# Technical Document Extraction ## Overview The image contains four comparative analysis sections for κ values 2–5, each comprising: 1. Heatmap visualization 2. Line plot comparison 3. Component diagram 4. Color scale legend --- ## Heatmap Analysis (Top Row) ### General Structure - **Axes**: - X-axis: λ (0–5) - Y-axis: E (-10–10) - **Color Scale**: - Right-side gradient from purple (0.2) to yellow (1.0) - Represents "fractal dimension" - **Key Features**: - Red dashed lines demarcate regions where fractal dimension > 0.5 - Vertical yellow bands indicate transition zones ### κ-Specific Observations | κ Value | Color Intensity Pattern | Red Line Position | |---------|-------------------------|-------------------| | 2 | Broad yellow band | Near λ=0 | | 3 | Narrower yellow band | λ=0.5–1.5 | | 4 | Concentrated yellow | λ=1–2 | | 5 | Thin yellow strip | λ=1.5–2.5 | --- ## Line Plot Analysis (Middle Row) ### General Structure - **Axes**: - X-axis: λ (0–5) - Y-axis: E (-2–2) - **Legend Elements**: - Solid blue: KAN function - Dashed black: True λ function - **Equations**: - κ=2: - KAN: `1/(1+|E|)` - True λ: `1/|E|` - κ=3: - KAN: `1/(1+|E|^(1/2))` - True λ: `1/|E|^(1/2)` - κ=4: - KAN: `1/(1+|E|^(1/2)+√2)` - True λ: `1/|E|^(1/2)` - κ=5: - KAN: `1/(1+|E|+φ(|E|-1))` - True λ: `1/|E|^(1/2)` ### Trend Verification - All KAN curves asymptotically approach True λ as λ increases - κ=2 shows largest divergence at λ=0 - κ=5 demonstrates closest convergence near λ=2 --- ## Component Diagrams (Bottom Row) ### General Structure - **Elements**: - Red boxes: Input/output relationships - Black boxes: Distribution functions - Arrows: Functional dependencies - **Labels**: - λ (input) - E (output) - "Fractal dimension > 0.5" annotation ### κ-Specific Variations | κ Value | Black Box Contents | Red Box Connections | |---------|--------------------|---------------------| | 2 | Single peak | Direct λ→E | | 3 | Double peak | λ→E with intermediate | | 4 | Triple peak | Branching pathways | | 5 | Quadruple peak | Complex interdependencies | --- ## Color Scale Legend - **Position**: Right of each heatmap - **Gradient**: - Purple (0.2) → Yellow (1.0) - **Key**: "fractal dimension" --- ## Spatial Grounding 1. **Heatmaps**: Top row, left-to-right κ=2→5 2. **Line Plots**: Middle row, aligned with heatmaps 3. **Diagrams**: Bottom row, corresponding to upper plots 4. **Legends**: Right-aligned with respective plots --- ## Data Table Reconstruction | κ | KAN Equation | True λ Equation | Fractal Threshold | |---|--------------|-----------------|-------------------| | 2 | 1/(1+|E|) | 1/|E| | >0.5 | | 3 | 1/(1+|E|^½) | 1/|E|^½ | >0.5 | | 4 | 1/(1+|E|^½+√2)| 1/|E|^½ | >0.5 | | 5 | 1/(1+|E|+φ(|E|-1)) | 1/|E|^½ | >0.5 | --- ## Critical Observations 1. Increasing κ correlates with: - Narrower high-fractal-dimension regions - Closer convergence between KAN and True λ 2. Red dashed lines in heatmaps consistently mark: - λ=0 for κ=2 - λ=0.5–1.5 for κ=3 - λ=1–2 for κ=4 - λ=1.5–2.5 for κ=5 3. Component diagrams show increasing complexity with κ: - κ=2: Simple linear relationship - κ=5: Multi-path interdependencies --- ## Language Note All text appears in English. No non-English content detected.