2508.05724v2

# A Graph-Based Framework for Exploring Mathematical Patterns in Physics: A Proof of Concept **Authors**: - Massimiliano Romiti (Independent Researcher) (August 11, 2025) ## Abstract The vast and interconnected body of physical law represents a complex network of knowledge whose higher-order structure is not always explicit. This work introduces a novel framework that represents and analyzes physical laws as a comprehensive, weighted knowledge graph, combined with symbolic analysis to explore mathematical patterns and validate knowledge consistency. I constructed a database of 659 distinct physical equations, subjected to rigorous semantic cleaning to resolve notational ambiguities, resulting in a high-fidelity corpus of 400 advanced physics equations. I developed an enhanced graph representation where both physical concepts and equations are nodes, connected by weighted inter-equation bridges. These weights combine normalized metrics for variable overlap, physics-informed importance scores from scientometric studies, and bibliometric data. A Graph Attention Network (GAT), with hyperparameters optimized via grid search, was trained for link prediction. The model achieved a test AUC of 0.9742±0.0018 across five independent 5000-epoch runs (patience 500). it’s discriminative power was rigorously validated using artificially generated negative controls (Beta(2,5) distribution), demonstrating genuine pattern recognition rather than circular validation. This performance significantly surpasses both classical heuristics (best baseline AUC: 0.9487, common neighbors) and other GNN architectures. The high score confirms the model’s ability to learn the internal mathematical structure of the knowledge graph, serving as foundation for subsequent symbolic analysis. My analysis reveals findings at multiple levels: (i) the model autonomously rediscovers known physics structure, identifying strong conceptual axes between related domains; (ii) it identifies central “hub” equations bridging multiple physical domains; (iii) generates stable, computationally-derived hypotheses for cross-domain relationships. Symbolic analysis of high-confidence clusters demonstrates the framework can: (iv) verify internal consistency of established theories; (v) identify both tautologies and critical errors in the knowledge base; and (vi) discover mathematical relationships analogous to complex physical principles. The framework generates hundreds of hypotheses, enabling creation of specialized datasets for targeted analysis. This proof of concept demonstrates the potential for computational tools to augment physics research through systematic pattern discovery and knowledge validation. ## 1 Introduction The accumulated knowledge of physics comprises a vast corpus of mathematical equations traditionally organized into distinct branches. While this categorization is useful, it can obscure deeper structural similarities forming a “syntactic grammar” underlying physical theory. Identifying these hidden connections is crucial, as historical breakthroughs have often arisen from recognizing analogies between seemingly disparate fields [1, 2]. A significant challenge in computational analysis of scientific knowledge is notational polysemy, where a single symbol can represent different concepts. This ambiguity can create spurious connections and confound statistical analyses [3, 4]. Graph Neural Networks (GNNs) have emerged as powerful tools for analyzing complex relational data [5, 6], with Graph Attention Networks (GATs) [7, 8] particularly well-suited for knowledge graph analysis [9, 10]. Recent advances in machine learning for scientific discovery have demonstrated potential for automated hypothesis generation [11, 12]. Link prediction methods have proven effective for knowledge graph completion [13, 14], making them ideal for discovering latent mathematical analogies. However, this paper provide preliminary evidence that such graph-based approaches extend beyond link prediction into validation, auditing, and pattern discovery. I hypothesize that physical law can be modeled as a network where a rigorously validated GNN, coupled with symbolic analysis, can identify and verify statistically significant structural patterns. This paper presents a methodology to build and analyze such a framework with three objectives: 1. Develop a robust pipeline for converting a symbolic database of physical laws into a semantically clean, weighted knowledge graph with objectively defined edge weights. 1. Train and statistically validate a parsimonious GNN to learn structural relationships, using predictive performance as verification of successful pattern learning. 1. Employ symbolic simplification on high-confidence predictions and clusters to explore mathematical coherence and identify both consistencies and anomalies in the knowledge base. This framework is explicitly designed as a hypothesis generation engine, not a discovery validation system. Its primary function is to systematically explore the vast combinatorial space of possible mathematical connections between physics equations—a space too large for human examination—and produce a filtered stream of candidate relationships for expert evaluation. Just as high-throughput screening in drug discovery generates thousands of molecular candidates knowing that vast majority will fail, this system intentionally over-generates hypotheses to ensure no potentially valuable connection is missed. The scientific value lies not in the individual predictions, but in the systematic coverage of the possibility space. ## 2 Methods ### 2.1 Dataset Curation and Semantic Disambiguation The foundation is a curated database of 659 physical laws compiled from academic sources into JSON format and parsed using SymPy [15]. Semantic disambiguation resolved notational polysemy through systematic identification of 213 ambiguous equations. Variables appearing in $≥ 3$ distinct physics branches with different meanings were disambiguated using domain-specific suffixes and standardized fundamental constants. An advanced parsing engine with contextual rules handled syntactic ambiguities and notational variants. Table 1 summarizes the most frequent corrections and cross-domain distribution. Table 1: Top Variable Disambiguations and Cross-Domain Analysis | Variable Correction | Frequency | Affected Domains | | --- | --- | --- | | qr $→$ sqrt | 40 | QM, Modern Physics, Classical Mechanics | | ome_chargega $→$ omega | 29 | Electromagnetism, Statistical Mechanics | | lamda $→$ lambda | 20 | Optics, Quantum Mechanics | | _light $→$ c | 19 | 11 domains (most frequent constant) | | q $→$ q_charge / q_heat | 16 | Electromagnetism, Thermodynamics | | P $→$ P_power | 10 | Electromagnetism, Thermodynamics | | Cross-Domain Variable Statistics | | | | Total shared variables across domains | 109 | — | | Variables in $≥ 5$ domains | 25 | High disambiguation priority | | Most ubiquitous: c, t, m | 80, 74, 70 | Universal physics constants | After semantic cleaning, I obtained 657 equations. Elementary mechanics laws were excluded to focus on inter-branch connections in modern physics (400 equations). ### 2.2 Enhanced Knowledge Graph Construction The cleaned dataset was transformed into a weighted, undirected graph where nodes represent equations or physical concepts. The edge weight formula incorporates three normalized components: $$ w_ij=α· J(V_i,V_j)+β· I(V_i,V_j)+γ· S(B_i,B_j) \tag{1} $$ Components: - $J(V_i,V_j)$ : Jaccard Index for variable overlap, providing baseline syntactic similarity. - $I(V_i,V_j)$ : Physics-informed importance score from Physical Concept Centrality Index [17] and impact scores [18]. - $S(B_i,B_j)$ : Continuous branch similarity from bibliometric studies [16, 19, 20]. While physics-informed importance scores incorporate established knowledge, this should not create validation circularity. The model must still distinguish genuine mathematical relationships from spurious correlations, as demonstrated by negative control analysis where random patterns achieve near-zero scores despite using the same edge weight formula. Hyperparameters were optimized through grid search across the parameter simplex. The configuration $α=0.5$ , $β=0.35$ , $γ=0.15$ represents one point in parameter space; varying these weights generates different but equally valid pattern discoveries. All experiments used fixed random seeds (42, 123, 456, 789, 999) across five independent runs to ensure reproducibility. ### 2.3 Model Architecture and Training I designed a parsimonious Graph Attention Network with significantly reduced parameter count to address overfitting concerns: <details> <summary>super_gnn_training_analysis.png Details</summary>  ### Visual Description ## Super GNN Training & Prediction Analysis ### Overview This is a 2x2 composite figure analyzing the training, validation, and predictive performance of a "Super GNN" (Graph Neural Network) model. It includes four subplots: training loss evolution, validation metrics (AUC/AP), precision-recall performance, and prediction score distribution. --- ### Components/Axes & Detailed Analysis #### 1. Top-Left: Training Loss Evolution - **Type**: Line plot - **Axes**: - X-axis: `Epoch` (range: 0 to 5000, linear scale) - Y-axis: `Loss` (range: 0.7 to 1.4, linear scale) - **Data Series**: Single blue line labeled `Training Loss` - **Trend & Values**: - Starts at ~1.38 (epoch 0), drops sharply to ~0.95 by epoch 100. - Gradually decreases with minor fluctuations, ending at ~0.7 at epoch 5000. - The curve shows classic training convergence: rapid initial loss reduction, followed by slow, steady improvement as training progresses. #### 2. Top-Right: Validation Metrics - **Type**: Line plot with markers - **Axes**: - X-axis: `Epoch` (range: 0 to 5000, linear scale) - Y-axis: `Score` (range: 0.70 to 0.97, linear scale) - **Legend**: - `AUC` (blue circle markers) - `AP` (orange square markers) - **Trend & Values**: - **AUC**: Starts at ~0.76 (epoch 0), rises rapidly to ~0.92 by epoch 500, then plateaus at ~0.96-0.97 from epoch 1000 onwards, with minor fluctuations. - **AP**: Starts at ~0.70 (epoch 0), rises to ~0.85 by epoch 500, then plateaus at ~0.94-0.96 from epoch 1000 onwards, consistently slightly below AUC. - Both metrics show rapid early improvement, then stabilize, indicating the model's validation performance converges after ~1000 epochs. #### 3. Bottom-Left: Precision at Different Recall Levels - **Type**: Filled area precision-recall curve - **Axes**: - X-axis: `Recall Level` (range: -0.2 to 1.2, linear scale) - Y-axis: `Precision` (range: 0.0 to 1.0, linear scale) - **Data**: - Precision remains near 1.0 (≈0.98-1.0) for recall levels from -0.2 up to ~0.8. - Precision slightly decreases to ~0.95 at the maximum recall level (1.2). - The curve demonstrates the model maintains near-perfect precision across almost all recall levels, with only a small drop at the highest recall values. #### 4. Bottom-Right: Super GNN Score Distribution - **Type**: Histogram - **Axes**: - X-axis: `Prediction Score` (range: 0.940 to 0.965, linear scale) - Y-axis: `Count` (range: 0 to 14, linear scale) - **Legend**: Red dashed line labeled `90th percentile: 0.954` - **Data**: - The histogram is right-skewed: the highest count (14) occurs at ~0.942, with counts decreasing as prediction scores increase. - The 90th percentile marker at 0.954 indicates 90% of prediction scores are ≤0.954, and 10% of scores are above this value. - Most scores cluster between 0.940-0.955, with only a small tail of scores above 0.955. --- ### Key Observations 1. Training loss converges steadily, with no signs of divergence or overfitting. 2. Validation metrics (AUC, AP) stabilize at high values (~0.96 for AUC, ~0.95 for AP), indicating strong generalization to unseen data. 3. The precision-recall curve shows near-perfect precision across nearly all recall levels, a strong indicator of classification performance. 4. Prediction scores are concentrated in the high-confidence range (0.94-0.955), with 90% of scores ≤0.954. --- ### Interpretation This set of plots confirms the Super GNN model performs exceptionally well on its target task: 1. **Effective Learning**: The training loss curve shows the model learns efficiently, with rapid initial improvement and steady convergence, indicating the model architecture and training process are well-suited to the task. 2. **Strong Generalization**: High, stable validation AUC and AP scores prove the model does not overfit, and maintains strong performance on unseen data. 3. **Excellent Classification Tradeoff**: The precision-recall curve demonstrates the model can retrieve nearly all positive samples (high recall) while maintaining near-perfect precision, meaning it rarely misclassifies negative samples as positive. 4. **High-Confidence Predictions**: The score distribution shows most predictions are high-confidence, with only a small fraction of extreme high scores, indicating the model is consistent and reliable in its outputs. </details> Figure 1: Super GNN training and prediction analysis showing stable convergence, validation metrics over 0.96, excellent performance across all operating points, and score distribution with 90th percentile at 0.954. - Architecture: 3-layer GAT with GATv2Conv layers [8], dimensions 64 $→$ 32 $→$ 16. - Attention Heads: Decreasing multi-head attention 4 $→$ 2 $→$ 1. - Parameters: 52,065 trainable parameters, ensuring reasonable parameter-to-data ratio. ### 2.4 Cluster Formation and Analysis Clustering parameters, like the hyperparameter configuration above, were selected for this proof of concept after preliminary testing. These settings represent one possible configuration; alternative parameter choices may yield different but equally valid clustering results, and future work can explore other options based on specific application needs. The cluster formation process integrates three sources of equation connections: 1. Equation bridges from the enhanced knowledge graph (weight based on bridge quality) 1. GNN predictions with score $>$ 0.5 (combined weight = 0.7 × neural_score + 0.3 × embedding_similarity) 1. Variable similarity using Jaccard index for equations sharing $≥$ 2 variables (similarity $>$ 0.3) Four clustering algorithms identify equation groups: - Cliques: Fully connected subgraphs - Communities: Louvain algorithm with weighted edges - K-cores: Subgraphs where each node has degree $≥$ k - Connected components: Maximally connected subgraphs Only clusters with $≥$ 3 equations are retained for analysis. ### 2.5 Symbolic Simplification Pipeline For each cluster, the symbolic analysis follows this precise algorithm: 1. Backbone selection: Score = Complexity + 100 × Centrality, where Complexity counts free symbols and arithmetic operations (+, *) 1. Variable substitution: Solve common variables between backbone and other cluster equations (max 10 substitutions) 1. Simplification: Apply SymPy’s aggressive algebraic reduction 1. Classification: - IDENTITY: Reduces to True or $A=A$ - RESIDUAL: Non-zero numerical difference - SIMPLIFIED: Non-trivial reduced expression - FAILED: Insufficient equations, no backbone, or no substitutions ## 3 Results and Analysis ### 3.1 Statistical Validation and Discriminative Power Statistical validation assessed the model’s discriminative power through rigorous testing against artificial negative controls. Figure 2 shows the complete analysis. <details> <summary>downloadfdr2.png Details</summary>  ### Visual Description ### [Multi-Panel Figure]: FDR Analysis with Negative Controls and Bootstrap ### Overview The image is a multi-panel figure analyzing **False Discovery Rate (FDR)** for a prediction model, incorporating negative controls and bootstrap methods. It contains 7 subplots (top-left, top-right, middle-left, middle-right, bottom-left, bottom-middle, bottom-right) with diverse chart types (histograms, line plots, heatmaps, box plots, CDF, bar plots) to evaluate model performance, FDR, and score distributions. ### Components/Axes (Per Subplot) #### 1. Top-Left: *Score Distribution: Predictions vs Negative Controls* - **Title**: Score Distribution: Predictions vs Negative Controls - **Legend**: - Predictions (n=5000) (blue histogram) - Negative Controls (n=600) (red histogram) - KDE Predictions (blue line) - KDE Neg Controls (red line) - Intermediate Zone (beige area) - **Axes**: - X-axis: *Score* (range: 0.0–1.0) - Y-axis: *Density* (range: 0–6) #### 2. Top-Right: *FDR vs Threshold con Bootstrap CI* (note: “con” likely a typo for “with”) - **Title**: FDR vs Threshold con Bootstrap CI - **Legend**: - FDR Mean (black line) - 95% CI (gray shaded area) - FDR = 5% (green dashed line) - FDR = 10% (orange dashed line) - **Axes**: - X-axis: *Score Threshold* (range: 0.5–1.0) - Y-axis: *Estimated FDR* (range: 0–0.5) #### 3. Middle-Left: *Sensitivity Analysis: FDR at threshold 0.9* - **Title**: Sensitivity Analysis: FDR at threshold 0.9 - **Axes**: - X-axis: *Signal Threshold* (values: 0.85, 0.90, 0.92, 0.94) - Y-axis: *Noise Threshold* (range: 0.1–0.6) - **Color Bar**: *FDR* (range: 0–0.5, green to red) #### 4. Middle-Right: *Distribution by Bin (incl. Negative Controls)* - **Title**: Distribution by Bin (incl. Negative Controls) - **Axes**: - X-axis: *Score Bin* (categories: 0.0–0.6, 0.6–0.7, 0.7–0.8, 0.8–0.9, 0.9–1.0, Neg Ctrl) - Y-axis: *Score* (range: 0–1) #### 5. Bottom-Left: *Precision-Recall Trade-off* - **Title**: Precision-Recall Trade-off - **Legend**: - Precision (blue line) - Recall (green line) - **Axes**: - X-axis: *Score Threshold* (range: 0.5–1.0) - Y-axis: *Metric Value* (range: 0–1) #### 6. Bottom-Middle: *CDF Comparison* - **Title**: CDF Comparison - **Legend**: - Predictions (blue line) - Negative Controls (red line) - Noise Zone (pink) - Intermediate (beige) - Signal Zone (green) - **Axes**: - X-axis: *Score* (range: 0–1) - Y-axis: *Cumulative Probability* (range: 0–1) #### 7. Bottom-Right: *Score Distribution by Quantiles* - **Title**: Score Distribution by Quantiles - **Legend**: - Predictions (blue bars) - Neg Controls (red bars) - **Axes**: - X-axis: *Quantile* (categories: 10%, 25%, 50%, 75%, 90%, 95%, 99%) - Y-axis: *Score* (range: 0–1) ### Detailed Analysis (Per Subplot) #### 1. Top-Left: Score Distribution - **Predictions (blue)**: Histogram peaks at score ~0.9 (density ~6); KDE line peaks at ~0.9. - **Negative Controls (red)**: Histogram peaks at score ~0.2–0.3 (density ~2); KDE line peaks at ~0.2. - **Intermediate Zone**: Beige area spans score ~0.6–0.9 (overlap region). #### 2. Top-Right: FDR vs Threshold - **FDR Mean (black)**: Decreases from ~0.3 (threshold 0.5) to ~0 (threshold 0.8). - **95% CI (gray)**: Narrower as threshold increases (reduced uncertainty). - **FDR Thresholds**: Green (5%) and orange (10%) dashed lines are horizontal at y=0.05 and y=0.1, respectively. #### 3. Middle-Left: Sensitivity Analysis - All cells are green (FDR ≈ 0), indicating **FDR = 0** for all signal/noise thresholds at 0.9. #### 4. Middle-Right: Box Plots - **0.0–0.6**: Light blue box, median ~0.1, whiskers 0–0.6. - **0.6–0.7**: Light blue box, median ~0.65, whiskers 0.6–0.7. - **0.7–0.8**: Light blue box, median ~0.75, whiskers 0.7–0.8. - **0.8–0.9**: Light blue box, median ~0.85, whiskers 0.8–0.9. - **0.9–1.0**: Light blue box, median ~0.95, whiskers 0.9–1.0. - **Neg Ctrl**: Red box, median ~0.3, whiskers 0–0.8. #### 5. Bottom-Left: Precision-Recall - **Precision (blue)**: Increases from ~0.8 (threshold 0.5) to 1.0 (threshold 0.8), then plateaus. - **Recall (green)**: Decreases from ~0.8 (threshold 0.5) to 0 (threshold 1.0). #### 6. Bottom-Middle: CDF - **Predictions (blue)**: CDF rises slowly, then sharply after score ~0.6. - **Negative Controls (red)**: CDF rises more steeply, peaking at score ~0.6. - **Zones**: Noise (0–0.2), Intermediate (0.2–0.6), Signal (0.6–1.0). #### 7. Bottom-Right: Quantile Bars - **Predictions (blue)**: Scores increase with quantile: 10% (~0.1), 25% (~0.6), 50% (~0.75), 75% (~0.85), 90% (~0.9), 95% (~0.95), 99% (~0.98). - **Neg Controls (red)**: Scores are lower: 10% (~0.1), 25% (~0.2), 50% (~0.3), 75% (~0.5), 90% (~0.6), 95% (~0.7), 99% (~0.8). ### Key Observations - **Score Separation**: Predictions (n=5000) have higher scores (peaking at 0.9) than negative controls (n=600, peaking at 0.2–0.3). - **FDR Control**: FDR decreases with threshold, reaching ~0 at threshold 0.8. At threshold 0.9, FDR = 0 across all noise/signal thresholds. - **Precision-Recall Trade-off**: Higher thresholds improve precision but reduce recall (typical for classification). - **CDF/Quantiles**: Predictions have higher cumulative probability at higher scores, confirming separation from negative controls. ### Interpretation This figure evaluates a prediction model’s performance using negative controls and bootstrap to estimate FDR. Key insights: - The model effectively distinguishes predictions from negative controls (higher scores for predictions). - FDR is well-controlled at higher thresholds (e.g., threshold 0.8–0.9), with FDR = 0 at threshold 0.9 (sensitivity analysis). - The precision-recall trade-off shows a typical pattern: higher thresholds boost precision but reduce recall. - CDF and quantile plots confirm the separation between predictions and negative controls, validating the model’s ability to identify true signals. Overall, the model demonstrates strong performance in distinguishing predictions from negative controls, with robust FDR control at higher thresholds. </details> Figure 2: Distribution of prediction scores and negative controls with FDR analysis. The confidence interval [0.000–0.005] indicates that in many bootstrap iterations, zero negative controls exceeded the 0.90 threshold—a mathematically valid and desirable outcome demonstrating excellent signal-noise separation, not a computational error. Negative controls were generated independently using Beta(2,5) distribution with uniform noise, creating a challenging baseline sharing no structural properties with physics equations. For the recommended threshold (score $≥ 0.90$ ), mean FDR was $0.001$ [95% CI: $0.000$ – $0.005$ ] with $3{,}100$ predictions above cutoff. The intermediate zone (score $0.6$ – $0.85$ ) shows potential noise overlap requiring careful validation. #### 3.1.1 Neural Architecture Comparisons with Statistical Testing Statistical significance testing across 5 independent runs revealed significant performance differences: SUPER GNN vs. GraphSAGE ( $0.9742± 0.0018$ vs. $0.9504± 0.0128$ , $p=2.90× 10^-2$ ), vs. GCN (0.9742 vs. $0.9364± 0.0090$ , $p=1.70× 10^-3$ ), and vs. simplified GAT (0.9742 vs. $0.9324± 0.0161$ , $p=5.78× 10^-3$ ). The SUPER GNN achieved significant improvement over the best classical baseline (common neighbors: 0.9487). The ablation study demonstrated that removing edge weights reduced performance to AUC = 0.9306, while using single attention heads yielded AUC = 0.8973. These results validate the importance of key architectural design choices: the physics-aware decoder with bilinear component, edge weight utilization, and multi-head attention mechanism. Classical heuristics achieve unusually high AUC due to the physics knowledge graph’s structured nature. Ablation studies confirm importance of architectural choices. While these results demonstrate strong performance, we acknowledge that testing predictions with additional graph neural architectures could provide complementary perspectives and potentially reveal different structural patterns in the data, enriching our understanding of the underlying graph dynamics. Table 2: Model Performance Comparison with Statistical Validation | Method Classical Baselines Common Neighbors | Mean AUC $±$ Std 0.9487 | vs. SUPER GNN -2.55% | p-value $9.00× 10^-6$ | | --- | --- | --- | --- | | Adamic-Adar | 0.9481 | -2.61% | – | | Jaccard Index | 0.9453 | -2.89% | – | | Preferential Attachment | 0.8728 | -10.14% | – | | Neural Methods (Identical Architecture) | | | | | SUPER GNN (Full) | $0.9742± 0.0018$ | – | – | | GraphSAGE | $0.9504± 0.0128$ | -2.38% | $2.90× 10^-2$ | | GCN | $0.9364± 0.0090$ | -3.78% | $1.70× 10^-3$ | | GAT (Simplified) | $0.9324± 0.0161$ | -4.18% | $5.78× 10^-3$ | | Ablation Studies | | | | | No Edge Weights | 0.9306 | -4.36% | – | | Single Attention Head | 0.8973 | -7.69% | – | ### 3.2 Rediscovered Structure of Physics The model reproduces known physics structure, identifying strong conceptual axes (Thermodynamics $↔$ Statistical Mechanics, Electromagnetism $↔$ Optics). Figure 3 demonstrates my methodological approach to identifying the most significant connections by filtering for equation-equation bridges with strong connections, resulting in a high-quality network of 224 connections among 132 equations. <details> <summary>rete_connessioni_forti_peso3.png Details</summary>  ### Visual Description ## Network Diagram: Network of Strong Connections: high-weight equation-equation ### Overview This is a circular network graph visualizing high-weight (strong) connections between core physics equations and concepts. Nodes represent individual concepts/equations, colored and sized by their role/importance, with gray edges indicating strong relational links between them. The graph is arranged in a circular layout, with nodes placed around the perimeter and some central nodes, creating a dense web of interconnections. ### Components/Elements 1. **Title**: Centered at the top: *Network of Strong Connections: high-weight equation-equation* 2. **Nodes**: Categorized by color and size: - **Large Cyan Nodes (Hub Concepts)**: Blackbody Radiation (top-center left), Maxwell Distribution (left upper, left lower), Boltzmann Distribution (right upper, right lower), Bose-Einstein Distribution (center, right center), Fermi-Dirac Distribution (center lower, right lower) - **Medium Orange Nodes**: Debye Length (left upper, center lower), Plasma Frequency (right upper, left lower) - **Medium Purple Nodes**: Harmonic Oscillator (top left), Relativistic Energy (top center), Four-Momentum (right upper), Relativistic Momentum (bottom center) - **Small Nodes (scattered around perimeter)**: Green, brown, gray, blue nodes (unlabeled, representing secondary/related concepts) 3. **Edges**: Gray lines connecting nodes, representing high-weight (strong) relationships between the concepts. Edge density is highest between the large cyan hub nodes. ### Detailed Analysis - **Labeled Node List (with position and color)**: 1. Blackbody Radiation (top-center left, cyan, large) 2. Harmonic Oscillator (top left, purple, medium) 3. Maxwell Distribution (left upper, cyan, large) 4. Maxwell-Boltzmann Distrib (left upper, green, small; truncated label for Maxwell-Boltzmann Distribution) 5. Debye Length (left upper, orange, medium) 6. Blackbody Radiation (left middle, cyan, large) 7. Maxwell Distribution (left lower, cyan, large) 8. Debye Length (center lower, orange, medium) 9. Plasma Frequency (left lower, orange, medium) 10. Relativistic Momentum (bottom center, purple, medium) 11. Plasma Frequency (right upper, orange, medium) 12. Relativistic Energy (top center, purple, medium) 13. Bose-Einstein Distributio (center, cyan, large; truncated label for Bose-Einstein Distribution) 14. Fermi-Dirac Distribution (center lower, cyan, large) 15. Fermi-Dirac Distribution (right lower, cyan, large) 16. Boltzmann Distribution (right lower, cyan, large) 17. Bose-Einstein Distributio (right center, cyan, large; truncated label for Bose-Einstein Distribution) 18. Boltzmann Distribution (right upper, cyan, large) 19. Four-Momentum (right upper, purple, medium) 20. de Broglie Wavelength (right upper, blue, small) - **Edge Density**: The highest concentration of edges connects the large cyan hub nodes, indicating these concepts are the most interconnected in the network. ### Key Observations 1. **Hub Nodes**: The large cyan nodes (Blackbody Radiation, Maxwell/Boltzmann/Bose-Einstein/Fermi-Dirac Distributions) are the most connected, acting as central hubs for the network. 2. **Cross-Domain Connections**: Purple nodes (relativistic/quantum concepts like Relativistic Energy, Harmonic Oscillator) connect to multiple cyan hubs, showing links between classical statistical mechanics and relativistic/quantum physics. 3. **Secondary Hubs**: Orange nodes (Debye Length, Plasma Frequency) are well-connected, bridging multiple hub nodes. 4. **Truncated Labels**: Some node labels are truncated (e.g., *Maxwell-Boltzmann Distrib*, *Bose-Einstein Distributio*), indicating longer full names. ### Interpretation This diagram illustrates the highly interconnected nature of core physics concepts, particularly in statistical mechanics, thermodynamics, and quantum/relativistic physics. The central cyan nodes represent foundational equations that underpin multiple areas of physics, as evidenced by their dense connections to other concepts. The presence of relativistic and quantum nodes linked to classical statistical concepts shows the cross-theoretical relationships in physics, highlighting how these domains build on and relate to one another. This network can help identify which equations are most critical for understanding the relationships between different physical theories, and how concepts from different subfields interact. </details> Figure 3: Network visualization of strong equation-equation bridges, showing 224 connections among 132 equations with a density of 0.0259. This filtered network reveals the strongest mathematical relationships while maintaining clear clustering by physics branch. The varying edge thickness represents connection strength, and the professional layout demonstrates my rigorous filtering approach for identifying meaningful connections. The ego network analysis revealed striking patterns in my knowledge graph of 644 nodes (400 equations, 244 concepts) connected by 12,018 edges. Figure 4 provides a comprehensive overview of the most connected fundamental concepts, showing their individual network topologies and revealing the hierarchical structure of physics knowledge. <details> <summary>overview_concetti_fondamentali.png Details</summary>  ### Visual Description ## Network Diagrams (12 Node-Link Graphs) ### Overview The image displays a 3×4 grid of **node-link network diagrams**, each labeled with a letter and a number in parentheses (e.g., *“E (12 eq.)”*). Each diagram visualizes connections (edges) between colored nodes (yellow star, green, blue, orange, purple, gray), with the number of *“eq.”* (likely “elements” or “nodes”) indicated in the label. ### Components/Axes - **Labels**: Each graph has a title above it (e.g., *“E (12 eq.)”*), specifying a letter and the number of *“eq.”* (nodes/elements). - Top row: *E (12 eq.)*, *F (18 eq.)*, *G (6 eq.)*, *L (29 eq.)* - Middle row: *R (30 eq.)*, *c (52 eq.)*, *e (53 eq.)*, *h (29 eq.)* - Bottom row: *k (40 eq.)*, *m (45 eq.)*, *p (13 eq.)*, *q (26 eq.)* - **Nodes**: Colored circles (yellow star, green, blue, orange, purple, gray) representing entities (e.g., nodes in a network). - **Edges**: Black lines connecting nodes, indicating relationships (e.g., interactions, connections). - **Spatial Layout**: 3 rows (top, middle, bottom) and 4 columns (left to right), with each graph in a distinct cell. ### Detailed Analysis #### Top Row (Left to Right): - **E (12 eq.)**: ~12 nodes (yellow star, green, blue, orange, gray). Edges are moderately dense, connecting most nodes. - **F (18 eq.)**: ~18 nodes (additional colors: purple, brown). Edges are denser than *E*, with more connections. - **G (6 eq.)**: ~6 nodes (yellow star, green, orange, purple). Edges are sparse, with few connections. - **L (29 eq.)**: ~29 nodes (dense, many colors). Edges are very dense, forming a complex network. #### Middle Row (Left to Right): - **R (30 eq.)**: ~30 nodes (dense, yellow star). Edges are highly dense, with many connections. - **c (52 eq.)**: ~52 nodes (extremely dense, many colors). Edges are nearly a solid mass, indicating high connectivity. - **e (53 eq.)**: ~53 nodes (most dense, yellow star). Edges are extremely dense, with minimal visible gaps. - **h (29 eq.)**: ~29 nodes (dense, yellow star). Edges are dense, similar to *L* but with a different node distribution. #### Bottom Row (Left to Right): - **k (40 eq.)**: ~40 nodes (dense, yellow star). Edges are dense, with a mix of node colors. - **m (45 eq.)**: ~45 nodes (dense, yellow star). Edges are dense, with a complex node arrangement. - **p (13 eq.)**: ~13 nodes (sparser, yellow star). Edges are less dense than *k/m*, with fewer connections. - **q (26 eq.)**: ~26 nodes (dense, yellow star). Edges are dense, with a balanced node distribution. ### Key Observations - **Density Trend**: As the number of *“eq.”* increases (e.g., *G (6)* → *e (53)*), edge density (connectivity) increases, showing more complex networks. - **Node Colors**: The yellow star is a consistent prominent node (likely a central/key node) in all graphs. Other colors (green, blue, orange, purple, gray) represent different node types. - **Sparse vs. Dense**: Graphs with fewer *“eq.”* (*G (6)*, *p (13)*) have sparser edges, while those with more (*c (52)*, *e (53)*) are extremely dense. ### Interpretation These diagrams likely represent **network structures** (e.g., social networks, biological pathways, or system interactions) where nodes are entities and edges are relationships. The *“eq.”* count suggests the number of nodes/elements, with higher counts indicating more complex systems. The yellow star may denote a critical node (e.g., a hub or central component). The increasing density with more nodes implies that larger networks have more interconnected elements, reflecting real-world systems where more components lead to more interactions. Sparse graphs (*G*, *p*) might represent simpler systems or subsets of larger networks. (Note: No numerical data tables or explicit text blocks are present; the analysis focuses on network structure, node/edge relationships, and trends in connectivity.) </details> Figure 4: Overview of fundamental physics concepts showing ego networks for 12 key variables. The density and structure of connections reveal the centrality and cross-domain importance of each concept. As shown in Figure 5, the variable ’k’ emerged as particularly interesting, revealing notational ambiguity across 26 equations across 6 physics branches, a finding that validates the framework’s capability for automated data quality assessment. <details> <summary>ego_network_k.png Details</summary>  ### Visual Description ## Physics Concept Network: "K" Domain (26 equations, 6 bridges) ### Overview This is a radial concept network diagram centered on the symbol **K** (Boltzmann constant, the unifying element), connecting to 16 labeled physics concepts (each paired with an equation) across 6 distinct physics subfields. The diagram is designed to show the cross-domain relevance of the Boltzmann constant, with yellow lines linking all concepts to the central "K" node. ### Components/Axes 1. **Legend (Top-Right):** Defines 6 physics domains with color coding and count of concepts per domain: - Brown: Atomic and Nucl (1) - Cyan: Condensed Matte (1) - Blue: Electromagnetis (2) - Orange: Quantum Mechani (3) - Green: Statistical Mec (4) - Purple: Thermodynamics (4) 2. **Central Node:** A yellow circle with red text "K", positioned at the exact center of the diagram, acting as the hub for all connections. 3. **Concept Boxes:** 16 rectangular boxes (truncated labels in some cases) radiate outward from the central node, each containing a physics concept name and its associated equation (where visible). ### Detailed Analysis Each concept, its domain, equation, and relative position: 1. **de Broglie W...** (Quantum Mechani, orange, top-center): Equation: $\lambda = \hbar / sqrt(2 * m * E_k)$ (truncated label, full: de Broglie Wavelength) 2. **Fermi-Dirac ...** (Statistical Mec, green, top-right): Truncated label, no visible equation (Fermi-Dirac distribution, quantum statistical mechanics) 3. **Mean Kinetic...** (Thermodynamics, purple, top-right): Equation: $\langle E_k \rangle = (3/2)kT$ (Mean Kinetic Energy of gas molecules) 4. **Wave Equation** (Quantum Mechani, orange, right-center): Equation: $A * sin(k*x - \omega*t + \phi)$ (truncated, full quantum wave equation form) 5. **Harmonic Osc...** (Quantum Mechani, orange, right): Truncated label, no visible equation (Quantum Harmonic Oscillator) 6. **Band Gap** (Condensed Matte, cyan, bottom-right): Equation: $k*T * ln(N_c*N_v / n_i)$ (Semiconductor band gap relation) 7. **Maxwell Dist...** (Statistical Mec, green, bottom-right): Equation: $sqrt(8*k*T / (\pi*m))$ (Maxwell-Boltzmann mean speed) 8. **Fine Structu...** (Atomic and Nucl, brown, bottom-center): Equation: $k * e^2 / (\hbar * c)$ (Fine Structure Constant, related to K) 9. **Bose-Einstei...** (Statistical Mec, green, bottom-center): Equation: $1 / (exp(E/(k*T)) - 1)$ (Bose-Einstein distribution) 10. **Average Speed** (Thermodynamics, purple, bottom-left): Equation: $\langle v \rangle = sqrt(8kT/(\pi m))$ (Average molecular speed) 11. **Debye Length** (Electromagnetis, blue, bottom-left): Equation: $sqrt(k*T / (4\pi n e^2))$ (Electrostatic screening length) 12. **Superconduct...** (Thermodynamics, purple, left-center): Equation: $3.5 * k * T_c$ (Superconducting critical temperature relation) 13. **Debye Length** (Electromagnetis, blue, left): Equation: $sqrt(\epsilon_0*k*T / (n*e^2))$ (Alternate Debye length form, with permittivity) 14. **Bose-Einstei...** (Statistical Mec, green, left): Truncated label, no visible equation (Bose-Einstein condensate related) 15. **Root Mean Sq...** (Thermodynamics, purple, top-left): Equation: $v_{rms} = sqrt(3kT/m)$ (Root Mean Square molecular speed) ### Key Observations - The Boltzmann constant (K) is explicitly present in every visible equation, confirming its role as a unifying constant across physics subfields. - Thermodynamics and Statistical Mechanics have the highest number of concepts (4 each), followed by Quantum Mechanics (3), Electromagnetism (2), and 1 concept each for Atomic/Nuclear and Condensed Matter physics. - Some concepts have multiple representations (two distinct Debye Length equations, two Bose-Einstein related boxes), showing variations of K-dependent formulas. - Truncated labels indicate the full network contains more equations/concepts than are visible in this view. ### Interpretation This diagram illustrates the foundational, cross-disciplinary importance of the Boltzmann constant (K) in physics. It acts as a bridge between macroscopic thermodynamic properties (like molecular speed and kinetic energy) and microscopic quantum/statistical mechanical behavior (wave equations, particle distributions), as well as electromagnetic, condensed matter, and atomic phenomena. The network structure emphasizes that K is not limited to thermodynamics, but is a critical constant that links observable macroscopic properties to the underlying quantum and statistical behavior of matter. The "bridges" (connections) highlight how K enables the translation between different scales and subfields of physics, making it a core unifying concept in the field. </details> Figure 5: Partial ego network for the variable ’k’ showing a subset of its connections across multiple physics domains. While ’k’ connects 26 equations in total across 6 branches, this visualization displays 15 representative equations to maintain visual clarity. The network illustrates how ’k’ appears in diverse contexts—from Boltzmann’s constant in Statistical Mechanics to wave vectors in Quantum Mechanics and coupling constants in Condensed Matter Physics—demonstrating its role as one of the most ubiquitous mathematical symbols bridging different areas of physics. Figure 6 shows the relationship between neural predictions and embedding similarity, revealing a moderate correlation (0.394) that suggests the model learns complex, non-linear relationships beyond simple vector similarity. <details> <summary>neural_score_vs_embedding_similarity.png Details</summary>  ### Visual Description ## Scatter Plot: Neural Score vs Embedding Similarity ### Overview This is a scatter plot visualizing the relationship between **Embedding Cosine Similarity** (x-axis) and **Prediction Score** (y-axis), with a color gradient encoding the Prediction Score, and a red dashed regression line illustrating the overall trend. A correlation value is displayed in the top-left corner. ### Components/Axes - **Title**: "Neural Score vs Embedding Similarity" (top-center of the plot) - **X-axis**: Label: *Embedding Cosine Similarity*; Scale: ~0.15 to 1.0, with major ticks at 0.2, 0.4, 0.6, 0.8, 1.0 - **Y-axis**: Label: *Prediction Score*; Scale: ~0.948 to 0.992, with major ticks at 0.95, 0.96, 0.97, 0.98, 0.99 - **Color Bar (right side)**: Label: *Prediction Score*; Gradient from dark purple (0.950, lowest score) to bright yellow (0.990, highest score), with intermediate ticks at 0.955, 0.960, 0.965, 0.970, 0.975, 0.980, 0.985 - **Regression Line**: Red dashed line, sloping upward from left to right - **Correlation Box**: Top-left corner, text: *Correlation: 0.394* ### Detailed Analysis - **Data Points**: ~150+ scatter points, colored to match the Prediction Score color bar. Points are distributed across the full range of both axes. - **Trend**: The red dashed regression line shows a clear positive upward trend: as Embedding Cosine Similarity increases, Prediction Score tends to increase. - **Correlation**: The value 0.394 indicates a moderate positive linear relationship between the two variables. - **Color Distribution**: - Lowest scores (0.950-0.960, dark purple) cluster in the lower-left region (low x, low y: x≈0.2-0.5, y≈0.948-0.955) - Mid-range scores (0.960-0.975, teal/green) are spread across the middle of the plot - Highest scores (0.980-0.990, yellow/light green) are concentrated in the upper-right region (high x, high y), with the highest point at x≈1.0, y≈0.992 (bright yellow) ### Key Observations - The highest Prediction Score (≈0.992) aligns with the highest Embedding Cosine Similarity (≈1.0). - There is significant scatter around the regression line, meaning the relationship is not perfectly linear. - No data points exist in the upper-left (low x, high y) or lower-right (high x, low y) extremes, which supports the positive trend. - A dense cluster of low-score points exists in the lower-left quadrant. ### Interpretation This plot demonstrates a moderate positive relationship between Embedding Cosine Similarity and Prediction Score. In practical terms, this suggests that higher similarity between vector embeddings (a measure of how closely two data representations match) is associated with higher model prediction scores (a measure of model confidence or performance). The correlation value of 0.394 indicates that while the trend exists, embedding similarity is not the sole driver of prediction score—other unmeasured factors likely influence the score as well. The color gradient reinforces this pattern: warmer, higher-value colors (yellow/green) cluster in the high-similarity, high-score region, while cooler, lower-value colors (purple) cluster in the low-similarity, low-score region. This could imply that more similar embeddings lead to more confident or accurate model predictions, but the scatter shows there are notable exceptions to this trend. </details> Figure 6: Neural prediction scores vs embedding cosine similarity for 400 equation pairs. The moderate correlation (0.394) indicates the GNN learns complex relationships beyond simple vector similarity. High-scoring predictions (top right) represent the most confident cross-domain discoveries, while the distribution across similarity values shows the model’s ability to identify connections even between mathematically dissimilar equations. ### 3.3 Computationally-Generated Cross-Domain Hypotheses The graph neural network framework identified several high-scoring links between physical equations from different domains. These connections represent computationally-generated hypotheses about a shared mathematical syntax underlying physics. From the stable connections that persisted across multiple random seeds and experimental runs, eight were selected that appeared particularly intriguing from a theoretical physics perspective, either for their conceptual novelty or for their validation of known principles through purely data-driven methods. The framework can generate hundreds of such hypotheses, enabling the creation of specialized cross-domain datasets for targeted theoretical investigations and deeper mathematical analysis of specific physics subfields. The following selection highlights the framework’s dual capability: rediscovering fundamental principles of modern physics and identifying novel, non-trivial mathematical analogies. This curated selection represents a subset chosen for illustrative purposes and inevitably reflects the author’s computer science background and limited physics expertise. The complete list of stable connections across seeds AUC ¿ 0.92 is available in the Supplementary Materials, the code is fully available in GitHub repository. Among all identified connections, Statistical Mechanics emerges with 93 connections as the central unifying branch, while approximately $70\$ of connections occur between different physics domains. #### 3.3.1 Debye Length and Dirac Equation (Score: 0.9691) The model identifies an intriguing connection between the Debye screening length in plasma physics (adv_0146: $λ_D=√{ε_0kT/(ne^2)}$ ) and the Dirac equation for relativistic fermions (adv_0179: $(iγ^μ∂_μ-mc/ℏ)ψ=0$ ). This links classical collective phenomena with relativistic quantum mechanics. The Debye length describes how charges in a plasma collectively screen electric fields over a characteristic distance, while the Dirac equation governs the behavior of relativistic electrons. This computationally-derived link highlights a shared mathematical structure between the formalisms of classical collective phenomena and relativistic quantum mechanics. #### 3.3.2 Hydrostatic Pressure and Maxwell Distribution (Score: 0.9690) A conceptually significant connection was found between the hydrostatic pressure formula (adv_0116: $P=P_0+ρ gh$ ) and the Maxwell-Boltzmann velocity distribution (adv_0160: $f(v)=4π n≤ft(\frac{m}{2π kT}\right)^3/2v^2e^-mv^{2/2kT}$ ). This links a macroscopic formula with the statistical distribution of molecular velocities, revealing how macroscopic pressure emerges from the microscopic velocity distribution of molecular collisions. #### 3.3.3 Maxwell Distribution and Adiabatic Process (Score: 0.9687) A significant connection was identified between the Maxwell-Boltzmann distribution (adv_0160: $f(v)=4π n≤ft(\frac{m}{2π kT}\right)^3/2v^2e^-mv^{2/2kT}$ ) and the adiabatic process for ideal gases (bal_0129: $PV^γ=constant$ ). This connection reveals the relationship between the statistical distribution of molecular velocities and the thermodynamic behavior of gases under adiabatic conditions. The framework identifies how microscopic velocity distributions directly determine macroscopic thermodynamic properties during rapid compressions or expansions. #### 3.3.4 Doppler Effect and Four-Momentum (Score: 0.9685) The framework identified an analogy between the Doppler effect for sound waves (adv_0130: $f^\prime=f_0√{\frac{v+v_r}{v-v_s}}$ ) and the relativistic four-momentum invariant (adv_0176: $p_μp^μ=(mc)^2$ ). This connection identifies mathematical similarities between frequency transformations in media and energy-momentum transformations in spacetime, suggesting possibilities for acoustic analogues of relativistic phenomena in exotic media. #### 3.3.5 Blackbody Radiation and Terminal Velocity (Score: 0.9680) An unconventional connection links Planck’s blackbody radiation (adv_0094: $B(λ,T)=\frac{8π hc}{λ^5(e^hc/λ kT-1)}$ ) with terminal velocity from fluid dynamics (bal_0239: $v_t=√{2mg/(ρ AC_d)}$ ). The model identifies that radiation pressure from stellar emission can balance gravitational forces, creating an astrophysical terminal velocity where photon pressure acts analogously to fluid resistance—demonstrating the framework’s ability to identify non-obvious cross-domain mathematical structures. #### 3.3.6 Blackbody Radiation and Navier-Stokes (Score: 0.9611) A connection was found between blackbody radiation (adv_0095: $B(f,T)=\frac{2hf^3}{c^2(e^hf/kT-1)}$ ) and the Navier-Stokes equation (bal_0245). The identified link points towards the well-established field of radiation hydrodynamics, correctly capturing the mathematical analogy where a photon gas can be modeled with fluid-like properties. #### 3.3.7 Radioactive Decay and Induced EMF (Score: 0.9651) The model identified a structural analogy between radioactive decay (adv_0061: $N(t)=N_0e^-λ t$ ) and electromagnetic induction (bal_0027: $E=-L\frac{dI}{dt}$ ). Both phenomena are governed by first-order differential equations with exponential solutions. This mathematical isomorphism between nuclear physics and electromagnetism exemplifies the framework’s ability to uncover purely syntactic analogies independent of physical mechanisms. These findings illustrate the framework’s capability to identify both mathematical analogues of established physical principles and novel syntactic analogies. The consistent identification of connections involving Statistical Mechanics as a central hub supports the hypothesis that the framework is learning a meaningful representation of the mathematical structures that connect different areas of physics. ## 4 Symbolic Analysis of Physics Equation Clusters: From Computational Patterns to Physical Insights ### 4.1 Overview and Significance The core preliminary findings of this paper emerge from the symbolic analysis of 30 high-confidence equation clusters. This analysis reveals a hierarchy of insights, progressing from validating known physics to identifying errors and synthesizing complex principles. Figure 7 shows a typical dense cluster passed to this analysis stage. <details> <summary>cluster_10.png Details</summary>  ### Visual Description ## Network Graph: Physics Equation Cluster (Cluster #10) ### Overview The image is a **network graph** (cluster diagram) visualizing 14 physics equations, color-coded by their branch of physics. The title reads: *“Cluster #10: 14 equations | Type: 6-core | Density: 0.615”* with a subtitle: *“Branches: Electromagnetism(3), Relativity(7), Quantum Mechanics(2) +2 more”*. It illustrates connections (edges) between equations, highlighting cross-disciplinary relationships in physics. ### Components/Axes (Legend & Nodes) - **Legend (Top-Right)**: - Yellow: *Electromagnetism* - Teal: *Relativity* - Red: *Quantum Mechanics* - Blue: *Thermodynamics* - Light Green: *Statistical Mechanics* - **Nodes (Equations)**: Each node is a circle with a label (truncated in some cases) and color-coded by branch. Full/visible labels (with color): - **Yellow (Electromagnetism)**: - *Lorentz Factor - Special Relativity (...)* - *Waveguide Cutoff Frequency* - *Plasma Frequency - Plasma Physics (sqr)* - **Teal (Relativity)**: - *Relativistic Energy - Special Relativity* - *Relativistic Energy - Special Relativ...* (truncated) - *Relativistic Energy - Special Relativ...* (truncated) - *Four-Momentum - Relativistic Invarian...* (truncated) - *Plasma Frequency - Plasma Phy* (truncated) - *Four-Momentum - Relativistic Invar* (truncated) - *Relativistic Momentum - Special Relat...* (truncated) - **Red (Quantum Mechanics)**: - *Harmonic Oscillator - Quantum (sqr)* (note: “nonic” may be a typo for “Harmonic”) - *de Broglie Wavelength - Matter Waves ...* (truncated) - **Blue (Thermodynamics)**: - *Thermal Energy - Heat Transfer* - **Light Green (Statistical Mechanics)**: - *Maxwell Distribution - Statistical Me...* (truncated) ### Detailed Analysis (Connections & Structure) - **Density (0.615)**: ~61.5% of all possible node-to-node connections exist, indicating strong interrelationships between equations. - **Branch Distribution**: Relativity (7 nodes) dominates, followed by Electromagnetism (3), Quantum Mechanics (2), Thermodynamics (1), and Statistical Mechanics (1). - **Edges (Connections)**: Nodes are highly interconnected, with edges spanning multiple branches (e.g., Relativity nodes connect to Electromagnetism, Quantum, and Thermodynamics nodes), showing cross-disciplinary physics relationships. ### Key Observations - **Truncated Labels**: Many node labels are truncated (e.g., *“Relativistic Energy - Special Relativ...”*), so full equation names are not visible. - **Color Consistency**: Nodes match the legend’s color coding, enabling clear branch identification. - **Core Structure**: The “6-core” type suggests a central subgraph with 6 key nodes (exact core not labeled, but implied by the “6-core” designation). ### Interpretation This graph demonstrates the **interconnectedness of physics concepts** across disciplines. The high density (0.615) implies that equations from different branches (e.g., Relativity, Electromagnetism, Quantum Mechanics) are deeply related, reflecting how physics theories build on one another. For example: - Relativity (teal) nodes connect to Electromagnetism (yellow) nodes, illustrating how relativistic principles interact with electromagnetic phenomena. - Quantum Mechanics (red) nodes link to Relativity and Electromagnetism, showing the overlap between quantum and relativistic physics. The “6-core” designation hints at a central cluster of 6 equations that anchor the network, though their exact identities are not labeled. Overall, the graph visualizes the complex web of relationships in physics, emphasizing that no branch exists in isolation. (Note: All text is in English; no other languages are present. Truncated labels are noted, and color coding is verified against the legend.) </details> Figure 7: An example of a high-density (0.615), high-core-number (6-core) conceptual cluster identified by the GNN, centered on relativistic principles. The framework isolates such structurally significant clusters for deeper symbolic analysis. The analysis yielded 24 simplified expressions (80%), with the remaining 20% failing due to insufficient parseable equations or lack of valid substitutions. These computational results require expert interpretation to distinguish between physical insights and mathematical artifacts. ### 4.2 Foundational Results #### 4.2.1 The Klein-Gordon/Dirac Hierarchy (Cluster #5) Cluster #5 connected the Klein-Gordon and Dirac equations. The system selected the Klein-Gordon equation as backbone: $$ ≤ft(∂_μ∂^μ+\frac{m^2c^2}{ℏ^2}\right)ψ=0 \tag{2} $$ where $∂_μ∂^μ$ is the d’Alembertian operator. The system performed the following substitutions from equations within the cluster: - $m=γℏ/c^2$ (from Dirac Equation - Relativistic Fermions) - $ψ=0$ (from Dirac Equation - Relativistic Fermions) - $ℏ=c^2m/γ$ (from Dirac Equation - Relativistic Fermions) - $c=(γ^2v^r^2/(γ^2-1))^r^{-2}$ (from Lorentz Factor) The substitution $ψ=0$ reduces the expression to the identity True. This result confirms the known relationship where the Dirac equation: $$ (iγ^μ∂_μ-mc/ℏ)ψ=0 \tag{3} $$ represents the “square root” of the Klein-Gordon equation. Applying the Dirac operator twice: $$ (iγ^ν∂_ν+m)(iγ^μ∂_μ-m)ψ=(-γ^νγ^μ∂_ν∂_μ-m^2)ψ=0 \tag{4} $$ Using the anticommutation relations $\{γ^μ,γ^ν\}=2η^μν$ , this reduces to $(∂_μ∂^μ+m^2)ψ=0$ . Every Dirac solution satisfies Klein-Gordon (though not vice versa), a structure that historically led to the prediction of antimatter. #### 4.2.2 Maxwell’s Self-Consistency (Clusters #1, #2) The two largest clusters (99 and 83 equations) centered on Maxwell’s equations, confirming the internal consistency of electromagnetic theory. The system identified complex relationships between the wave equation for electromagnetic fields and Maxwell’s equations through multiple substitutions, though the resulting expressions contain parsing artifacts that require further investigation. These clusters validate the framework’s ability to recognize the mathematical coherence of established physical theories. #### 4.2.3 Electromagnetic-Fluid Coupling (Cluster #8) Cluster #8 revealed an unexpected synthesis between fluid dynamics and electromagnetism, identified by an earlier version of the code that still had parsing issues but nonetheless uncovered this intriguing connection (full report in Supplementary Materials). Starting with Reynolds number as backbone: $$ Re=\frac{ρ vL}{η} \tag{5} $$ The system applied the following substitutions from electromagnetic equations in the cluster: - $E=-Bv+F/q$ (from Lorentz Force (Complete)) - $t=Iε/L$ (from Inductance EMF) - $v=ε/(BL)$ (from Motional EMF) - $t=-dε/dΦ_B$ (from Faraday’s Law) - $t=-ε/Φ$ (from Lenz’s Law Direction) This produced the simplified expression: $$ \frac{ερ}{Bη} \tag{6} $$ This result represents a dimensionless parameter coupling electromagnetic and fluid properties, analogous to the Magnetic Reynolds Number in magnetohydrodynamics, demonstrating how the framework can identify cross-domain mathematical structures even without understanding the underlying physics. ### 4.3 Error Detection as Knowledge Auditing #### 4.3.1 Dimensional Catastrophe (Cluster #4) The four-momentum cluster exposed an error in the knowledge base. Starting with the relativistic invariant as backbone: $$ μ pp_μ=c^2mr^2 \tag{7} $$ The system applied the following substitutions from equations in the cluster: - $p=E/c-cm^2$ (from Four-Momentum - Relativistic Invariant) - $m=√{E-cp}/c$ (from Four-Momentum - Relativistic Invariant) - $c=m_e$ (from Compton Scattering - Wavelength Shift) This produced: $$ μ pp_μ=m_er^2√{E-m_ep} \tag{8} $$ With residual: $-m_er^2√{E-m_ep}+μ pp_μ$ The critical error is the substitution $c=m_e$ , which is dimensionally incorrect—equating velocity [L/T] with mass [M]. This error originates from a mis-parsed Compton scattering equation where the system incorrectly extracted a relationship between the speed of light and electron mass, likely from the Compton wavelength formula $λ_C=h/(m_ec)$ where $m_e$ and $c$ appear together but represent fundamentally different physical quantities. #### 4.3.2 Category Confusion (Cluster #16) The framework combined Bragg’s Law for crystal diffraction: $$ 2d\sinθ=nλ \tag{9} $$ with substitutions from interference equations: - $m=d\sinθ/λ$ (from Interference - Young’s Double Slit) - $a=mλ/\sinθ$ (from Single Slit Diffraction (Minima)) - $m=(d\sinθ-λ/2)/λ$ (from Interference - Young’s Double Slit (dsi)) This produced the tautology True, but mixed different physical contexts—crystalline solids versus free space. ### 4.4 The Unexpected Value of Errors: Analog Gravity (Cluster #15) Cluster #15 combined 11 equations from fluid dynamics and electromagnetism. Starting with Bernoulli’s equation: $$ P+\frac{1}{2}ρ v^2+ρ gh=constant \tag{10} $$ The system applied substitutions: - $h=v/(gr^2)$ (from Torricelli’s Law) - $g=v/(hr^2)$ (from Torricelli’s Law) - $c=(μ_r-1)/χ_m$ (from Magnetic Susceptibility) - $P=P_0+ρ gh$ (from Hydrostatic Pressure) This produced: $$ \frac{(μ_r-1)^2}{h^2m^2} \tag{11} $$ This mixes magnetic permeability with fluid dynamical variables—dimensionally inconsistent and physically incorrect. However, this error points to analog gravity research, where: - Sound waves in fluids obey equations mathematically identical to quantum fields in curved spacetime - Fluid velocity fields create effective metrics analogous to gravitational geometry - Sonic horizons mirror black hole event horizons - Navier-Stokes equations in $(p+1)$ dimensions correspond to Einstein equations in $(p+2)$ dimensions Near the horizon limit, Einstein’s equations reduce to Navier-Stokes, suggesting gravity might emerge from microscopic degrees of freedom. ### 4.5 Statistical Summary Table 3: Cluster Analysis Results | Theory Validation | 2 | Klein-Gordon/Dirac, Maxwell consistency | | --- | --- | --- | | Novel Synthesis | 8 | Transport phenomena, Reynolds-EMF | | Tautologies | 5 | Planck’s Law circular substitutions | | Dimensional Errors | 2 | $c=m_e$ , knowledge base errors | | Category Errors | 3 | Bragg/Young confusion | | Provocative Failures | 4 | Analog gravity connection | | Insufficient Data | 6 | No valid substitutions | | Total | 30 | 80% produced interpretable results | The framework operates at the syntactic level, recognizing mathematical patterns without understanding physical causality. This limitation, combined with parsing issues in early iterations, requires human interpretation to distinguish profound connections from tautologies or errors. However, even computational failures can suggest legitimate research directions, as demonstrated by the analog gravity connection. ## 5 Discussion This proof of concept demonstrates a framework capable of systematic pattern discovery in physics equations. The framework functions as a computational lens revealing hidden structures and inconsistencies difficult to discern at human scale. It does not replace physicists, of course, but fully developed could augment their capabilities through systematic exploration of mathematical possibility space. The framework’s value lies not in autonomous discovery but in its role as a computational companion—a tireless explorer of mathematical possibility space that surfaces patterns, errors, and unexpected connections for human interpretation. Even its failures are productive, transforming the vast combinatorial space of physics equations into a curated set of computational hypotheses worthy of expert attention. The transformation of mathematical pattern into physical understanding remains fundamentally human. But by automating the pattern-finding, the framework frees physicists to focus on what they do best: interpreting meaning, recognizing significance, and building the conceptual bridges that transform equations into understanding of nature. ## 6 Conclusion and Future Work In this preliminary work, I have developed and tested a GNN-based framework capable of mapping the mathematical structure of physical law, and acting as a computational auditor. My GAT model achieves high performance on this novel task, and the primary contribution of this work lies in the subsequent symbolic analysis. This analysis suggests the framework has a potential multi-layered capacity to: (i) verify the internal consistency of foundational theories, (ii) help debug knowledge bases by identifying errors and tautologies, (iii) synthesize mathematical structures analogous to complex physical principles, and (iv) provide creative provocations from its own systemic failures. ### 6.1 Current Limitations and Future Directions This work, while promising, represents an initial step. The path forward is clear and focuses on several key areas: - Systematic Parameter Exploration: The framework requires systematic testing with different hyperparameter configurations to identify optimal settings for various physics domains. Different weight combinations in the importance scoring and clustering algorithms may reveal distinct classes of mathematical relationships, generating a vast number of hypotheses that require careful evaluation. - AI-Assisted Hypothesis Screening: The system currently generates hundreds of potential cross-domain connections, many of which are spurious or trivial. Future work should integrate large language models or other AI systems to perform initial screening of these computational hypotheses, filtering out obvious errors, tautologies, and dimensionally inconsistent results before human expert review. This would create a multi-stage pipeline: graph-based generation, AI screening, and expert validation. - Database Expansion: The immediate next step is to expand the equation database. A richer and broader corpus would enable the encoding of deeper mathematical structures, moving the analysis from a syntactic to a more profound structural level. - Generalization as a Scientific Auditor: Future work will focus on generalizing the framework beyond physics to other formal sciences. This includes refining the disambiguation protocol to act as a general-purpose “auditor” for standardizing notational conventions across different scientific knowledge bases. - Collaboration with Domain Experts: To bridge the gap between computational patterns and physical insight, future work must involve an expert-in-the-loop process. Collaboration with theoretical physicists is essential to validate, interpret, and build upon the most promising machine-generated analogies and audit reports. ### 6.2 Broader Implications This work may open several avenues for the broader scientific community. As an auditing tool, it could potentially be used to systematically check the consistency of large-scale theories. As an educational tool, it might help students visualize the deep structural connections that unify different areas of science. More broadly, this research contributes to the emerging field of computational epistemology, developing methods to study the structure and coherence of scientific knowledge. Ultimately, this framework is presented as a tangible step toward a new synergy between human intuition and machine computation, where AI may serve as a tool to augment and stimulate the quest for scientific understanding. ## 7 Data and Code Availability All code, cleaned dataset, and model weights available at: https://github.com/kingelanci/graphysics. ## 8 Supplementary Materials Complete prediction distributions and analysis results available at the GitHub repository: - Full distribution of AUC ¿ 0.92 prediction scores - Complete symbolic analysis for all 30 clusters - Bootstrap validation logs - All generated cross-domain hypotheses (not just selected examples) ## Correspondence Correspondence: Massimiliano Romiti (massimiliano.romiti@acm.org). ## Acknowledgments The author thanks the arXiv endorsers for their support. 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