2508.05724

# 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: $$ w_{ij}=\alpha\cdot J(V_{i},V_{j})+\beta\cdot I(V_{i},V_{j})+\gamma\cdot S(B_{i},B_{j}) \tag{1} $$ 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 $\alpha=0.5$ , $\beta=0.35$ , $\gamma=0.15$ 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 ## Chart Compilation: Super GNN Training & Prediction Analysis ### Overview The image presents a compilation of four charts analyzing the training and prediction performance of a Super GNN model. The charts depict training loss evolution, validation metrics (AUC and AP), precision at different recall levels, and the distribution of Super GNN scores. ### Components/Axes **Top-Left Chart: Training Loss Evolution** * **Title:** Training Loss Evolution * **X-axis:** Epoch (ranging from 0 to 5000) * **Y-axis:** Loss (ranging from 0.7 to 1.4) * **Legend:** Training Loss (blue line) **Top-Right Chart: Validation Metrics** * **Title:** Validation Metrics * **X-axis:** Epoch (ranging from 0 to 5000) * **Y-axis:** Score (ranging from 0.7 to 0.95) * **Legend:** * AUC (blue line with circle markers) * AP (orange line with square markers) **Bottom-Left Chart: Precision at Different Recall Levels** * **Title:** Precision at Different Recall Levels * **X-axis:** Recall Level (ranging from approximately -0.2 to 1.2) * **Y-axis:** Precision (ranging from 0.0 to 1.0) * **Data:** A series of overlapping blue bars, each representing precision at a specific recall level. **Bottom-Right Chart: Super GNN Score Distribution** * **Title:** Super GNN Score Distribution * **X-axis:** Prediction Score (ranging from 0.940 to 0.965) * **Y-axis:** Count (ranging from 0 to 14) * **Data:** A green histogram showing the distribution of prediction scores. * **Additional Element:** A vertical dashed red line indicating the 90th percentile at a score of 0.954. ### Detailed Analysis **Training Loss Evolution** * The blue "Training Loss" line starts at approximately 1.38 and rapidly decreases until around epoch 1000, where it reaches approximately 0.8. From epoch 1000 to 5000, the loss continues to decrease gradually, fluctuating around 0.75. * Trend: Decreasing loss over epochs, indicating learning. **Validation Metrics** * The blue "AUC" line starts at approximately 0.76 and increases rapidly until around epoch 1000, where it reaches approximately 0.93. From epoch 1000 to 5000, the AUC continues to increase slightly, fluctuating around 0.97. * The orange "AP" line starts at approximately 0.70 and increases rapidly until around epoch 1000, where it reaches approximately 0.94. From epoch 1000 to 5000, the AP fluctuates around 0.96. * Trend: Both AUC and AP increase rapidly initially and then plateau, indicating model convergence. **Precision at Different Recall Levels** * The blue bars show the precision at different recall levels. The precision is high (close to 1.0) for lower recall levels and gradually decreases as the recall level increases. * Trend: Decreasing precision with increasing recall. **Super GNN Score Distribution** * The green histogram shows the distribution of prediction scores. The distribution appears to be somewhat normal, with a peak around 0.940-0.945. * The red dashed line indicates the 90th percentile at a score of 0.954. * Trend: The distribution is centered around 0.945, with a long tail towards higher scores. ### Key Observations * The training loss decreases over time, indicating that the model is learning. * The validation metrics (AUC and AP) increase rapidly initially and then plateau, indicating model convergence. * Precision decreases as recall increases, which is a common trade-off in machine learning. * The distribution of Super GNN scores is somewhat normal, with a peak around 0.940-0.945. ### Interpretation The charts provide a comprehensive analysis of the Super GNN model's performance. The decreasing training loss and increasing validation metrics indicate that the model is learning effectively. The precision-recall curve shows the trade-off between precision and recall. The distribution of Super GNN scores provides insights into the model's prediction behavior. The 90th percentile score can be used to identify high-confidence predictions. Overall, the charts suggest that the Super GNN model is performing well. </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 ## FDR Analysis: Predictions vs Negative Controls and Bootstrap ### Overview The image presents a comprehensive analysis of False Discovery Rate (FDR) using predictions and negative controls, incorporating bootstrap methods. It includes several plots illustrating score distributions, sensitivity analysis, precision-recall trade-offs, CDF comparisons, and score distributions by quantiles and bins. ### Components/Axes **1. Score Distribution: Predictions vs Negative Controls** * **Type**: Histogram with KDE (Kernel Density Estimate) * **X-axis**: Score (0.0 to 1.0) * **Y-axis**: Density (0 to 6) * **Data**: * Predictions (n=5000) - Blue histogram and KDE * Negative Controls (n=600) - Red histogram and KDE * **Zones**: * Intermediate Zone - Light beige vertical band between approximately 0.6 and 0.8 on the x-axis. **2. FDR vs Threshold con Bootstrap CI** * **Type**: Line plot with confidence interval * **X-axis**: Score Threshold (0.5 to 1.0) * **Y-axis**: Estimated FDR (0.0 to 0.5) * **Data**: * FDR Mean - Black line * 95% CI - Gray shaded area around the mean * FDR = 5% - Green dashed horizontal line * FDR = 10% - Orange dashed horizontal line **3. Sensitivity Analysis: FDR at threshold 0.9** * **Type**: Heatmap * **X-axis**: Signal Threshold (0.85 to 0.94) * **Y-axis**: Noise Threshold (0.0 to 0.6) * **Color Scale**: FDR (0.0 to 0.5), ranging from dark green to red. All values appear to be dark green, indicating a low FDR. **4. Distribution by Bin (incl. Negative Controls)** * **Type**: Boxplot * **X-axis**: Score Bin (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 (0.0 to 1.0) * **Data**: Boxplots showing the distribution of scores for each bin, including a separate boxplot for negative controls. **5. Precision-Recall Trade-off** * **Type**: Line plot * **X-axis**: Score Threshold (0.5 to 1.0) * **Y-axis**: Metric Value (0.0 to 1.0) * **Data**: * Precision - Blue line * Recall - Green line **6. CDF Comparison** * **Type**: Line plot * **X-axis**: Score (0.0 to 1.0) * **Y-axis**: Cumulative Probability (0.0 to 1.0) * **Data**: * Predictions - Blue line * Negative Controls - Red line * **Zones**: * Noise Zone - Light red vertical band from 0.0 to approximately 0.6 on the x-axis. * Intermediate - Light beige vertical band from approximately 0.6 to 0.8 on the x-axis. * Signal Zone - Light green vertical band from approximately 0.8 to 1.0 on the x-axis. **7. Score Distribution by Quantiles** * **Type**: Bar plot * **X-axis**: Quantile (10%, 25%, 50%, 75%, 90%, 95%, 99%) * **Y-axis**: Score (0.0 to 1.0) * **Data**: * Predictions - Blue bars * Neg Controls - Red bars ### Detailed Analysis **1. Score Distribution: Predictions vs Negative Controls** * The blue histogram (Predictions) is skewed towards higher scores, peaking around 0.9-1.0. The KDE (blue line) confirms this trend. * The red histogram (Negative Controls) is skewed towards lower scores, with a peak around 0.2. The KDE (red line) confirms this trend. * The intermediate zone (0.6-0.8) shows an overlap between the two distributions. **2. FDR vs Threshold con Bootstrap CI** * The black line (FDR Mean) slopes downward as the score threshold increases. * The gray area (95% CI) narrows as the score threshold increases. * The FDR drops below 5% (green dashed line) at a score threshold of approximately 0.75. * The FDR drops below 10% (orange dashed line) at a score threshold of approximately 0.65. **3. Sensitivity Analysis: FDR at threshold 0.9** * The heatmap is uniformly dark green, indicating a very low FDR across all combinations of signal and noise thresholds at a score threshold of 0.9. All values are approximately 0.0. **4. Distribution by Bin (incl. Negative Controls)** * The boxplots show an increasing median score as the score bin increases. * The "Neg Ctrl" boxplot shows a lower median score compared to the other bins. * The 0.0-0.6 bin has a median score around 0.1, with several outliers above 0.4. * The 0.9-1.0 bin has a median score around 0.95. * The Neg Ctrl bin has a median score around 0.3. **5. Precision-Recall Trade-off** * The blue line (Precision) generally increases with the score threshold, reaching 1.0 at a threshold of approximately 0.8. * The green line (Recall) generally decreases with the score threshold, approaching 0.0 at a threshold of approximately 1.0. **6. CDF Comparison** * The red line (Negative Controls) rises more rapidly than the blue line (Predictions) for lower scores. * The blue line (Predictions) rises sharply for higher scores, especially after 0.8. * The noise zone (0.0-0.6) shows a higher cumulative probability for negative controls. * The signal zone (0.8-1.0) shows a higher cumulative probability for predictions. **7. Score Distribution by Quantiles** * For all quantiles, the blue bars (Predictions) are higher than the red bars (Neg Controls). * Both predictions and negative controls show increasing scores as the quantile increases. * The difference between predictions and negative controls is most pronounced at higher quantiles. ### Key Observations * Predictions tend to have higher scores than negative controls. * Increasing the score threshold reduces the FDR but also reduces recall. * A score threshold of 0.9 results in a very low FDR, according to the sensitivity analysis. * The CDF comparison highlights the separation between predictions and negative controls across different score ranges. ### Interpretation The data suggests that the predictions are well-separated from the negative controls, with predictions generally having higher scores. The FDR analysis indicates that a higher score threshold can be used to reduce the false discovery rate. However, this comes at the cost of reduced recall, as shown in the precision-recall trade-off. The CDF comparison provides a visual representation of the score distributions for predictions and negative controls, further supporting the conclusion that the two groups are distinct. The sensitivity analysis at a threshold of 0.9 confirms that a high threshold can effectively minimize the FDR. The distribution by bin and quantile plots provide additional insights into the score distributions across different categories. </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 $\leftrightarrow$ Statistical Mechanics, Electromagnetism $\leftrightarrow$ 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: Strong Connections between Equations ### Overview The image is a network diagram illustrating strong connections between various physics-related equations or concepts. Nodes represent these equations/concepts, and edges (lines) indicate a strong relationship or connection between them. The nodes are colored differently, but there is no explicit legend to define what the colors represent. The size of the nodes may indicate the relative importance or degree of connectivity. ### Components/Axes * **Title:** "Network of Strong Connections: high-weight equation-equation" * **Nodes:** Represented by colored circles, each labeled with a specific equation or concept. * Examples: Blackbody Radiation, Relativistic Energy, Plasma Frequency, Boltzmann Distribution, de Broglie Wavelength, Bose-Einstein Distribution, Fermi-Dirac Distribution, Debye Length, Maxwell Distribution, Harmonic Oscillator, Four-Momentum, Maxwell-Boltzmann Distrib, Relativistic Momentum * **Edges:** Represented by gray lines connecting the nodes, indicating a strong connection. * **Node Colors:** The nodes are colored in a variety of colors including: orange, green, light blue, purple, gray, and brown. There is no legend to indicate what the colors represent. * **Node Size:** The nodes are of varying sizes. ### Detailed Analysis or Content Details Here's a breakdown of the nodes and their approximate positions, along with observations about their connections: * **Blackbody Radiation:** Located in the top-left quadrant, connected to multiple nodes. Node color is light blue. * **Relativistic Energy:** Located near the top-center, connected to multiple nodes. Node color is green. * **Plasma Frequency:** Located in the top-right quadrant, connected to multiple nodes. Node color is orange. * **Boltzmann Distribution:** Located in the right quadrant, connected to multiple nodes. Node color is light blue. * **de Broglie Wavelength:** Located in the right quadrant, connected to multiple nodes. Node color is light blue. * **Bose-Einstein Distribution:** Located near the center, connected to multiple nodes. Node color is light blue. * **Fermi-Dirac Distribution:** Located near the center-bottom, connected to multiple nodes. Node color is light blue. * **Debye Length:** Located in the left quadrant, connected to multiple nodes. Node color is orange. * **Maxwell Distribution:** Located in the left quadrant, connected to multiple nodes. Node color is light blue. * **Harmonic Oscillator:** Located in the top-left quadrant, connected to multiple nodes. Node color is gray. * **Four-Momentum:** Located in the right quadrant, connected to multiple nodes. Node color is purple. * **Maxwell-Boltzmann Distrib:** Located in the left quadrant, connected to multiple nodes. Node color is gray. * **Relativistic Momentum:** Located in the bottom-center, connected to multiple nodes. Node color is purple. **Observations on Connections:** * The network appears densely connected, with most nodes having multiple connections to other nodes. * There doesn't appear to be a clear hierarchical structure; the connections seem relatively distributed. * The node sizes vary, suggesting some equations/concepts have a higher degree of connectivity or importance within the network. ### Key Observations * The diagram illustrates relationships between various physics concepts. * The connections are "high-weight," suggesting strong relationships. * Node size appears to correlate with the number of connections. * Node color is not explained. ### Interpretation The network diagram visualizes the interconnectedness of various equations and concepts in physics. The "high-weight" connections suggest that the relationships depicted are significant and fundamental. The varying node sizes likely indicate the relative importance or influence of each equation/concept within the network. Without a legend for the node colors, it's difficult to determine if the colors represent specific categories or properties. The diagram suggests that these equations/concepts are not isolated but rather form a complex web of relationships, highlighting the underlying unity of physics. The diagram could be used to understand the relationships between different concepts in physics, or to identify key areas for further research. </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 Diagram: Graph Visualizations ### Overview The image presents a collection of twelve network diagrams, each labeled with a letter (E, F, G, L, R, c, e, h, k, m, p, q) and a number followed by "eq.". Each diagram consists of nodes (represented by colored circles) connected by edges (represented by gray lines). A yellow star is present in each diagram, seemingly connected to other nodes. The number following the letter likely represents the number of nodes in the graph. ### Components/Axes * **Nodes:** Represented by colored circles. The colors observed are orange, green, blue, gray, purple, and brown. * **Edges:** Represented by gray lines connecting the nodes. * **Labels:** Each graph is labeled with a letter and a number in parentheses, e.g., "E (12 eq.)". * **Star:** A yellow star is present in each graph. ### Detailed Analysis or ### Content Details Here's a breakdown of each graph: * **E (12 eq.):** 12 nodes. Colors present: orange, green, blue, gray. The yellow star is present. * **F (18 eq.):** 18 nodes. Colors present: orange, green, gray, brown. The yellow star is present. * **G (6 eq.):** 6 nodes. Colors present: orange, green, purple. The yellow star is present. * **L (29 eq.):** 29 nodes. Colors present: orange, green, blue, gray, purple, brown. The yellow star is present. * **R (30 eq.):** 30 nodes. Colors present: orange, green, blue, gray, brown. The yellow star is present. * **c (52 eq.):** 52 nodes. Colors present: orange, green, blue, gray, purple, brown. The yellow star is present. * **e (53 eq.):** 53 nodes. Colors present: orange, green, blue, gray. The yellow star is present. * **h (29 eq.):** 29 nodes. Colors present: orange, green, blue, gray. The yellow star is present. * **k (40 eq.):** 40 nodes. Colors present: orange, green, blue, gray, purple. The yellow star is present. * **m (45 eq.):** 45 nodes. Colors present: orange, green, blue, gray, purple, brown. The yellow star is present. * **p (13 eq.):** 13 nodes. Colors present: orange, green, blue, gray, purple. The yellow star is present. * **q (26 eq.):** 26 nodes. Colors present: orange, green, blue, gray. The yellow star is present. ### Key Observations * The number of nodes varies significantly across the graphs, ranging from 6 to 53. * The colors of the nodes are not consistent across all graphs. Some graphs have a wider variety of colors than others. * The yellow star is a consistent element in all graphs. * The edges (gray lines) appear to connect nodes in a complex and seemingly random manner. ### Interpretation The image likely represents different network configurations or states, possibly related to some underlying system or process. The "eq." likely stands for "entities". The number of entities varies across the different states. The colored nodes could represent different types of entities or different states of the same entity. The yellow star could represent a central node or a node with a special property. The connections between the nodes (edges) represent relationships or interactions between the entities. Without further context, it's difficult to determine the specific meaning of these graphs, but they likely visualize the structure and connectivity of a complex system. </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 ### Overview The image is a concept network diagram centered around the variable "K", likely representing Boltzmann constant. It illustrates the relationships between various physics concepts and equations within different domains of physics. The diagram consists of labeled rectangular nodes representing physics concepts, connected to a central node "K" with lines. The concepts are grouped by color, indicating their respective physics domain, as defined in the legend. The diagram indicates 26 equations and 6 bridges. ### Components/Axes * **Central Node:** A yellow circle labeled "K" at the center of the diagram. * **Concept Nodes:** Rectangular nodes surrounding the central node, each labeled with a physics concept and an associated equation. * **Connecting Lines:** Yellow lines connecting the central node "K" to each of the concept nodes. * **Legend (Top-Right):** * Atomic and Nucl (1) - Brown * Condensed Matte (1) - Cyan * Electromagnetis (2) - Blue * Quantum Mechani (3) - Orange * Statistical Mec (4) - Green * Thermodynamics (4) - Purple ### Detailed Analysis or ### Content Details Here's a breakdown of each concept node, its associated equation, and its corresponding physics domain based on color: 1. **Atomic and Nucl (Brown):** * Concept: Fine Structu... * Equation: k * e^2 / (ħ * c) * Position: Bottom-Right 2. **Condensed Matte (Cyan):** * Concept: Band Gap * Equation: k*T * ln(N\_c*N\_v/n) * Position: Right 3. **Electromagnetis (Blue):** * Concept: Debye Length * Equation: sqrt(k*T / (4*π*n*e^2)) * Position: Bottom-Left 4. **Electromagnetis (Blue):** * Concept: Debye Length * Equation: sqrt(ε\_0*k*T / (n*e^2)) * Position: Left 5. **Quantum Mechani (Orange):** * Concept: Wave Equation * Equation: A * sin(k*x - w*t + * Position: Top-Right 6. **Quantum Mechani (Orange):** * Concept: Harmonic Osc... * Equation: (No equation visible) * Position: Right 7. **Quantum Mechani (Orange):** * Concept: de Broglie W... * Equation: h/sqrt(2 * m * E\_k) * Position: Top 8. **Statistical Mec (Green):** * Concept: Fermi-Dirac ... * Equation: (No equation visible) * Position: Top-Right 9. **Statistical Mec (Green):** * Concept: Maxwell Dist... * Equation: sqrt(8*k*T / (π*m)) * Position: Bottom-Right 10. **Statistical Mec (Green):** * Concept: Bose-Einstei... * Equation: 1 / (exp(E/(k*T)) - * Position: Bottom 11. **Statistical Mec (Green):** * Concept: Bose-Einstei... * Equation: (No equation visible) * Position: Left 12. **Thermodynamics (Purple):** * Concept: Mean Kinetic... * Equation: (E\_k) = (3/2)kT * Position: Top-Right 13. **Thermodynamics (Purple):** * Concept: Root Mean Sq... * Equation: v\_rms = √(3kT/m) * Position: Top-Left 14. **Thermodynamics (Purple):** * Concept: Average Speed * Equation: (v) = √(8kT/πm) * Position: Bottom-Left 15. **Thermodynamics (Purple):** * Concept: Superconduct... * Equation: 3.5 * k * T\_c * Position: Left ### Key Observations * The central node "K" represents a key concept (Boltzmann constant) that links various physics domains. * The diagram visually represents the interconnectedness of different physics concepts through the common variable "K". * The number of concepts from each domain varies, with Statistical Mechanics and Thermodynamics having the most representations. * The equations associated with each concept demonstrate how "K" is used in different contexts. ### Interpretation The concept network illustrates how the Boltzmann constant "K" is a fundamental constant that appears in various equations across different domains of physics. The diagram emphasizes the unifying role of "K" in connecting seemingly disparate concepts such as electromagnetism, quantum mechanics, statistical mechanics, and thermodynamics. The number of equations associated with each domain suggests the relative importance or frequency of use of "K" within those fields. The diagram serves as a visual aid for understanding the relationships between these concepts and the central role of the Boltzmann constant in physics. </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 The image is a scatter plot showing the relationship between "Embedding Cosine Similarity" on the x-axis and "Prediction Score" on the y-axis. The data points are colored according to their "Prediction Score," as indicated by a color bar legend on the right. A red dashed line represents a linear trend line through the data. The plot also displays the correlation coefficient between the two variables. ### Components/Axes * **Title:** Neural Score vs Embedding Similarity * **X-axis:** Embedding Cosine Similarity * Scale: 0.2 to 1.0, with gridlines at intervals of 0.2. * **Y-axis:** Prediction Score * Scale: 0.95 to 0.99, with gridlines at intervals of 0.01. * **Color Bar (Legend):** Prediction Score * Scale: 0.950 (purple) to 0.990 (yellow), with intermediate values at 0.955, 0.960, 0.965, 0.970, 0.975, 0.980, 0.985. * **Correlation:** A text box in the top-left corner indicates "Correlation: 0.394". * **Trend Line:** A red dashed line shows a positive linear trend. ### Detailed Analysis * **Data Points:** The scatter plot contains numerous data points, each represented by a circle. The color of each circle corresponds to its Prediction Score, ranging from purple (low) to yellow (high). * **Trend Line:** The red dashed line indicates a positive correlation between Embedding Cosine Similarity and Prediction Score. * The line starts at approximately (0.2, 0.955) and ends at approximately (1.0, 0.972). * **Data Distribution:** * For Embedding Cosine Similarity values between 0.2 and 0.4, the Prediction Scores are mostly between 0.95 and 0.96. * For Embedding Cosine Similarity values between 0.6 and 0.8, the Prediction Scores are mostly between 0.97 and 0.98. * For Embedding Cosine Similarity values near 1.0, the Prediction Scores are close to 0.99. * **Color Mapping:** * Data points with Prediction Score around 0.95 are purple. * Data points with Prediction Score around 0.96 are blue/purple. * Data points with Prediction Score around 0.97 are teal/green. * Data points with Prediction Score around 0.98 are light green. * Data points with Prediction Score around 0.99 are yellow. ### Key Observations * There is a positive correlation between Embedding Cosine Similarity and Prediction Score. * The correlation coefficient is 0.394, indicating a moderate positive correlation. * The data points are more densely clustered in the lower-left region of the plot (low Embedding Cosine Similarity and low Prediction Score). * There is a wider spread of Prediction Scores for lower Embedding Cosine Similarity values. ### Interpretation The scatter plot suggests that as the Embedding Cosine Similarity increases, the Prediction Score tends to increase as well. However, the moderate correlation coefficient (0.394) indicates that the relationship is not very strong, and there are other factors influencing the Prediction Score. The color gradient provides an additional layer of information, showing how the Prediction Score varies across different Embedding Cosine Similarity values. The trend line visually confirms the positive correlation, but the scatter of the data points around the line suggests that the relationship is not perfectly linear. The clustering of points in the lower-left region indicates that low Embedding Cosine Similarity values are more frequently associated with low Prediction Scores. </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: $\lambda_{D}=\sqrt{\epsilon_{0}kT/(ne^{2})}$ ) and the Dirac equation for relativistic fermions (adv_0179: $(i\gamma^{\mu}∂_{\mu}-mc/\hbar)\psi=0$ ). 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: $P=P_{0}+\rho gh$ ) and the Maxwell-Boltzmann velocity distribution (adv_0160: $f(v)=4\pi n\left(\frac{m}{2\pi kT}\right)^{3/2}v^{2}e^{-mv^{2}/2kT}$ ). 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: $f(v)=4\pi n\left(\frac{m}{2\pi kT}\right)^{3/2}v^{2}e^{-mv^{2}/2kT}$ ) and the adiabatic process for ideal gases (bal_0129: $PV^{\gamma}=\text{constant}$ ). 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: $f^{\prime}=f_{0}\sqrt{\frac{v+v_{r}}{v-v_{s}}}$ ) and the relativistic four-momentum invariant (adv_0176: $p_{\mu}p^{\mu}=(mc)^{2}$ ). 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: $B(\lambda,T)=\frac{8\pi hc}{\lambda^{5}(e^{hc/\lambda kT}-1)}$ ) with terminal velocity from fluid dynamics (bal_0239: $v_{t}=\sqrt{2mg/(\rho AC_{d})}$ ). 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: $N(t)=N_{0}e^{-\lambda t}$ ) and electromagnetic induction (bal_0027: $\mathcal{E}=-L\frac{dI}{dt}$ ). 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 Diagram: Physics Equations Cluster ### Overview The image is a network diagram visualizing the relationships between different physics equations within a cluster. The nodes represent equations, and the edges represent connections or relationships between them. The nodes are colored based on the branch of physics they belong to. The diagram provides an overview of how different concepts and equations are interconnected within the cluster. ### Components/Axes * **Title:** Cluster #10: 14 equations | Type: 6-core | Density: 0.615 * **Subtitle:** Branches: Electromagnetism(3), Relativity(7), Quantum Mechanics(2) +2 more * **Nodes:** Each node represents a physics equation or concept. * **Edges:** The lines connecting the nodes represent relationships between the equations. * **Node Colors (Legend, located top-right):** * Yellow: Electromagnetism * Teal: Relativity * Red: Quantum Mechanics * Salmon: Thermodynamics * Light Green: Statistical Mechanics ### Detailed Analysis or ### Content Details **Node Descriptions and Connections:** * **Electromagnetism (Yellow Nodes):** * Waveguide Cutoff Frequency (bottom-left) * Plasma Frequency - Plasma Physics (sqr) (bottom-left) * Lorentz Factor - Special Relativity (...) (top) * **Relativity (Teal Nodes):** * Relativistic Energy - Special Relativity (top-left) * Relativistic Energy - Special Relati... (top-center) * Four-Momentum - Relativistic Invarian... (center-right) * Plasma Frequency - Plasma Phy (center-right) * Four-Momentum - Relativistic Invar (bottom-right) * Relativistic Energy - Special Relativ... (bottom-left) * Thermal Energy - Heat Transfer (bottom-center) * Relativistic Momentum - Special Relat... (bottom-center) * **Quantum Mechanics (Red Nodes):** * nonic Oscillator - Quantum (sqr) (left) * de Broglie Wavelength - Matter Waves ... (bottom-right) * **Statistical Mechanics (Light Green Nodes):** * Maxwell Distribution - Statistical Me... (top-right) **Connections:** The nodes are interconnected with gray lines, indicating relationships between the equations. The density of connections varies across the diagram. ### Key Observations * The "Relativity" branch has the most nodes (7), indicating a central role in this cluster. * The "Electromagnetism" branch has 3 nodes. * The "Quantum Mechanics" branch has 2 nodes. * The diagram shows how different branches of physics are interconnected through specific equations. * The density of connections suggests that certain equations have more relationships with others. ### Interpretation The network diagram visualizes the relationships between physics equations within a specific cluster. The connections between nodes suggest how different concepts and equations are related to each other. The dominance of "Relativity" suggests that this branch plays a central role in the cluster, with connections to other branches like "Electromagnetism," "Quantum Mechanics," and "Statistical Mechanics." The diagram provides a high-level overview of the interconnectedness of physics concepts within this cluster, potentially highlighting areas where different branches of physics intersect and influence each other. The "Density: 0.615" likely refers to the network density, indicating the ratio of actual connections to possible connections. </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: $$ \left(\partial_{\mu}\partial^{\mu}+\frac{m^{2}c^{2}}{\hbar^{2}}\right)\psi=0 \tag{2} $$ where $∂_{\mu}∂^{\mu}$ is the d’Alembertian operator. The system performed the following substitutions from equations within the cluster: - $m=\gamma\hbar/c^{2}$ (from Dirac Equation - Relativistic Fermions) - $\psi=0$ (from Dirac Equation - Relativistic Fermions) - $\hbar=c^{2}m/\gamma$ (from Dirac Equation - Relativistic Fermions) - $c=(\gamma^{2}v^{r^{2}}/(\gamma^{2}-1))^{r^{-2}}$ (from Lorentz Factor) The substitution $\psi=0$ reduces the expression to the identity True. This result confirms the known relationship where the Dirac equation: $$ (i\gamma^{\mu}\partial_{\mu}-mc/\hbar)\psi=0 \tag{3} $$ represents the “square root” of the Klein-Gordon equation. Applying the Dirac operator twice: $$ (i\gamma^{\nu}\partial_{\nu}+m)(i\gamma^{\mu}\partial_{\mu}-m)\psi=(-\gamma^{\nu}\gamma^{\mu}\partial_{\nu}\partial_{\mu}-m^{2})\psi=0 \tag{4} $$ Using the anticommutation relations $\{\gamma^{\mu},\gamma^{\nu}\}=2\eta^{\mu\nu}$ , this reduces to $(∂_{\mu}∂^{\mu}+m^{2})\psi=0$ . 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: $$ Re=\frac{\rho vL}{\eta} \tag{5} $$ The system applied the following substitutions from electromagnetic equations in the cluster: - $E=-Bv+F/q$ (from Lorentz Force (Complete)) - $t=I\epsilon/L$ (from Inductance EMF) - $v=\epsilon/(BL)$ (from Motional EMF) - $t=-d\epsilon/d\Phi_{B}$ (from Faraday’s Law) - $t=-\epsilon/\Phi$ (from Lenz’s Law Direction) This produced the simplified expression: $$ \frac{\epsilon\rho}{B\eta} \tag{6} $$ 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: $$ \mu pp_{\mu}=c^{2}mr^{2} \tag{7} $$ The system applied the following substitutions from equations in the cluster: - $p=E/c-cm^{2}$ (from Four-Momentum - Relativistic Invariant) - $m=\sqrt{E-cp}/c$ (from Four-Momentum - Relativistic Invariant) - $c=m_{e}$ (from Compton Scattering - Wavelength Shift) This produced: $$ \mu pp_{\mu}=m_{e}r^{2}\sqrt{E-m_{e}p} \tag{8} $$ With residual: $-m_{e}r^{2}\sqrt{E-m_{e}p}+\mu pp_{\mu}$ 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 $\lambda_{C}=h/(m_{e}c)$ 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: $$ 2d\sin\theta=n\lambda \tag{9} $$ with substitutions from interference equations: - $m=d\sin\theta/\lambda$ (from Interference - Young’s Double Slit) - $a=m\lambda/\sin\theta$ (from Single Slit Diffraction (Minima)) - $m=(d\sin\theta-\lambda/2)/\lambda$ (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: $$ P+\frac{1}{2}\rho v^{2}+\rho gh=\text{constant} \tag{10} $$ The system applied substitutions: - $h=v/(gr^{2})$ (from Torricelli’s Law) - $g=v/(hr^{2})$ (from Torricelli’s Law) - $c=(\mu_{r}-1)/\chi_{m}$ (from Magnetic Susceptibility) - $P=P_{0}+\rho gh$ (from Hydrostatic Pressure) This produced: $$ \frac{(\mu_{r}-1)^{2}}{h^{2}m^{2}} \tag{11} $$ 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. Special gratitude to the ACM community for professional development resources and to online physics communities for discussions that helped clarify physical interpretations. 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