Cognitive Phase Transitions in Semantically Constrained Vector Quantization: Emergent Retrieval/Inference Separation Under Physical Bandwidth Constraints

We study the emergence of cognitive structure in Vector Quantization (VQ) systems operating under physical bandwidth constraints. Through systematic experimentation on operational maintenance language, we identify a phase transition governed by the ratio N/K (number of training samples to codebook size): below a critical threshold γ_c ≈ 25–30, VQ codebooks spontaneously separate retrieval-type and inference-type communications into distinct Voronoi regions, without explicit supervision. We formalize this as the Cognitive Emergence Law: N/K < C · d_cog, where d_cog measures the intrinsic cognitive separability of the input domain (Cohen's d = 2.45 on raw BPE tokens) and C_emp = 0.391 ≈ 1/e = 0.368 (6.3% deviation explained by mean-field corrections). We show that this emergence is driven by the inputs themselves rather than the reward model — random reward outperforms MiniLM reward (2/3 vs 1/3 seeds significant at K=128, N/K=1.55), confirming that cognitive structure is a property of operational language under compression, not an artifact of supervision. These results have direct implications for bandwidth-constrained AI inference systems (SMS, 2G, LoRa, satellite), where semantic compression must preserve functional distinctions rather than topical similarity. The critical ratio N/K < C · d_cog provides a deployable design parameter for such systems. Preprint. HAL identifier: hal-05596229.