comparison from Lagrange-only to full shape sphere.
Not Started
1.5
Level-4 Comparison Chart
Bar chart + convergence curves of d(4) across configuration types.
Not Started
1.6
Analytical Prediction of SV #116
Predict weakest generator landscape from symbolic structure. Most complex free item.
Not Started
▶ Phase 2: Light Compute (hours, local/small AWS) Hours
2.1
N=5 Level 1–2
Third data point for d_N(k) vs N. HIGH impact — headline universality result.
Not Started
2.2
N=4 with 1/r², 1/r³, log(r)
Paper 3 falsifiable prediction #2. CRITICAL — directly falsifiable.
Not Started
2.3
r⁴ Potential (Regular)
Prediction: finite algebra. Tests regular/singular dichotomy for higher degree.
Not Started
2.4
Charge Sweep Phase 3 (+1/+q/−1)
Phase 3 of charge sweep crashed. Needs AWS script fix.
Crashed
2.5
S₄ Tier Decomposition (N=4)
Does S₃ tier structure generalize to S₄? Integer-quantized scaling?
Not Started
2.6
Harmonic Dimension 15 — Rep Theory Derivation
Derive dim=15 from isotropic oscillator Lie algebra (sp(4,ℝ) or similar).
Not Started
2.7
H₃⁺ and Ozone Molecular Systems
Molecular triatomic reconfirmation. High outreach value.
Not Started
▶ Phase 3: Medium Compute ($10–$200 AWS) $10–$200
3.1
Parametric Exponent Sweep (1,015 values of n)
Gap ratio vs n continuous curve. Script ready. Est. $13–50. VERY HIGH impact.
Not Started
3.2
Yukawa Potential (3 Scenarios)
Lambdification/OOM issues. Recursion fix deployed but not confirmed.
Blocked
3.3
Re-run 7 Retracted Gravitational Configs
2 of 7 directly validated; 5 inferred from mass sweep. Completeness.
Not Started
3.4
Complete Interrupted Atlases
Sun-Earth-Moon (11%), Sun-Jupiter-Asteroid (7%). Checkpoints on S3.
Not Started
3.5
Lagrange Hires 1000×1000 Scan
Resolve concentric ring features and isosceles beads. Est. $3–12.
Not Started
▶ Phase 4: Verification & Theory Theory
4.1
SageMath Independent Verification
Essential: independently confirm [3, 6, 17, 116] using a second CAS.
Not Started
4.2
Growth Rate Formula / Generating Function
OEIS search, recurrence relations, exponential fits for d(k).
Not Started
4.3
Level-4 Bound Improvement
Current: d(4) ≥ 5,604 at 200K samples. mpmath at 4.4% when reclaimed.
Not Started
Physical Systems Catalog
Gravitational
SystemMassesSequenceAtlas
Equal-mass 3-body1:1:1[3, 6, 17, 116]✓ Complete
Sun-Earth-Moon1 : 3×10⁻⁶ : 3.7×10⁻⁸[3, 6, 17, 116]†11%
Sun-Jupiter-Asteroid1 : 9.5×10⁻⁴ : 10⁻¹⁰[3, 6, 17, 116]†7%
Binary Star + Planet1:1:0.001[3, 6, 17, 116]‡✓ Complete
Triple BH (LISA)1:0.01:10⁻⁵[3, 6, 17, 116]†✓ Complete
Atomic / Coulomb
SystemChargesMassesSequenceAtlas
Helium atom(+2,−1,−1)7294:1:1[3, 6, 17, 116]✓
Li⁺ ion(+3,−1,−1)12789:1:1[3, 6, 17, 111]✓
H⁻ ion(+1,−1,−1)1836:1:1[3, 6, 17, 116]✓
Positronium Ps⁻(+1,−1,−1)1:1:1[3, 6, 17, 116]✓
Muonic Helium(+2,−1,−1)7294:1:207[3, 6, 17, 116]✓
H₂⁺ molecular ion(+1,+1,−1)1836:1836:1[3, 6, 17, 115]✓
Alternative Potentials
SystemPotentialSequenceNotes
Calogero-Moser1/r²[3, 6, 17, 116]Integrable in 1D — same sequence
Cubic potential1/r³[3, 6, 17, 116]Jahn-Teller ring observed
2D Vorticeslog(r)[3, 6, 17, 116]Transcendental singularity
Composite1/r + 1/r²[3, 6, 17, 116]Multi-pole universal
Harmonic oscillatorr²[3, 6, 13, 15]FINITE — saturates at dim 15
Penning trap1/r + harmonic[3, 6, 17, 116]All-repulsive + external trap
Nuclear / Plasma (In Progress)
SystemPotentialStatusNotes
Tritium / He-3YukawaIn progressRecursion fix deployed
p-n-n ScatteringYukawaIn progress
Dusty PlasmaYukawaFailed — OOMLambdification issue
† Inferred from mass ratio sweep + diagnostic (SymPy 1.10.1 artifact corrected).
‡ Directly re-run on AWS with SymPy 1.13.3 and confirmed.
Paper Trilogy + Supplement
1
Super-exponential growth of the Poisson algebra generated by pairwise Hamiltonians of the planar three-body problem
preprint.tex
Dimension sequence [3, 6, 17, 116], mass invariance theorem, potential comparison (harmonic/Newton/CM), level-4 lower bound d(4) ≥ 4,501. Reframed from "non-integrability certificate" to "structural algebraic invariant" after CM reckoning.
Draft
2
S₃-equivariant jet filtration of the three-body Poisson algebra: tier decomposition, integer scaling exponents, and syzygy structure
paper2_s3_filtration.tex
52+44+16+4 = 116 tier decomposition. Clebsch-Gordan verification: 24A + 28A' + 52E. E-fraction = 2/3 at every bracket level. 32 syzygies + 8 true zeros. Integer-quantized scaling exponents α = 0, 1, 2, 3.
Draft
3
Universal dimension sequences of pairwise Poisson algebras: independence from spatial dimension, potential exponent, and charge sign
paper3_universality.tex
N=4 sequence [6, 14, 62]. d-independence for N=3 and N=4. Pole-order invariance (1/r, 1/r², 1/r³). Charge-sign invariance. Formal universality conjecture. Three falsifiable predictions.
Draft
4
The Poisson Algebra of Pairwise Interactions: A Calogero-Moser Integrability Test
paper4_calogero_integrability.tex
1D Calogero-Moser (exactly integrable) gives [3, 6, 17, 116]. Singularity class invariance. Galperin superintegrable mass ratios all universal. revtex4-2, 4 pages.
Draft
Publication Blockers
arXiv endorsement needed math-ph / dynamical systems — no endorser yet
Current preprint host Zenodo (DOI: 10.5281/zenodo.18899804)
Existing arXiv publication cs.RO/math.GN (topological robotics) — doesn't carry over
SageMath verification (4.1) Essential for credibility — not started
Conjectures & Open Questions

Universality Conjecture (Revised, March 2026)

For N ≥ 3 particles in d ≥ 1 spatial dimensions, interacting via a singular central potential: (1) the pairwise Poisson algebra has infinite GK dimension; (2) the cumulative dimension sequence d_N(n) depends only on N; (3) the sequence is independent of spatial dimension, potential type (within singular class), mass ratios, and charge configurations. For regular potentials (V polynomial in r), the algebra is finite-dimensional.

✓ N=3 sequence proved
✓ N=4 sequence [6,14,62]
✓ Mass invariance (25+ configs)
✓ d-independence (d=1,2,3)
✓ Potential-type invariance (5 types)
✓ Charge-sign invariance
○ N=5 not yet computed
○ N=4 with 1/r² not tested

Critical Locus Conjecture

The local rank of the Poisson algebra, viewed on the reduced configuration space (shape sphere for N=3), has its critical locus at exactly the fixed-point set of the S_N action. Rank drops occur only at symmetric configurations — Lagrange (S₃), Euler (Z₂), isosceles (Z₂).

✓ 99,000 grid points, 11 configs
✓ No non-symmetric anomalies found
✓ Cross-potential consistency
○ S₄ extension untested (N=4)

Cosmological Perspective (Speculative)

If the universality conjecture holds for general N, the S_N symmetric fixed point of the cosmic N-body problem corresponds to the homogeneous, isotropic universe. The critical locus conjecture predicts a rank drop at this point — an algebraic explanation for the dimensional reduction from ~3×10⁸⁰ degrees of freedom to the Friedmann equation's single scale factor a(t).

✓ d=3 explicitly computed
✓ Charge-sign invariant (EM too)
○ Only N=3,4 tested
○ GR not modeled

Falsifiable Predictions (Paper 3)

Three predictions that would falsify or strengthen the universality conjecture:

○ P1: N=5, d=1, 1/r → predict d(0)=10, universal growth
○ P2: N=4 with 1/r² → predict [6, 14, 62] (potential invariance at N=4)
○ P3: d(4) for 1/r² → predict matches d(4) for 1/r (level-4 universality)
Infrastructure & Compute

Core Engines

exact_growth.py — N=3 symbolic Poisson bracket engine (levels 0–3)
exact_growth_nbody.py — Generalized engine (N, d, potential, charges)
stability_atlas.py — Shape-sphere atlas scanner (exact engine)
multi_epsilon_atlas.py — Adaptive + multi-epsilon scanning
targeted_adaptive_scan.py — Regional high-resolution scans

Lambdify Pipeline (4-layer fallback)

Layer 1: Standard sp.lambdify (fast numpy)
Layer 2: Flat-file no-CSE (pycode → temp .py)
Layer 3: Flat-file with CSE (slower, smaller code)
Layer 4: Point-by-point xreplace (last resort, ~1000× slower)
1/r² result: 92 via L1, 63 via L2, 0 xreplace

AWS Campaign

S3 bucket: s3://3body-compute-290318/
Atlas instances: r6i.4xlarge (16 vCPU, 128 GB)
Level-4 mpmath: r6i.8xlarge → r6a.8xlarge (spot)
Multiprocessing: --workers N (fork-based, 8× speedup)
Total data: 9.04 GB synced locally
S3 sync caveat: Use cp --recursive for numpy arrays (size-identical trap)

In-Progress Compute

Sun-Earth-Moon atlas: 11/100 rows (spot reclaimed)
11%Checkpoint on S3
Sun-Jupiter-Asteroid: 7/100 rows
7%Checkpoint on S3
Level-4 mpmath rank: 667/15,000 rows
4.4%ETA ~574h · rank=667, plateau=0

Cost History

Original campaign: ~$800–1,000 (19 instances, single-threaded, 48h)
With multiprocessing: ~$100–125 equivalent
1/r² atlas (w/ fixes): $2.50 (was projected $28)
Parametric sweep estimate: $13–50 (Tier 3 hybrid)
Atlas fleet peak: $435/day (16 atlas + 1 mpmath)

Known Issues

Yukawa lambdification: OOM/recursion on deeply nested exp(-μ/u)
SymPy version sensitivity: 1.10.1 fails 63/156 L3 expressions
Level-4 mpmath derivatives cache: 243 MB, must sync to S3
Li⁺ dim=111, H₂⁺ dim=115: SVD artifact or real physics? Under investigation