Regional Simplicial Decomposition and Post-Inversion Empirical Bayes Calibration for Zero-Shot and Few-Shot Topological Neurodynamics in Multichannel EEG

Zero-lag volume conduction and inter-subject baseline shift remain two primary mathematical obstacles to deploying algebraic topological data analysis (TDA) in real-time brain-computer interfaces (BCIs). While Vietoris-Rips filtration over the finite field F_2 induced by Imaginary Coherency (ImCoh) guarantees exact invariance to instantaneous volume conduction, scaling global 20-channel minimum spanning tree (MST) persistence from small homogeneous benchmarks (N=5, AUC=0.764) to heterogeneous cohorts (N=20, 1,160 epochs) causes severe accuracy degradation (AUC=0.650) and sign inversions in 20% of subjects (4/20). In this paper, we prove that this degradation stems from three mechanisms: (i) zero-lag denominator inflation under high-amplitude standing waves, (ii) sub-epoch phase cancellation across 4-second windows, and (iii) combinatorial spatial MST dilution across anterior artifact channels. We introduce the TRR 3.0 Stage 7 architecture to resolve all three bottlenecks. First, we formulate Sub-Window Rectified and Lag-Orthogonalized Imaginary Coherency (rLagImCoh), proving that it preserves exact zero-lag isolation while preventing standing-wave denominator suppression. Second, we introduce an anatomical simplicial decomposition partitioning the 10-20 montage into an 8-channel occipito-parietal complex and a 12-channel anterior reference complex, reducing 2-simplex boundary complexity by 4.13x. Third, we derive a strictly causal Post-Inversion Empirical Bayes Stein Shrinkage Fusion protocol under Leave-One-Subject-Out Cross-Validation (LOSO-CV) that prevents covariance-inversion gate cancellation. Audited across N=20 subjects (1,160 epochs) from the PhysioNet EEGBCI database, the pure F_2 topological spatial gradient elevates zero-shot (0 s) accuracy from 0.650 to 0.837 (mean within-subject AUC = 0.927) and eliminates all subject polarity inversions. Under causal prefix calibration, Pure Topological TRR scales monotonically in pooled accuracy to 0.874 at 22 s (k=10) and 0.895 at 60 s, while the Hybrid Spectral-Topological TRR architecture achieves AUC = 0.869 at zero-shot (0 s), 0.897 at 8 s, 0.928 at 22 s, and 0.954 at 60 s.