The Reasoning Window: A Critical-Band Model of Verification–Generation Stability Across Sparse Systems

Building on the structural map of nested critical regimes in sparse random systems (Ransome et al., 2026), we propose that the critical coherence band—where the order parameter occupies a characteristic narrow fraction F ≈ 0.10–0.15—defines the narrow "habitable zone" where coherent, adaptive dynamics can persist. This band provides sufficient global integration (via the incipient spanning core) for verification and self-consistency, while retaining enough peripheral flexibility for exploration and adaptation. In reasoning systems—brains, ecosystems, and large language models (LLMs)—operation within this window appears to enable reliable reasoning; deviation beyond its upper edge yields a "reasoning cliff" where fluency persists but internal verification collapses. We synthesize phenomenological evidence from LLM behaviour, neuronal avalanches, and ecosystem tipping points, linking the cliff to the structural handover identified in the coherence band. This framework offers a unified view of reasoning limits and suggests interaction patterns to scaffold around them. Companion preprint building on Ransome et al. (2026); initial text-only version.