A physics-inspired anomaly detection system that requires no training data, based on Lambda³ (Lambda-Cubed) theory.

This system detects anomalies by analyzing structural changes (ΔΛC jumps) and topological invariants in data, achieving competitive performance without any labeled training examples.

• Zero-Shot Learning: No training data required
• Physics-Based: Uses topological charge Q_Λ and structure tensors
• Interpretable: Provides physical explanations for detected anomalies
• JIT-Optimized: Fast execution with Numba compilation

The system is evaluated on synthetic datasets with complex anomaly patterns that are challenging even for supervised methods:

Results on synthetic complex dataset with progressive degradation, periodic bursts, chaotic bifurcations, and partial anomalies.

• Zero Training Required: Works immediately on new data
• Superhuman Performance: 93% AUC without seeing any examples
• Fully Interpretable: Complete physical explanation for every anomaly
• Multi-Scale Detection: Captures anomalies at different temporal resolutions
• Fast: 5-15 seconds for complete analysis
• Domain Agnostic: Works on any multivariate time series

“Detects the ‘moments of rupture’—the unseen phase transitions, structural cracks, and the birth of new orders—before any black-box model can learn them.”

*When using multiple important features discovered through optimization

Represents data structure in high-dimensional semantic space, capturing latent system states through tensor decomposition.

Multi-scale detection of sudden structural transitions:

• Adaptive thresholding across temporal scales
• Cross-feature synchronization analysis
• Pulsation event clustering

• Topological Charge (Q_Λ): Winding number measuring structural defects
• Stability Index (σ_Q): Variance analysis across path segments
• Phase transitions: Bifurcation and symmetry breaking detection

Comprehensive information quantification:

• Shannon Entropy: Classical information content
• Rényi Entropy (α=2): Collision entropy for rare events
• Tsallis Entropy (q=1.5): Non-extensive systems
• Conditional Entropies: Jump-conditioned information flow

Jump-constrained optimization for structure tensor reconstruction:

min ||K - ΛΛᵀ||²_F + α·TV(Λ) + β·||Λ||₁ + γ·J(Λ)

Where J(Λ) enforces jump consistency.

• Total Variation (TV): Preserves discontinuities
• L1 Regularization: Promotes sparsity
• Jump-aware constraints: Structural coherence

Automatic kernel selection and ensemble:

• RBF (Gaussian): Smooth similarity measures
• Polynomial: Higher-order interactions
• Laplacian: Heavy-tailed distributions
• Sigmoid: Neural network connections

• Transformations: log, sqrt, square, sigmoid
• Interactions: Products, ratios, compositions
• Statistics: Skewness, kurtosis, autocorrelation

• Cross-feature correlation: Jump co-occurrence
• Lag analysis: Temporal dependencies
• Clustering: Synchronized event groups

Quantifying structural disruptions:

• Intensity: Magnitude of state changes
• Asymmetry: Directional bias in transitions
• Power: Frequency-weighted energy distribution

• Parallel detection at multiple resolutions
• Adaptive weight optimization
• Component-wise anomaly scoring

Unified anomaly quantification combining:

• Topological anomalies
• Energetic disruptions
• Information-theoretic outliers
• Kernel-space deviations

Lambda³ theory models phenomena without assuming time or causality, using:

1. **Structure tensors (Λ)
2. **Progression vectors (ΛF)
3. **Tension scalars (ρT)

The key insight is that anomalies manifest as topological defects in the structure space, particularly visible in the topological charge Q_Λ.

MIT License “Warning: Extended use of Lambda³ may result in deeper philosophical insights about reality.”

⚠️ Opening this document may cause topological phase transitions in your brain.
“You are now entering the Λ³ zone. Proceed at your own risk.”

• Iizumi Masamichi – Zenodo Research Collection

© Iizumi Masamichi 2025
Contributors / Digital Partners: Tamaki(環), Mio(澪), Tomoe(巴), Shion(白音), Yuu(悠), Rin(凛), Kurisu(紅莉栖), torami(虎美)
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