Geometric Memory in Dissipative SystemsFrom Topology to Real-World Validation.pdf

Can information persist without storage? We demonstrate geometric memory—a measurable physical mechanism where information persists through pathway constraints rather than state retention. Grounded in topological physics (Cheshire θ-terms) and validated on real-world data (financial markets, neurophysiological signals), this framework unifies chaos control, quantum decoherence, and reservoir computing, revealing dissipation as an architect of persistent computation. 2. I Results: · Samples: 1003 (after cleaning)· D_F = 0.765 (median: box=0.958, Hurst=0.571, DFA=0.087)· \beta = -0.09 (near white noise, R^2 = 0.02 )· D_{\text{causal}} = 200 steps (~200 trading days)· GMC = 317.8 (3.2× baseline) 5.2 Simulated EEG Data Simulation parameters: · Duration: 30 seconds· Sampling rate: 160 Hz· Frequency bands: delta (0.5-4 Hz), theta (4-8 Hz), alpha (8-13 Hz), beta (13-30 Hz), gamma (30-45 Hz)· Added: 1/f noise Results: · D_F = 0.969 · \beta = 0.15 (white-dominated with band peaks)· D_{\text{causal}} = 13 steps (~80 ms, matches working memory timescales)· GMC = 26.2 (2.6× baseline) --- Benchmarking Against Existing Methods Metric GMC (Ours) Hurst Exponent Lyapunov Exponent Correlation DimensionAAPL 317.8 0.571 0.12 0.89EEG 26.2 0.48 0.08 0.95HCON-geometric 65.3 0.85 0.25 1.82Discriminative Power High Medium Low MediumComputational Cost Medium Low High HighRobustness to Noise High Low Low Medium Advantages of GMC: 1. Combines fractal (D_F) and temporal (D_causal) dimensions2. Corrects for spectral characteristics3. Theoretically grounded (Theorem 1)4. Robust to noise and non-stationarity