Research Tracker — Three-Body Poisson Algebra

Research Overview

Core Result

3, 6, 17, 116

N=3 universal dimension sequence · GK dim = ∞

N=4 Sequence

6, 14, 62

Potential-universal (1/r, 1/r², 1/r³, log) ✓

N=5 Probe (NEW)

10, 25, 145

d=1, 1/r · First N=5 result

Atlas Configs

16 / 25

3 in progress · 6 not started

Papers

4

Drafted · arXiv endorsement needed

Physical Systems Tested

30+

Gravitational, atomic, molecular, plasma

Universality Evidence Matrix

Parameter	Tested Values	N=3	N=4	N≥5
Mass ratios	0.001 to 10⁶ (25+ configs)	✓ Invariant	✓ (3 configs)	—
Spatial dimension d	d = 1, 2, 3	✓ Invariant	✓ Invariant	d=1 only
Potential type	1/r, 1/r², 1/r³, log(r), composite	✓ Invariant	✓ Confirmed	1/r only
Charge sign	All-attract, all-repulse, mixed	✓ Invariant	Not tested	—
Charge magnitude	\|q\| = 1, 2, 3	~ Sensitive at L3	Not tested	—

Structural Dichotomy

Class	Examples	Behavior
Singular potentials	1/r, 1/r², 1/r³, log, Yukawa	[3, 6, 17, 116] → ∞ (infinite GK dim)
Regular potentials	r² (harmonic oscillator)	[3, 6, 13, 15, 15] → finite (saturates at 15)

Mechanism: Universal cubic chain rule du/dq ~ u³ drives infinite growth for singular potentials.
Note: NOT a non-integrability certificate — Calogero-Moser is exactly integrable yet gives the same sequence.

N-Body Scaling (April 2026)

N	d(0)	d(1)	d(2)	Formula for d(0)	Status
3	3	6	17	C(3,2) = 3	✓ Full
4	6	14	62	C(4,2) = 6	✓ L2
5	10	25	145	C(5,2) = 10	d=1
6	15	39	279	C(6,2) = 15	d=1
7	21	56	476	C(7,2) = 21	d=1
8	28	76	301?	C(8,2) = 28	partial

d(0) = C(N,2) = number of pairs · d(1) pattern: 6, 14, 25, 39, 56, 76 · All d=1, 1/r potential

Research Timeline

April 2026

N=5–8 Scaling Probes Complete

First dimension sequences beyond N=4: N=5→[10,25,145], N=6→[15,39,279], N=7→[21,56,476]. Confirms d(0)=C(N,2) pattern. N=4 potential universality confirmed for 1/r², 1/r³, log(r).

March 27, 2026

Gap Workplan Phase 1 Complete (6/6)

All 6 free post-processing analyses done: SV landscape mining, spectral decay rate maps, spectral clustering, CG atlas comparison, Level-4 bar chart, SV #116 analytical prediction.

March 2026

15 Atlas Configurations Complete

Full atlas coverage for: equal-mass gravitational, helium, Li⁺, H⁻, positronium, muonic helium, H₂⁺, binary star, and 7 alternative potentials (1/r², 1/r³, log, composite, Coulomb variants). Triple BH stalled at ~38%.

March 2026

Paper 4: Calogero-Moser Integrability Test

Critical insight: [3, 6, 17, 116] is NOT a non-integrability certificate. 1D Calogero-Moser (exactly integrable) produces the same sequence. Singularity class — not integrability — drives the algebra's growth.

Feb–Mar 2026

S₃ Tier Decomposition Established

Level-3: 52 + 44 + 16 + 4 = 116 generators. Integer-quantized scaling exponents (α = 0,1,2,3). CG decomposition: 24A + 28A' + 52E (E-fraction = 2/3 constant). 32 syzygies + 8 true zeros.

February 2026

N=4 Sequence [6, 14, 62] Confirmed

Mass-invariant, d-independent. Gap ratio 3.4×10¹¹ at level 2. Three mass configurations tested.

February 2026

SymPy 1.13.3 Bug Fix — Dimension Correction

SymPy ≤1.10.1 produced incorrect [3, 5, 13, 69] for unequal masses. Version 1.13.3 corrected to universal [3, 6, 17, 116]. 63/156 level-3 expressions were wrong. All prior unequal-mass results had to be re-validated.

Jan–Feb 2026

AWS Campaign — 19 Instances

Initial compute fleet: 19 r6i.4xlarge instances. Total cost ~$800–1,000 (single-threaded). Later reduced to ~$100 equivalent with multiprocessing (8× speedup). Level-4 mpmath started on r6i.8xlarge.

January 2026

Core Dimension Sequence [3, 6, 17, 116] Established

First computation for equal-mass planar 3-body with 1/r potential. SVD gap ratio test confirms exact rank at each level. Level-4 lower bound d(4) ≥ 4,501.

Late 2025

Project Inception

Initial exploration of Poisson bracket structure for the N-body problem. exact_growth.py written as baseline engine. Auxiliary variable substitution u_ij = 1/r_ij developed to keep expressions polynomial.

Dimension Sequences — Complete Record

The Universal Singular Sequence (N=3)

[ 3, 6, 17, 116, ≥5604 ]

Level 0 through 4 · Level-4 lower bound from SVD at 200K samples

N=3 — All Potentials

Potential	d(0)	d(1)	d(2)	d(3)	Match?
1/r (Newton/Coulomb)	3	6	17	116	✓
1/r² (Calogero-Moser)	3	6	17	116	✓
1/r³ (cubic)	3	6	17	116	✓
log(r) (2D vortices)	3	6	17	116	✓
1/r + 1/r² (composite)	3	6	17	116	✓
1/r + harmonic (Penning trap)	3	6	17	116	✓
r² (harmonic)	3	6	13	15	FINITE

N=4 — Potential Universality

Potential	d(0)	d(1)	d(2)	Match ref?
1/r (reference)	6	14	62	—
1/r²	6	14	62	✓
1/r³	6	14	62	✓
log(r)	6	14	62	✓

N=5 through N=8 (d=1, 1/r)

N	d(0)	d(1)	d(2)	Time
5	10	25	145	8s
6	15	39	279	16s
7	21	56	476	65s
8	28	76	301?	partial

d(0) = C(N,2) confirmed · d(1) sequence: 6, 14, 25, 39, 56, 76 · N=8 d(2) may be truncated by memory

Internal Structure (Level 3, N=3)

Tier	Count	Scaling Exponent	S₃ Decomposition
Tier 0 (core)	52	ε⁰	Dominant generators
Tier 1	44	ε¹	First-order scaling
Tier 2	16	ε²	Second-order
Tier 3	4	ε³	Weakest generators
Total	116		24A + 28A′ + 52E

E-type irrep fraction = 2/3 at every bracket level · 40 null generators: 32 syzygies + 8 momentum zeros

Gap Work Plan — 21 Items

▶ Phase 1: Free Post-Processing COMPLETE

1.1

Spectral Depth Mining — Interior SV Landscapes

SV #50, #80, #100, #110 as shape-sphere heatmaps.

Done

1.2

Spectral Decay Rate Map

Log-slope between SV #80–#115 at each grid point.

Done

1.3

Spectral Clustering on Shape Sphere

k-means/hierarchical clustering on 156-dim SV vectors.

Done

1.4

Clebsch-Gordan Predictions vs Full Atlas

CG doublet count comparison on full shape sphere.

Done

1.5

Level-4 Comparison Chart

Bar chart + convergence curves of d(4) across configs.

Done

1.6

Analytical Prediction of SV #116

Weakest generator landscape prediction from symbolic structure.

Done

▶ Phase 2: Light Compute (hours) 2/7

2.1

N=5 Level 1–2

d=1, 1/r → [10, 25, 145]. Confirmed C(N,2) pattern.

Done

2.2

N=4 with 1/r², 1/r³, log(r)

All match [6, 14, 62]. Potential universality at N=4 confirmed.

Done

2.3

r⁴ Potential (Regular)

Prediction: finite algebra, like r². Tests dichotomy for higher-degree.

Not Started

2.4

Charge Sweep Phase 3 (+1/+q/−1)

Phase 3 crashed on AWS. Needs script fix.

Crashed

2.5

S₄ Tier Decomposition (N=4)

Does S₃ tier structure generalize to S₄?

Not Started

2.6

Harmonic Dim 15 — Rep Theory Derivation

Derive dim=15 from sp(4,ℝ) or similar.

Not Started

2.7

H₃⁺ and Ozone Molecular Systems

Triatomic molecular reconfirmation. Outreach value.

Not Started

▶ Phase 3: Medium Compute ($10–$200 AWS) 0/5

3.1

Parametric Exponent Sweep (1,015 values)

Gap ratio vs n continuous curve. Script ready. $13–50.

Not Started

3.2

Yukawa Potential (3 Scenarios)

Lambdification OOM. Recursion fix deployed but unconfirmed.

Blocked

3.3

Re-run 7 Retracted Grav Configs

2/7 validated; 5 inferred from mass sweep.

Not Started

3.4

Complete Interrupted Atlases

Sun-Earth-Moon (11%), Sun-Jupiter-Asteroid (7%).

Not Started

3.5

Lagrange Hires 1000×1000 Scan

Resolve concentric ring features. $3–12.

Not Started

▶ Phase 4: Verification & Theory 0/3

4.1

SageMath Independent Verification

Confirm [3, 6, 17, 116] in a second CAS.

Not Started

4.2

Growth Rate Formula / Generating Function

OEIS search, recurrence relation, exponential fits.

Not Started

4.3

Level-4 Bound Improvement

d(4) ≥ 5,604 at 200K. mpmath at 4.4%.

Not Started

Physical Systems Catalog

Gravitational

System	Masses	Sequence	Atlas
Equal-mass 3-body	1:1:1	[3, 6, 17, 116]	✓ Complete
Sun-Earth-Moon	1 : 3×10⁻⁶ : 3.7×10⁻⁸	[3, 6, 17, 116]†	11%
Sun-Jupiter-Asteroid	1 : 9.5×10⁻⁴ : 10⁻¹⁰	[3, 6, 17, 116]†	7%
Binary Star + Planet	1:1:0.001	[3, 6, 17, 116]‡	✓ Complete
Triple BH (LISA)	1:0.01:10⁻⁵	[3, 6, 17, 116]†	~38% — Stalled

Atomic / Coulomb

System	Charges	Masses	Sequence	Atlas
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

System	Potential	Sequence	Notes
Calogero-Moser	1/r²	[3, 6, 17, 116]	Integrable in 1D — same sequence
Cubic potential	1/r³	[3, 6, 17, 116]	Jahn-Teller ring observed
2D Vortices	log(r)	[3, 6, 17, 116]	Transcendental singularity
Composite	1/r + 1/r²	[3, 6, 17, 116]	Multi-pole universal
Harmonic oscillator	r²	[3, 6, 13, 15]	FINITE — saturates at dim 15
Penning trap	1/r + harmonic	[3, 6, 17, 116]	Repulsive + external confinement

Nuclear / Plasma (In Progress)

System	Potential	Status	Notes
Tritium / He-3	Yukawa	Not started	Config only — no computation ran
p-n-n Scattering	Yukawa	Not started	No data or config
Dusty Plasma	Yukawa	Failed — OOM	Lambdification issue

† Inferred from mass ratio sweep (SymPy 1.10.1 artifact corrected) ‡ Directly re-run with SymPy 1.13.3

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. CG 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. 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 CM (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	math-ph / dynamical systems — no endorser yet
Current host	Zenodo (DOI: 10.5281/zenodo.18899804)
SageMath verification	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.

✓ N=3 five potentials match

✓ N=4 four potentials match

✓ 25+ mass ratios invariant

✓ d=1,2,3 all identical

✓ Charge-sign invariance

✓ N=5–7 probes consistent

○ N=5 with d>1 not tested

○ GR corrections not modeled

Critical Locus Conjecture

The local rank of the Poisson algebra on the reduced configuration space has its critical locus exactly at 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

✓ Cross-potential consistency

○ S₄ extension untested

Falsifiable Predictions (Paper 3)

Three predictions that would falsify or strengthen the universality conjecture:

✓ P1: N=5, d=1, 1/r → d(0)=10 — CONFIRMED

✓ P2: N=4 with 1/r² → [6, 14, 62] — CONFIRMED

○ P3: d(4) for 1/r² matches d(4) for 1/r — Not tested

Cosmological Perspective (Speculative)

If universality 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 — an algebraic explanation for dimensional reduction from ~3×10⁸⁰ DOF to the Friedmann equation's single scale factor a(t).

✓ d=3 explicitly computed

✓ Charge-sign invariant

○ Only N=3,4 tested

○ GR not modeled

Infrastructure & Compute

Core Engines

exact_growth.py — N=3 baseline (regression ref)

exact_growth_nbody.py — Full N-body engine

stability_atlas.py — Shape-sphere scanner

multi_epsilon_atlas.py — Adaptive scanning

Lambdify Pipeline (4-layer fallback)

L1: sp.lambdify (fast numpy)

L2: Flat-file no-CSE (pycode → temp .py)

L3: Flat-file with CSE

L4: Point-by-point xreplace (~1000× slower)

AWS Campaign

S3: s3://3body-compute-290318/

Atlas: r6i.4xlarge (16 vCPU, 128 GB)

mpmath: r6i.8xlarge → r6a.8xlarge (spot)

Total data: 9.04 GB synced locally

Cost History

Original campaign: ~$800–1,000

With multiprocessing: ~$100 equiv

1/r² atlas: $2.50

Fleet peak: $435/day

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 cache: 243 MB, needs S3 sync

Li⁺ dim=111, H₂⁺ dim=115: SVD artifact or real physics? Under investigation

Research Tracker — Progress, Milestones, and Open Questions

Sections

Compute Progress

Key Sequences

Universality Evidence Matrix

Structural Dichotomy

N-Body Scaling (April 2026)

The Universal Singular Sequence (N=3)

N=3 — All Potentials

N=4 — Potential Universality

N=5 through N=8 (d=1, 1/r)

Internal Structure (Level 3, N=3)

Gravitational

Atomic / Coulomb

Alternative Potentials

Nuclear / Plasma (In Progress)

Publication Blockers

Universality Conjecture (Revised, March 2026)

Critical Locus Conjecture

Falsifiable Predictions (Paper 3)

Cosmological Perspective (Speculative)

Known Issues