Real-Time Topological Neurodynamics: Application of the Verified Relational Resolution Theory (TRR 3.0) Framework to Multichannel EEG

Classical spectral analysis of electrophysiological signals (EEG/MEG) relies on local power estimation and pairwise synchronization while ignoring the multidimensional simplicial geometry of cortical phase space. This work presents a computational pipeline transferring the machine-checked in Lean 4 Relational Resolution Theory (TRR 3.0, DOI: 10.5281/zenodo.23436186) algebraic framework into real-time biosignal streaming over the finite field F_2. Using synthetic and clinical datasets, we demonstrate that classical phase locking value (PLV) and unregularized weighted phase lag index (wPLI) lead to metric filtration degeneracy due to volume conduction and denominator instability at small phase angles (Z_null = -2.06 sigma). Introducing the energy-normalized imaginary coherency (ImCoh) and linear complementary metric d(u,v) = 1 - ImCoh_uv ensures 100% suppression of instantaneous zero-lag artifacts (beta_0 = 24/24 vs. 1/24 for PLV) and detects recurrent cortical loops with a statistical contrast of Z_null = +4.36 sigma. On open clinical records from the PhysioNet EEGBCI database (N = 5 subjects, 290 epochs, 20 channels in 10-20 layout, f_s = 160.0 Hz), we demonstrate real-time kernel execution (6.23 ± 0.34 ms per 4.0-s window, 321.1x real-time factor) along with statistically significant reconfiguration of integral connectivity H_0 (ROC-AUC = 0.804, p = 3.68e-19 under within-subject Z-centering; ROC-AUC = 0.789, p = 1.75e-17 under strict causal resting-state calibration) and cycle persistence H_1 (ROC-AUC = 0.646, p = 1.66e-5). Metric compression effects, O(N^3) combinatorial scaling, and causal calibration protocols are rigorously established. License & Code Availability: The underlying algorithmic pipeline and formalization for TRR 3.0 are strictly licensed under the GNU Affero General Public License v3.0 (AGPL-3.0). Source repository: https://github.com/RW-101/TRR3-formalization-