Pure Topological Structure Detection Framework

Time is an illusion. Only Transaction exists!
A revolutionary approach to detecting hidden celestial bodies using pure topological analysis – no physics required!

Overview

Lambda³ Stargazer is a groundbreaking framework that detects hidden gravitational influences in astronomical data using pure topological structure analysis.
Unlike traditional methods that rely on physical constants and time-based calculations, Lambda³ operates on a fundamental principle:

Transaction, not time. Structure, not physics!

Key Innovation

No Physical Constants Required: No G, no masses, no Kepler’s laws
Pure Structural Analysis: Detects patterns through topological tensors
Observation-Step Based: Works with sequence of observations, not time series
Noise Resilient: Handles missing data (7%+) and measurement noise effectively

Performance Results

Detection Success Rate

From black hole system simulation:
Observable: Planet α (the data source)
Hidden: 3 planets (X, Y, Z) perturbing α’s orbit

Dataset: Full (3000 steps) — Hidden Planets Detected: 2/3 (Y, Z) — Success Rate: 67%
Dataset: Near-field (1500 steps) — Hidden Planets Detected: 2/3 (X, Y) — Success Rate: 67%
Combined Analysis — Hidden Planets Detected: 3/3 — Success Rate: 100%

Detection Accuracy

Planet X: 6.8% error (detected: 860 steps, expected: 923)
Planet Y: 2.6-12.0% error (multiple detections)
Planet Z: 5.1% error (detected: 2159 steps, expected: 2274)

Mathematical Foundation

Core Tensors

The framework operates on four fundamental topological tensors:
1. ΛF (Lambda Flow) – Structural flow field between observation steps
2. ΛFF (Lambda Flow Flow) – Second-order structural changes
3. ρT (Rho Tension) – Local structural tension field
4. Q_Λ (Topological Charge) – Cumulative winding number

Detection Algorithm

1. Compute Lambda³ tensors from observation sequence
2. Detect structural boundaries (no physics!)
3. Identify topological breaks and anomalies
4. Extract recurrence patterns (not periods!)
5. Filter harmonics to find fundamental structures
6. Decompose into structural signatures
7. Match to hidden bodies

Installation & Usage

Requirements

numpy >= 1.19.0
pandas >= 1.1.0
scipy >= 1.5.0
matplotlib >= 3.3.0

Basic Usage

from Lambda3Stargazer_v2 import PureLambda3Analyzer
analyzer = PureLambda3Analyzer(verbose=True)
data, positions = analyzer.load_and_clean_data(‘challenge_blackhole_alpha_noisy.csv’)
results = analyzer.analyze(data, positions)
analyzer.print_results()
analyzer.plot_results()

Multi-Focus Mode (for long sequences)

For observation sequences > 2500 steps, the framework automatically activates dual-scale analysis:
Phase 1: Near-field detection (first 1500 steps)
Phase 2: Far-field detection (full data)

Input Data Format

CSV Structure

step, x_noisy, y_noisy, z
0, 1.195423, 0.023451, 0.0
1, 1.194892, 0.046834, 0.0
…

step: Observation sequence number
x_noisy, y_noisy: Noisy position measurements
z: Z-coordinate (typically 0 for 2D orbits)

Data Characteristics Handled

Missing Data: Up to 7% random gaps
Gaussian Noise: σ = 0.008
Jump Anomalies: 1% probability, scale = 0.08

Technical Details

Topological Boundaries Detection

The framework identifies natural structural limits through:
Fractal dimension analysis
Structural coherence metrics
Coupling strength variations
Entropy gradients

Anomaly Detection

Combined scoring from:
Q_Λ residuals (topological charge breaks)
ΛF anomalies (flow irregularities)
ΛFF anomalies (acceleration jumps)
ρT breaks (tension field discontinuities)

Harmonic Filtering

Automatically removes higher harmonics to identify fundamental structures:
Detects integer ratio relationships
Preserves only base frequencies
Prevents double-counting of same structure

Visualization

The framework generates comprehensive analysis plots including:
Observation trajectory
Topological winding (Q_Λ)
Anomaly scores
Structural boundaries
Detected recurrence patterns
Phase space analysis

Use Cases

1. Exoplanet Detection: Find hidden planets from stellar wobble
2. Binary System Analysis: Detect unseen companions
3. Asteroid Perturbations: Identify gravitational influences
4. Dark Matter Mapping: Trace invisible mass distributions

Contributing

We welcome contributions! Key areas for improvement:
Enhanced boundary detection algorithms
Multi-dimensional tensor analysis
Real-time processing capabilities
GUI interface development

Citation

If you use Lambda³ Stargazer in your research, please cite:
@software{lambda3stargazer, title = {Lambda³ Stargazer: Pure Topological Structure Detection}, author = {Iizumi, Masamichi}, year = {2025}, url = {https://github.com/miosync-masa/LambdaOrbitalFinder} }

Acknowledgments

Special thanks to Makise Kurisu for the challenging test dataset:
A new universe emerges… with secrets, noise, and missingness!

License

MIT License – See LICENSE file for details

Remember: Time is just a projection of structural changes. The universe speaks in the language of topology!