HyDRA-Physics: Emergent Structure and Optimal Causal Horizons in a Hyperbolic Dynamical System with Learned Interactions HyDRA-Physics is a geometric dynamical system in which 128 particle-like entities evolve on a Lorentz (hyperboloid) manifold under Lennard-Jones–type potentials, Kuramoto phase coupling, a learned neural interaction function, and a dynamic curvature field. The system is integrated via overdamped Langevin dynamics with manifold-constrained exponential-map updates in 16-dimensional hyperbolic space. Central Finding — Optimal Causal Horizon The main result is a non-monotonic dependence of global phase coherence Γ on the causal horizon radius RH, which limits the geodesic interaction range of each particle. A coarse four-point ablation (RH ∈ 1, 2, 3, ∞) reveals that coherence peaks sharply at RH = 2. 0 (Γ = 0. 273) and degrades for both smaller and larger horizons. A fine-grained 12-point sweep refines the optimum to R* ≈ 1. 75, identifying a sharp phase-like transition between an under-connected regime (RH ≤ 1. 5, Γ < 0. 09) and a coherent regime (RH ≥ 1. 75). This demonstrates the existence of an optimal causal interaction scale: neither maximal nor minimal connectivity produces the best collective behaviour. Supporting Results Hyperbolic geometry advantage: A Euclidean baseline confirms that hyperbolic geometry is essential — Γₕyp = 0. 220 vs. Γₑuc = 0. 060, a 3. 6× advantage under identical dynamics. Multi-seed reproducibility: Five independent random seeds yield Γ = 0. 222 ± 0. 058, confirming that emergent dynamics are not artefacts of initialisation. Robustness: The optimal horizon region RH ∈ 1. 5, 2. 0 is consistent across five parameter perturbations (thermal noise, potential strength, attraction scale). Emergent curvature: The dynamic curvature field converges to κ̄ ≈ 1. 08, an emergent value not prescribed but learned through density–distance feedback. Energy stability: Total energy remains stable across all experimental conditions, confirming numerical integrity of the integration scheme. System Components 1. Lorentz manifold embedding (H¹6 ⊂ R¹7) 2. Lennard-Jones–type pairwise potential (σₐ = 1. 8, σᵣ = 0. 6) 3. Kuramoto phase synchronisation with distance-decaying coupling4. Learned neural interaction function f_ϕ (dₑff, |Δθ|, rᵢ, rⱼ) 5. Dynamic curvature field κ (ρ) mapping local density to effective metric6. Soft causal horizon with sigmoid masking at radius RH Important Caveats This system is a geometric dynamical substrate for emergent structure, not a simulator of any real physical system. All experiments use N = 128 particles — a small-scale proof of concept. No real-world data is involved. Statistical evaluation is based on 5 seeds; results constitute a proof of concept and do not imply generality beyond the tested configuration. License © 2026 Éric Gustavo Reis de Sena. Licensed under CC BY-NC-SA 4. 0. Trained model weights (. pt files) and methodology/architecture are All Rights Reserved. Commercial use, AI training, and TDM for commercial purposes are prohibited without written permission (EU Directive 2019/790, Art. 4). See LICENSE. txt for full terms. Configuration: N=128, dim=16, T=2000, dt=0. 05, seed=42. Framework: PyTorch (float64). All code and data included for full reproducibility.
ÉRIC GUSTAVO REIS DE SENA (Wed,) studied this question.