The Precision Cliff: An Information-Theoretic Lower Bound on Quantization for Iterative Compositional Retrieval

Low-bit quantization degrades multi-hop reasoning faster than single-hop recall, yet no theory has said why deeper reasoning needs more bits or how many more. We answer both with a single closed-form information-theoretic floor. For any post-training compression scheme whose error placement is independent of the stored content and whose distortion is deterministic—conditions met by GPTQ, AQLM, AWQ, and QuIP#—the minimum bits per entry for reliable $T$-hop retrieval is $b_{\mathrm{any}}^{\ast }(T,N)=R({\delta }_T)$ with ${\delta }_T=1-2^{-1/T}$, and an explicit construction saturates this floor to within $0.002$ bits. A direct corollary is the $2\times$ separation law: deep reasoning demands twice the bit-rate of single-hop recall at capacity saturation (predicted $2.40$ for $N=4096$; measured $2.397$). Capacity utilization $\eta$ transfers this bits-per-entry floor to bits-per-parameter: a synthetic sweep traces a monotone depth penalty in $\eta$ (Spearman $\rho =-0.93$, $p<0.001$), and four LLM families (Llama-2-7B, Qwen3-8B, Phi-4-mini, InternLM3-8B) sit on the same $\eta$-curve at production scale with wall premiums of $1.05$–$1.55\times$—already forcing a $5$–$55\%$ upward correction to any $T=5$ quantization floor calibrated at $T=1$. A chain-given control (accuracy $0.99$ read-aloud vs. $0.46$ sequential lookup at 3 bits) isolates iterative retrieval as the causal mechanism, ruling out attention-routing and generation-noise alternatives.