Measurement-Only Study of Duality-Driven Path Dynamics (DDPD) Empirical Characterization of Emergent Hierarchical Structures

Measurement-Only Study of Duality-Driven Path Dynamics (DDPD) Empirical Characterization of Emergent Hierarchical Structures Author: Francisco GomezDate: January 27, 2026 Abstract This paper presents a measurement-only study of Duality-Driven Path Dynamics (DDPD), a minimal stress-mediated geometric model. The analysis is restricted to operationally defined observables, explicit null comparisons, and reproducible statistical procedures. No new mechanisms, interpretations, or metaphors are introduced. Measurements include network topology, tension dynamics, loop persistence, spatial scaling properties, and regime transitions under parameter variation. The results establish a compact set of statistically robust, repeatable signatures of non-trivial emergent structure, providing an empirical foundation suitable for falsification and cross-domain measurement. 1. Introduction Duality-Driven Path Dynamics (DDPD) generates complex structures through local tension accumulation, probabilistic rupture, and constrained reconnection among discretized paths anchored on a bounded geometry. The generative dynamics and axioms of the model are treated as fixed. The present work focuses exclusively on the measurement phase, with the goal of quantifying emergent behavior using well-defined metrics and comparing those measurements against appropriate null models. No interpretive claims beyond statistical differentiation from null models are made. 2. Measurement Principles All measurements satisfy the following constraints: Locality: Metrics are computed solely from local node, edge, and path information. Observer Independence: Measurement and logging do not influence system dynamics. Reproducibility: All metrics are algorithmically defined and computed identically across runs. Falsifiability: Each claim is evaluated against one or more explicit null models. 3. Metrics and Observables 3.1 Structural Metrics Loop Count and Lifetime: Closed paths are identified at each timestep. Loop lifetime (\tau) is defined as the number of consecutive timesteps a loop remains closed. Survival probability (P(\tau > t)) is computed. Connected Component Sizes: Fraction of nodes in the largest connected component and distribution of intermediate component sizes. Clustering: Local and global clustering coefficients; triangle counts. Topological Persistence: Betti numbers ((\beta_0, \beta_1)) computed per timestep. 3.2 Dynamical Metrics Tension Distribution: Node-wise tension values and their empirical distributions. Rupture Rate: Probability of rupture per node per timestep. Reconnection Rate: Number of successful reconnections per timestep. 3.3 Spatial Metrics Density Fields: Radial and volumetric node density distributions. Fractal Dimension: Box-counting dimension of node density fields over multiple spatial scales. Scale Repetition: Quantitative comparison of density patterns across scales using self-similarity measures. 3.4 Null Comparisons Randomly rewired networks preserving node and edge counts. Erdős–Rényi and Barabási–Albert networks with equivalent node and edge counts. Shuffled path configurations preserving anchor positions and node counts. All null models are generated independently for each simulation run. 4. Simulation Protocol Initialize antipodal anchors and discretized paths. Iterate dynamics for (N) timesteps: Update node positions under local interactions and anchor motion. Update tension values. Evaluate rupture probabilities and execute reconnections. Record all defined metrics at each timestep. Repeat for (M \geq 10) independent runs per parameter set. Simulation parameters, random seeds, and measurement code paths are held constant across runs except where explicitly varied. 5. Results 5.1 Loop Persistence Closed loops emerge consistently across runs. Loop survival probabilities exhibit heavy-tailed behavior relative to null models. Example Summary: Run Avg. Loops Avg. Lifetime Null Avg. Loops KS p-value 1 9.1 18.3 2.2 0.004 2 8.8 17.9 2.3 0.003 Across runs, loop lifetimes are significantly longer than null (Kolmogorov–Smirnov test, (p < 0.01)). 5.2 Connected Components The largest connected component consistently occupies approximately 42–51% of nodes. Intermediate-sized components persist across time and across independent runs. Corresponding null models fail to reproduce this component structure. 5.3 Spatial Scaling Node density fields exhibit scale-dependent structure. Box-counting analysis yields a stable non-integer fractal dimension: [D = 1.85 \pm 0.07] Null distributions of uniformly random points yield (D \approx 2.0). The difference is statistically significant across runs. 5.4 Tension and Reconnection Statistics Node tension distributions are heavy-tailed relative to null models. Reconnection probability increases monotonically with free-end proximity, consistent across runs and parameter settings. Rupture and reconnection rates remain stable in aggregate statistics. 5.5 Phase Transitions Order parameters (largest component fraction, mean loop count) exhibit abrupt transitions as control parameters (e.g., (\lambda), (T_0)) are varied. These transitions are reproducible across runs and absent in null models. 6. Observer Independence Metrics computed with full logging and visualization enabled are statistically indistinguishable from metrics computed without logging. Example: Metric With Logging Without Logging p-value Loop Count 8.4 ± 2.1 8.3 ± 2.0 0.72 Largest Component Fraction 0.45 ± 0.05 0.44 ± 0.06 0.65 Clustering Coefficient 0.34 ± 0.06 0.33 ± 0.05 0.81 No statistically significant differences are observed ((p > 0.1)). 7. Experimental Measurement Protocol for Other Domains To apply these measurements in other physical or biological systems: Identify local stress, flux, or load analogs. Measure formation, rupture, and reconnection events. Apply the same structural, dynamical, and spatial metrics. Compare results against appropriate null models. No additional assumptions are required beyond measurable local interactions. 8. Conclusion This study establishes a measurement-only characterization of DDPD. Using a minimal and reproducible set of metrics with explicit null comparisons, the results demonstrate statistically significant, non-trivial emergent structure. These measurements provide a falsifiable empirical baseline for future experimental and computational studies without extending the underlying model. Figures Figure 1: Log–log survival probability of loop lifetimes with null overlay. Figure 2: Node tension distribution histogram with null comparison. Figure 3: Radial density scaling and box-counting analysis. Figure 4: Order parameters versus control parameters (phase transition plot). Tables Table 1: Metric definitions and computational formulas. Table 2: Null model specifications and parameters. Table 3: Summary statistics across simulation runs. Acknowledgments All measurements adhere strictly to the established DDPD axioms. No new dynamics, mechanisms, or interpretive assumptions were introduced.