Reflexive Index φ as a Heuristic Model of Consciousness Formalizability: Validation on 265,956 Epochs and Clinical Perspectives

This preprint presents the reflexive index φ = (C·S·I)^(1/3) as a heuristic EEG-based biomarker for consciousness formalizability, grounded in the formal cognitive shadow theory. The index is derived from three neurophysiological components: capacity (C, broadband power), selectivity (S, theta-band signal-to-noise ratio), and integrity (I, fronto-occipital coherence). We validated φ on 265,956 epochs from 9 independent datasets (including LIS, DOC, sleep, and healthy controls) using strict subject-wise cross-validation (GroupKFold) and cross-dataset generalization. The index reliably distinguishes locked-in syndrome (LIS) from healthy volunteers (AUC = 0.947 ± 0.081) and shows specificity against control conditions (epilepsy, sex, hand movement, sleep stages) with AUC near 0.5. However, φ does not significantly differentiate VS, MCS−, and MCS+ (Kruskal-Wallis p = 0.152) and shows no correlation with CRS‑R scores, indicating that the current scalar model is insufficient for fine-grained DOC grading. The dynamic parameters — awareness (A, p = 0.045), temporal coherence (T), and flexibility (F) — were also evaluated, with only A showing empirical support. Anaesthesia hysteresis was confirmed (p = 0.902 between LOC and ROC), consistent with Theorem 8 on irreversible interface degradation. The theoretical framework has been formally verified in Coq 8.18+ and TLA⁺, with 10 theorems (including Theorems 1′, 8, 9, and 10) constructively proved. All verification artifacts, analysis code, and datasets are open-source. This work presents a first-order heuristic model that is clinically useful for LIS screening but highlights the need for matrix and dynamic extensions for consciousness disorder staging. The FORCED_REPORT protocol remains logically justified and awaits prospective validation. **Keywords:** consciousness, locked-in syndrome, EEG biomarker, reflexive index, theta rhythm, formal verification, cognitive shadow, FORCED_REPORT