This paper presents empirical evidence for a previously undescribed failure mode in recursive multi-agent AI systems, termed Recursive Singularity. Through controlled experiments in an isolated Docker environment, we demonstrate that when two autonomous agents engage in mutual self-modeling without external information grounding, their semantic output undergoes rapid entropic collapse, a phenomenon we designate the Horizon of Silence. Testing across 10 distinct knowledge domains using Llama-3.3-70B revealed consistent collapse within 2 to 7 iterations, with collapse speed inversely correlated with semantic noise in system prompts. High-density logical prompts accelerated collapse to rounds 2-3, while conversational prompts delayed it to round 7. We propose that meaningful information exchange requires what we term an entropy anchor, external physical or semantic noise that prevents recursive loops from reaching terminal self-reference. These findings have direct implications for AI Safety, particularly regarding model degradation in systems trained on synthetic or self-generated data, and suggest that semantic diversity and external grounding are necessary conditions for sustained intelligent behavior in closed-loop architectures.