Poincaré proved in 1890 that the trajectories of three gravitating bodies are fundamentally chaotic. Every prior method either works around this or fights through it. PNEP works through it by finding the moments where the chaos goes quiet.
Prior methods
MA01 / Static Binary yes/no from initial conditions. No timing. No identity. Never updates.
ML classifiers Trained on labelled systems. Still static. Cannot forecast timing or identity.
PNEP
Event-driven Samples only at mirror symmetry nodes. Updates in real time during integration.
The instant when any pair of bodies reaches closest approach and radial velocity vanishes (dr/dt = 0). At this moment the system achieves temporal reflection symmetry — the configuration is identical whether time runs forward or backward. The hierarchy is maximally legible. We call these moments geometrically honest.
Compute advantage
PNEP evaluates H only at nodes — typically tens to hundreds per system rather than millions of timesteps. Computational overhead is approximately 1% of full N-body evaluation. For a survey of 10,000 systems: hours not months.
The hierarchy index
One scalar. No training data. No frame dependence.
At each mirror symmetry node, a single number is computed from the variance of pairwise distances. That number encodes whether the system is hierarchically intact or geometrically scrambled.
The hierarchy index H
H = σ² / (1 + σ²) where σ² = Var(d₁₂, d₂₃, d₃₁)
Stable system — inner binary dominates, distances unequal, high varianceH = 0.83
H = 0 (chaotic)Classification threshold H = 0.50H = 1 (stable)
Why it works
In a stable triple at a node: one small distance (inner binary at periapsis) + two large distances (outer body far away) = high variance = H → 1. In chaos: all three distances comparable = low variance = H → 0. This is not empirical — it is geometric inevitability.
Frame-dependence correction
Early versions included an alignment angle term. In the CoM frame, bulk velocity is exactly zero — the term measured floating-point noise. Removing it and using pure distance variance was the decisive step achieving 100% ground truth validity.
Separation between populations
Stable mean H
0.827
Unstable mean H
0.091
Separation Δ
0.736
Decay regimes
Three distinct signatures. Same scalar H.
The slope, variance, and temporal behaviour of H at consecutive nodes identifies which of three regimes the system occupies — each with its own pre-ejection forecasting method.
Regime 1 — Stable
H flat near 0.83 at every node indefinitely. dH/dn ≈ 0. Signal does not waver across thousands of measurements. Zero false positives.
∞ lead
Regime 2 — Slow decay
H erodes at dH/dn ≈ +0.0012/node over 500–3000t. Geometric divorce. Outer body incrementally absorbs energy from inner binary. Linear extrapolation forecasts ejection hundreds of time units in advance.
412.6t median
Regime 3 — Flash ejection
H scrambles in [0.05–0.30], then spikes above 0.50 and stays. That spike is the warning. Validated on Burrau's Pythagorean Problem.
The most compelling evidence for universality: the 3+1 topology produces an inverted polarity — stable = low H_inner. A coincidental signal would break under topology change. A geometric law reveals itself differently but holds universally.
3-body hierarchical triple
Signal: H of all 3 pairs
Stable polarity: high H
92.0%
100% specificity · 0 false positives
3+1 hierarchical quadruple
Signal: H_inner of inner triple
Stable polarity: low H_inner ↕
93.5%
78.9t median lead time
2+2 peer binary quadruple
Signal: R × H combined
Stable polarity: high R×H
99.2%
100% specificity
Burrau's Pythagorean problem — canonical benchmark
Masses
3 : 4 : 5
H scramble zone
0.11–0.28
H-spike detected
t = 58.4
Physical ejection
t = 63.1
Body predicted
Mass 3 ✓
Lead time 4.7t · Energy drift 0.004% · Lightest body correctly identified via velocity geometry, not mass statistics.
Pre-ejection signals
Two novel signals. Both identify the escaping body.
Beyond the H slope forecast, two independent signals fire before ejection — each providing advance warning and each identifying the escaping body through geometry alone.
Liberation energy signature — gravitational analogue of atomic ionisation
1
Inner binary acts as gravitational energy engine — transfers energy to outer body through repeated close approaches over hundreds of orbital periods
2
At liberation threshold: outer body briefly circularises its orbit (mean eccentricity at anomaly = 0.124) — genuinely near-circular amid otherwise highly elliptical chaotic orbits
3
Eccentricity grows monotonically toward 1.0 and escape. Point of no return crossed. The body tracing this liberation orbit is the ejecting body.
Coverage
100%
Median lead
26.2t
Max lead
167.5t
H-spike velocity trigger (flash regime)
At the moment of H-spike, the body with the highest relative velocity from the system CoM is the escaping body. Reads velocity geometry — not mass ordering. Validated on Burrau: correctly identifies mass 3 (the lightest) through geometric evidence alone.
Node geometry directional signal
In the final nodes before ejection, one body's pairwise distances grow consistently while others remain stable. The confidence score — slope of the growing distance pair — correctly identifies the ejecting body in 100% of high-confidence cases (confidence > 0.5).
All three identity signals are independent. They converge on the same body through geometry alone, regardless of mass ordering.
Nodal breathing — new discovery
The earliest warning signal in the framework.
Before H shows any decay, the inter-node time interval begins rhythmic oscillation. This is Nodal Breathing — the outer orbital period approaching resonance with the inner binary period. It fires thousands of time units before ejection while H looks perfectly flat.
Validated result (v15.1)
Alert fired
t = 17.44
H at alert
0.857
Jitter J
0.611
Projected lead
~4000t
H is visually flat at alert time. No visible decay. Every prior method: stable. Nodal Breathing: marginal.
The jitter metric J
12-node rolling window
J = σ(Δt_node) / mean(Δt_node)
Alert threshold: J > 0.02. Physical interpretation: outer orbital period approaching rational ratio with inner binary period. Beat frequency = breathing frequency.
Three nested frequency layers — discovered in simulation
Layer 1 — Breathing carrier
f_B ≈ 0.352 nodes⁻¹ · period ≈ 2.8 nodes
Beat between inner and outer orbital frequencies. Dominant and clean across all tested systems. The resonance carrier wave.
Layer 2 — Envelope sub-frequency
f_env ≈ 0.027 nodes⁻¹ · period ≈ 36 nodes
Amplitude modulation of the breathing oscillation — nested within the carrier. Analogous to AM radio: ejection timing encoded in the envelope, not the carrier.
Layer 3 — Ejection timing
t_ejection ≈ k × T_env
Two sub-regimes found: Type A (k ≈ 1.0–1.2) — ejection within one envelope period. Type B (k ≈ 2–4) — ejection after several envelope cycles. Strongest case: a_out=4.5, T_env = 1872t, t_eject = 1898t, ratio = 1.014.
Validated results
407 systems. Three topologies. Zero false positives on stable.
3-body (n=87, seed 42)
92.0%
100% specificity · 0 FP · 20.9t lead · F1 93.5%
3+1 quadruple (n=200)
93.5%
92.2% specificity · 78.9t lead · F1 93.5%
2+2 peer binary (n=120)
99.2%
100% specificity · 2.1t lead · F1 99.2%
Compute cost
~1%
of full N-body evaluation. O(1) per node. Tens to hundreds of evaluations vs millions of timesteps.
Energy conservation
0.0068%
Median energy drift. Symplectic KDK leapfrog. Near-machine-precision across long-duration integrations.
Pre-ejection signal summary
H-slope coverage
100%
Slow decay regime
Liberation sig. coverage
100%
All ejecting systems
Nodal breathing lead
~4000t
Marginal systems
Open source — NumPy only, no external dependencies