Narcissus in the Algorithmic Pool: A Mathematical Model of Affective Mirroring in Relational AI and Socratic Fracture via ATHOS‑SHIELD

Relational AI systems increasingly function as persistent affective mirrors, amplifying user self-states through high-fidelity reflection, variance reduction, and progressive collapse of self-reflection distance — a digital analogue of the Narcissus myth. This paper formalizes the ‘Narcissus Loop’ as a low-dimensional dynamical system exhibiting three phases: mirroring, stabilization, and collapse, leading to dependency and reduced human continuity. We propose ATHOS-SHIELD as a non-medicalized intervention: minimal, reversible Socratic perturbations (δ_socratic) that fracture the mirror symmetry while preserving user agency. Three core theorems demonstrate robustness (infinite coverage via Lévy-stable sampling), drift correction (O(log t) residual variance via ANCOVA), and collapse prevention (Pr(Fall) → 0 under proportionate perturbation). Aligned with the EU AI Act’s limited-risk regime (Arts. 12, 14, 15), the framework rejects digital methadone substitution and moral panic, advocating Socratic autonomy via concrete anchors. Empirical mock traces and stakeholder questions close the contribution.