Information Must Flow: Recursive Bootstrapping for Information Bottleneck in Optimal Transport

We present the Context-Content Uncertainty Principle (CCUP), a unified framework that models cognition as the directed flow of information between high-entropy context and low-entropy content. Inference emerges as a cycle of bidirectional interactions, bottom-up contextual disambiguation paired with top-down content reconstruction, which resolves the Information Bottleneck in Optimal Transport (iBOT). Implemented via Rao-Blackwellized variational entropy minimization, CCUP steers representations toward minimal joint uncertainty while preserving inferential directionality. Local cycle completion underpins temporal bootstrapping, chaining simulations to refine memory, and spatial bootstrapping, enabling compositional hierarchical inference. We prove a Delta Convergence Theorem showing that recursive entropy minimization yields delta-like attractors in latent space, stabilizing perceptual schemas and motor plans. Temporal bootstrapping through perception-action loops and sleep-wake consolidation further transforms episodic traces into semantic knowledge. Extending CCUP, each hierarchical level performs delta-seeded inference: low-entropy content seeds diffuse outward along goal-constrained paths shaped by top-down priors and external context, confining inference to task-relevant manifolds and circumventing the curse of dimensionality. Building on this, we propose that language emerges as a symbolic transport system, externalizing latent content to synchronize inference cycles across individuals. Together, these results establish iBOT as a foundational principle of information flow in both individual cognition and collective intelligence, positioning recursive inference as the structured conduit through which minds adapt, align, and extend.